# Tails and probabilities for $p$-outside values

Pavlina Jordanova  Faculty of Mathematics and informatics, Konstantin
Preslavsky University of Shumen,
Universitetska 115, Shumen, Bulgaria

###### Abstract

The task for a general and useful classification of the tails of probability
distributions still has no satisfactory solution. Due to lack of information
outside the range of the data the tails of the distribution should be
described via many characteristics. Index of regular variation is a good
characteristic, but it puts too many distributions with very different tail
behavior in one and the same class. One can consider for example
Pareto($\alpha$), Fr$\acute{e}$chet($\alpha$) and Hill-horror($\alpha$) with
one and the same fixed parameter $\alpha>0$. The main disadvantage of VaR,
expectiles, and hazard functions, when we speak about the tails of the
distribution, is that they depend on the center of the distribution and on the
scaling factor. Therefore they are very appropriate for predicting ”big
losses”, but after a right characterization of the distributional type of ”the
payoff”. When analyzing the tail of the observed distribution we need some
characteristic which does not depend on the moments because in the most
important cases of the heavy-tailed distributions theoretical moments do not
exist and the corresponding empirical moments fluctuate too much. In this
paper, we show that probabilities for different types of outside values can be
very appropriate characteristics of the tails of the observed distribution.
They do not depend on increasing affine transformations and do not need the
existence of the moments. The idea origins from Tukey’s box plots, and allows
us to obtain one and the same characteristic of the tail of the observed
distribution within the whole distributional type with respect to all
increasing affine transformations. These characteristics answer the question:

At what extent we can observe ”unexpected” values?

###### keywords:

Tail inference , Point estimators

###### MSC:

[2010] 60E15 , 60E05 , 62G32

mytitlenotemytitlenotefootnotetext: The author is grateful to the bilateral
projects Bulgaria - Austria, 2016-2019, ”Feasible statistical modelling for
extremes in ecology and finance”, BNSF, Contract number 01/8, 23/08/2017 and
WTZ Project No. BG 09/2017..myfootnotemyfootnotefootnotetext: Corresponding
author<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1 Motivation and history of the problem

The task for a general and useful classification of probability distributions
with respect to the their tails seems to be still open. Embrechts et al.
(1997) [20] have made a very useful figure about the relations between
different subclasses of heavy-tailed distributions in the sense of the
infinite moment generating function for all positive arguments. However, this
classification puts too many distributions with very different tail behavior
in one and the same class. According to this classification for example
Pareto($\alpha$), Fr$\acute{e}$chet($\alpha$) and Hill-horror($\alpha$)
distributions, with one and the same fixed parameter $\alpha>0$ belong to one
and the same class of distributions with regularly varying tails with
parameter $\alpha$. However, the chance to observe ”unexpected” value in these
three cases is very different, especially for the Hill-Horror distribution.
See Figure 2. Comparison of the corresponding hazard rate functions
$r_{X}(x)$, cumulative distribution functions $F_{X}(x)$ (c.d.fs), and
probability density functions $f_{X}(x)$ (p.d.fs) when $x$ is fixed, and close
to the ends of the support of the corresponding distributions is a relatively
good approach, but all these characteristics depend on the center of the
distribution and the scale parameters. In order to delete these dependencies
usually, we normalize the considered random variable(r.v.) with the variance.
However, in order to do this, we need existence not only of the first but also
of the second moment of the observed distribution. In the most important cases
of heavy-tailed distributions, these moments do not exist and this approach is
not applicable. Therefore we need some characteristics which describe
separately the left and the right tail of the distribution and do not depend
on the moments. Index of regular variation, in cases when it is meaningful, is
not enough. These lead us to the idea about usage of quantiles, VaR and
expectiles, described e.g. in Daouia et al. (2018) [12] or Marinelli et al.
(2007) [38]. They are good characteristics, but also depend on the center of
the distribution and the scaling factor. Therefore they are very appropriate
for predicting ”big looses” within a fixed family of distributional type, but
first, you need a right characterization of the type of the ”the profit and
loss” distribution. A long critical review about the kurtosis can be seen in
Balanda and MacGillary (1988) [4]. Harter (1959) [29] uses sample quasi-ranges
in estimating population standard deviation. Mosteller (1946) [39] and Sarhan
(1954) [52] propose estimators of the mean and standard deviation which are
functions of order statistics. They give us the idea to work with something
related to the quantile spread.

Due to lack of information outside the range of the data the tails of the
distribution should be described via many characteristics. Through the paper,
we show that probabilities for different orders of outside values can be
appropriate characteristics for solving this task. Their properties outperform
the properties of the kurtosis, tail index and hazard function when speaking
about classification with respect to the tail of the observed distribution.
The main their advantages are that they do not depend on the center and the
scaling factor of the distribution, and do not need the existence of the
moments. They are useful for answering the question:

At what extend we should observe ”unexpected” values?

The idea origins from Tukey’s box plots (1977) [58] and Balanda and
MacGillary’s (1990) [5] spread-spread plot. However instead of the quantiles
here we use probabilities. This allows us to obtain one and the same
characteristic of the tail of the observed distribution within all
distributional type with respect to increasing affine transformations.

In Section 2 we define and investigate the general properties of probabilities
for $p$-outside values. In Section 3 their explicit forms are calculated and
plotted for the most popular probability distributions. Section 4 investigates
asymptotic properties of empirical left and right $p$-fences, and the
estimators of probabilities for left and right $p$-outside values. A result
about strong consistency of relative frequency estimator completes that part.

Different estimators of the exponent of regular variation are proposed in Hill
(1975) [30], Pickands (1975)[47] and Deckers-Einmahl-de Haan Dekkers (1989)
[15], Einmahl and Guillou (2008) [19], t-Hill Stehlik et al. (2010) [57],
Pancheva and Jordanova (2012) [44, 34], Jordanova et al. (2016) [33] among
others. The mean of order $p$ generalization of the t-Hill and Hill statistics
is introduced by Beran et al. (2014) [6], Caeiro et al. (2016) [8] and
Paulauskas and Vaiciulis (2017) [46]. Another approach can be seen in Huisman
et al. (2001) [31] who recommend to correct small-sample bias of Hill
estimators via weighted averages of its values for different thresholds. In
Sections 5 the previous results are applied and a completely new approach for
estimating the parameter of the heaviness of the tail of the observed
distribution is demonstrated. Four of the examples consider cumulative
distribution functions (c.d.fs.) with regularly varying right tail. These are
Pareto, Fr$\acute{e}$chet, Log-logistic and Hill-horror cases. It is easy to
realize that in order to estimate their indexes of regular variation working
with small samples only distribution sensitive estimators can be useful. The
main idea of this section is to show that our approach works also in the case
when the c.d.f. of the observed r.v. does not have regularly varying right
tail. We depict this result via an example about $H_{1}$ distribution (30)
which tail is not regularly varying. The paper finishes with some conclusive
remarks. The proofs are sent to the Appendix section. All plots and
computations are made via software R (2018)[48].

In this work we do not use the second order regular variation introduced in de
Haan and Stadtmueller (1996) [14] for distributions with regularly varying
tails, because it is applicable only for huge samples. A very comprehensive
study of analyzing extreme values under the second order regularly varying
condition can be found e.g. in de Haan and Ferreira (2006) [13]. The
corresponding properties of the convolutions and the central limit theorem are
obtained by Geluk and de Haan (1997) [24].

Through the paper we use the following notations: ${\mathop{=}\limits^{d}}$ is
for the equality in distribution, ${\mathop{\to}\limits^{d}}$ means
convergence in distribution, ${\mathop{\to}\limits^{a.s.}}$ denotes almost
sure convergence. $\in$ means that the considered r.v. belongs to the
corresponding probability type. $\sim$ is asymptotic equivalence.
$B(\alpha,\beta)=\int_{0}^{1}x^{\alpha-1}(1-x)^{\beta-1}dx$ is for the beta
function, and $Beta(\alpha,\beta)$ denotes Beta distribution with parameters
$\alpha>0$ and $\beta>0$. $\mathbf{X}\in Par(\alpha,\delta)$, means that a
r.v. $\mathbf{X}$ has Pareto c.d.f.

$F_{\mathbf{X}}(x)=\left\\{\begin{array}[]{ccc}0&,&x\leq\delta\\\
1-\left(\frac{\delta}{x}\right)^{\alpha}&,&x>\delta\end{array}\right..$ (1)

More general definitions of Pareto distributions, together with very useful
descriptions of the relations between them and the most important other
distributions could be seen e.g. in Arnold (2015) [3].

$\mathbf{X}\in Fr(\alpha,c)$, means that a r.v. $\mathbf{X}$ has
Fr$\acute{e}$chet c.d.f.

$F_{\mathbf{X}}(x)=\left\\{\begin{array}[]{ccc}0&,&x\leq 0\\\
e^{-(cx)^{-\alpha}}&,&x>0\end{array}\right..$ (2)

$\mathbf{X}\in NegWeibull(\alpha,\sigma,\mu)$ is an abbreviation of the fact
that the r.v. $\mathbf{X}$ has a Negative Weibull c.d.f.

$F_{\mathbf{X}}(x)=\left\\{\begin{array}[]{ccc}exp\left\\{-\left(-\frac{x-\mu}{\sigma}\right)^{\alpha}\right\\}&,&x\leq\mu\\\
1&,&x>\mu\end{array}\right..$ (3)

We consider only absolutely continuous distributions. For $p\in[0,1]$ the
theoretical quantile function of the c.d.f. $F$ is defined as

$F^{\leftarrow}(p)=inf\\{x\in R:F(x)\geq p\\}=sup\\{x\in R:F(x)\leq p\\}.$

Let $X_{1},X_{2},...,X_{n}$ be a sample of independent observations on a r.v.
$X$ with c.d.f. $F$. Here we denote the corresponding order statistics by
$X_{(1,n)}\leq X_{(2,n)}\leq...\leq X_{(n,n)}$. In Parzen (1979) [45], Hyndman
et al. (1996) [32], Langford (2006) [37] among others, one can find different
definitions of empirical $p$-quantiles,
$p\in\left[\frac{1}{n+1},\frac{n}{n+1}\right]$. We use the following one
$F^{\leftarrow}_{n}(p):=X_{([(n+1)p],n)}$, where $[a]$ means the integer part
of $a$. 111As it is noticed in Chu (1957) [10], for large samples, these
methods are equivalent because we consider only absolutely continuous
distributions.

##  2 Using probabilities for $p$-outside values for characterising the tails
of the observed distribution

The idea about classification of distributions based on quartiles and box
plots comes from Tukey (1977) [58], and recently was reminded by Devore (2015)
[18] and Jordanova and Petkova (2017) [35]. Here we generalize this concept
and introduce one more parameter in the definition of outside values, which
allows the researchers to decide at what extend atypical observations would be
called ”outside value”.

Denote by $R_{n}(F,p)=R_{n}(X,p)$

$=F^{\leftarrow}_{n}(1-p)+\frac{1-p}{p}(F^{\leftarrow}_{n}(1-p)-F^{\leftarrow}_{n}(p))=\frac{1}{p}F^{\leftarrow}_{n}(1-p)-\frac{1-p}{p}F^{\leftarrow}_{n}(p)$

and by $L_{n}(F,p)=L_{n}(X,p)$

$=F^{\leftarrow}_{n}(p)-\frac{1-p}{p}(F^{\leftarrow}_{n}(1-p)-F^{\leftarrow}_{n}(p))=\frac{1}{p}F^{\leftarrow}_{n}(p)-\frac{1-p}{p}F^{\leftarrow}_{n}(1-p),$

correspondingly empirical p-right- and empirical p-left-fence.

Their sum is equal to $F^{\leftarrow}_{n}(1-p)+F^{\leftarrow}_{n}(p)$. The
difference $F^{\leftarrow}_{n}(1-p)-F^{\leftarrow}_{n}(p)$ between these
quantiles is very well known. It is called empirical quantile spread(quasi
range), and is considered e.g. in Gumbel(1944)[26], Monsteller (1946) [39],
and Balanda and MacGillary (1990) [5]. The meaning of these values comes from
the expression

$p[R_{n}(F,p)-F^{\leftarrow}_{n}(p)]=F^{\leftarrow}_{n}(1-p)-F^{\leftarrow}_{n}(p)=p[F^{\leftarrow}_{n}(1-p)-L_{n}(F,p)].$

It is clear that analogously to Tukey’s box-plot (1977) [58], one can use
empirical box plot of order p. Its borders are determined via the values

$L_{n}(F,p),\,\,F^{\leftarrow}_{n}(p),\,\,F^{\leftarrow}_{n}(0.5),\,\,F^{\leftarrow}_{n}(1-p),\,\,R_{n}(F,p).$

The most frequently $p=0,25$. This case is partially investigated in the
supplementary material of Soza et al. (2019) [56].

Sample right or left $p$-outside values are the observations which fall
outside the interval $[L_{n}(F,p),R_{n}(F,p)]$. Their absolute frequencies,
strongly depend on the sample size.

###### Definition 1

Assume $p\in(0,0.5]$. We call on observation $Y$ sample(empirical)

  * 1.

right $p$-outside values if $Y>R_{n}(X,p);$

  * 2.

left $p$-outside values if $Y<L_{n}(X,p).$

For $p=0.25$, the definition coincides with the one in Devore (2015) [18]. He
calls them ”extreme right” and ”extreme left outliers”.

Denote by $n_{R}(p,n)$, and $n_{L}(p,n)$ the numbers of these outside values
in a sample of $n$ independent observations. According to the A.Kolmogorov’s
Zero-one law, never mind how small, but strictly positive is $p$ one can
almost sure observe such outside values in a large enough sample of
observations on a r.v. with many light tailed distributions, e.g. Gaussian.
Therefore the number of outside values is not too informative. In order to
classify distributions with respect to their tail behaviour we propose to
compare their theoretical probabilities an observation to be an outside value
of the considered type. Denote by

$\displaystyle p_{L,p}(X)$ $\displaystyle=$ $\displaystyle
p_{L,p}(F)=P(X<L(X,p)),$ $\displaystyle p_{R,p}(X)$ $\displaystyle=$
$\displaystyle p_{R,p}(F)=P(X>R(X,p))$

the observed r.v. $X$ to be left or right $p$-outside value. Here, analogously
to $R_{n}(F,p)$ and $L_{n}(F,p)$ we have denoted by

$\displaystyle R(F,p)$ $\displaystyle=$ $\displaystyle
R(X,p)=F^{\leftarrow}(1-p)+\frac{1-p}{p}[F^{\leftarrow}(1-p)-F^{\leftarrow}(p)]$
$\displaystyle=$
$\displaystyle\frac{1}{p}F^{\leftarrow}(1-p)-\frac{1-p}{p}F^{\leftarrow}(p)$

theoretical p-right fence and by

$\displaystyle L(F,p)$ $\displaystyle=$ $\displaystyle
L(X,p)=F^{\leftarrow}(p)-\frac{1-p}{p}[F^{\leftarrow}(1-p)-F^{\leftarrow}(p)]$
$\displaystyle=$
$\displaystyle\frac{1}{p}F^{\leftarrow}(p)-\frac{1-p}{p}F^{\leftarrow}(1-p)$

theoretical p-left fence. Their properties are analogous to the properties of
empirical p-right- and p-left-fences.

Let us note that for any absolutely continuous c.d.f. $F$

$R(X,0.5)=L(X,0.5)=F^{\leftarrow}(0.5),$

is the median and $p_{R,0.5}(X)=p_{L,0.5}(X)=0.5.$

It is not difficult to check that $R(F,p)$ and $L(F,p)$ are monotone.
Therefore by monotonicity of probability measures, the characteristics
$p_{L,p}(X)$ and $p_{R,p}(X)$ are also monotone.

###### Theorem 1

For a fixed $p\in(0,0.5]$ if there exist $f(F^{\leftarrow}(p))\in(0,\infty)$,
and $f(F^{\leftarrow}(1-p))\in(0,\infty)$, then,

a)

$L(F,p)$ is increasing in $p$;

b)

$R(F,p)$ is decreasing in $p$;

c)

$p_{L,p}(X)$ and $p_{R,p}(X)$ are non-decreasing in $p$.

For $p=0.25$, $p_{L,p}(X)$ and $p_{R,p}(X)$ are correspondingly the
probabilities an observation to be left- or right- extreme outlier. Jordanova
and Petkova (2018) [36] and Soza et al.(2019) [56] denote these probabilities
by $p_{eL}=p_{L,0.25}$ and $p_{eR}=p_{R,0.25}$. In that case, the authors
obtain them for Pareto, Fr$\acute{e}$chet, $H_{1}$, $H_{2}$, and Hill-Horror
distributions. Further on, we generalize these results.

###### Theorem 2

Assume $p\in(0,0.5]$, and $g(x)$ and $F:=F_{X}$ are strictly monotone, well
defined, and continuous function in the considered values. The characteristics
$p_{L,p}(X)$, $p_{R,p}(X)$, possess the following properties:

a)

$p_{L,p}(X)\in[0,p]$ and $p_{R,p}(X)\in[0,p]$.

b)

$p_{L,p}(X)=p_{L,p}(X+c)$, $p_{R,p}(X)=p_{R,p}(X+c)$, $c\in R$.

c)

If $c>0$, then $p_{L,p}(cX)=p_{L,p}(X)$, $p_{R,p}(cX)=p_{R,p}(X)$.

d)

If $c<0$, then $p_{L,p}(cX)=p_{R,p}(X)$, $p_{R,p}(cX)=p_{L,p}(X)$.

e)

If $g(x)$ is continuous and strictly increasing

$\displaystyle p_{L,p}(g(X))$ $\displaystyle=$ $\displaystyle
F\left\\{g^{\leftarrow}\left[\frac{1}{p}g[F^{\leftarrow}(p)]-\frac{1-p}{p}g[F^{\leftarrow}(1-p)]\right]\right\\},$
(4) $\displaystyle p_{R,p}(g(X))$ $\displaystyle=$ $\displaystyle
1-F\left\\{g^{\leftarrow}\left[\frac{1}{p}g[F^{\leftarrow}(1-p)]-\frac{1-p}{p}g[F^{\leftarrow}(p)]\right]\right\\}$
(5)

and

$\frac{1}{p}g[F^{\leftarrow}(1-p)]-\frac{1-p}{p}g[F^{\leftarrow}(p)]\geq
g[R(X,p)]\iff p_{R,p}(X)\geq p_{R,p}[g(X)],$ (6)
$\frac{1}{p}g[F^{\leftarrow}(p)]-\frac{1-p}{p}g[F^{\leftarrow}(1-p)]\geq
g[L(X,p)]\iff p_{L,p}(X)\leq p_{L,p}[g(X)].$ (7)

f)

If $g(x)$ is continuous and strictly decreasing

$\displaystyle p_{L,p}(g(X))$ $\displaystyle=$ $\displaystyle
1-F\left\\{g^{\leftarrow}\left[\frac{1}{p}g[F^{\leftarrow}(1-p)]-\frac{1-p}{p}g[F^{\leftarrow}(p)]\right]\right\\},$
(8) $\displaystyle p_{R,p}(g(X))$ $\displaystyle=$ $\displaystyle
F\left\\{g^{\leftarrow}\left[\frac{1}{p}g[F^{\leftarrow}(p)]-\frac{1-p}{p}g[F^{\leftarrow}(1-p)]\right]\right\\},$
(9)

and

$\frac{1}{p}g[F^{\leftarrow}(p)]-\frac{1-p}{p}g[F^{\leftarrow}(1-p)]\leq
g[L(X,p)]\iff p_{L,p}(X)\leq p_{R,p}[g(X)],$ (10)
$\frac{1}{p}g[F^{\leftarrow}(1-p)]-\frac{1-p}{p}g[F^{\leftarrow}(p)]\leq
g[R(X,p)]\iff p_{L,p}(g(X))\leq p_{R,p}(X).$ (11)

g)

$p_{R,p}(X_{(n,n)})=1-F_{X}^{n}\left[\frac{1}{p}F^{\leftarrow}_{X}(\sqrt[n]{1-p})-\frac{1-p}{p}F^{\leftarrow}_{X}(\sqrt[n]{p})\right]$

$p_{L,p}(X_{(n,n)})=F_{X}^{n}\left[\frac{1}{p}F^{\leftarrow}_{X}(\sqrt[n]{p})-\frac{1-p}{p}F^{\leftarrow}_{X}(\sqrt[n]{1-p})\right]$

h)

$p_{R,p}(X_{(1,n)})=\left\\{1-F_{X}\left[\frac{1}{p}F^{\leftarrow}_{X}(1-\sqrt[n]{p})-\frac{1-p}{p}F^{\leftarrow}_{X}(1-\sqrt[n]{1-p})\right]\right\\}^{n}$

$p_{L,p}(X_{(1,n)})=1-\left\\{1-F_{X}\left[\frac{1}{p}F^{\leftarrow}_{X}(1-\sqrt[n]{1-p})-\frac{1-p}{p}F^{\leftarrow}_{X}(1-\sqrt[n]{p})\right]\right\\}^{n}$

i)

For all $t>0$,

$p_{R,p}(X-t|X>t)=p_{R,p}(X|X>t),\quad p_{L,p}(X-t|X>t)=p_{L,p}(X|X>t).$

k)

Let $l<u$, and $l,u\in R$. Denote correspondingly the right-, left-, and
double-truncated r.vs. by $X_{RT}=(X|X<u)$, $X_{LT}=(X|X>l)$,
$X_{DT}=(X|l<X<u)$. If $0<F(u)$, $F(l)<1$ and $F(l)\not=F(u)$, then the
relations between their probabilities for $p$-outside values,
$F^{\leftarrow}:=F_{X}^{\leftarrow}$, and $F:=F_{X}$ are the following:

$\displaystyle p_{L,p}(X_{RT})$ $\displaystyle=$
$\displaystyle\frac{F\left\\{\frac{1}{p}F^{\leftarrow}[pF(u)]-\frac{1-p}{p}F^{\leftarrow}[(1-p)F(u)]\right\\}}{F(u)}$
$\displaystyle p_{R,p}(X_{RT})$ $\displaystyle=$ $\displaystyle
1-\frac{F\left\\{\frac{1}{p}F^{\leftarrow}[(1-p)F(u)]-\frac{1-p}{p}F^{\leftarrow}[pF(u)]\right\\}}{F(u)}$
$\displaystyle p_{L,p}(X_{LT})$ $\displaystyle=$
$\displaystyle\frac{F\left\\{\frac{1}{p}F^{\leftarrow}[p+(1-p)F(l)]-\frac{1-p}{p}F^{\leftarrow}[(1-p)+pF(l)]\right\\}-F(l)}{1-F(l)}$
$\displaystyle p_{R,p}(X_{LT})$ $\displaystyle=$
$\displaystyle\frac{1-F\left\\{\frac{1}{p}F^{\leftarrow}[1-p+pF(l)]-\frac{1-p}{p}F^{\leftarrow}[p+(1-p)F(l)]\right\\}}{1-F(l)}$
$\displaystyle p_{L,p}(X_{DT})$ $\displaystyle=$
$\displaystyle\frac{F\left\\{\frac{1}{p}F^{\leftarrow}[pF(u)+(1-p)F(l)]-\frac{1-p}{p}F^{\leftarrow}[(1-p)F(u)+pF(l)]\right\\}-F(l)}{F(u)-F(l)}$
$\displaystyle p_{R,p}(X_{DT})$ $\displaystyle=$
$\displaystyle\frac{F(u)-F\left\\{\frac{1}{p}F^{\leftarrow}[(1-p)F(u)+pF(l)]-\frac{1-p}{p}F^{\leftarrow}[pF(u)+(1-p)F(l)]\right\\}}{F(u)-F(l)}.$

If $P(X>0)=1$, and

  * 1.

$a>1$, then

$\displaystyle p_{L,p}(log_{a}(X))$ $\displaystyle=$ $\displaystyle
F\left\\{\sqrt[p]{\frac{F^{\leftarrow}(p)}{[F^{\leftarrow}(1-p)]^{1-p}}}\right\\},$
$\displaystyle p_{R,p}(log_{a}(X))$ $\displaystyle=$ $\displaystyle
1-F\left\\{\sqrt[p]{\frac{F^{\leftarrow}(1-p)}{[F^{\leftarrow}(p)^{1-p}]}}\right\\}.$

  * 1.

$0<a<1$, then

$\displaystyle p_{L,p}(log_{a}(X))$ $\displaystyle=$ $\displaystyle
1-F\left\\{\sqrt[p]{\frac{F^{\leftarrow}(1-p)}{[F^{\leftarrow}(p)]^{1-p}}}\right\\},$
$\displaystyle p_{R,p}(log_{a}(X))$ $\displaystyle=$ $\displaystyle
F\left\\{\sqrt[p]{\frac{F^{\leftarrow}(p)}{[F^{\leftarrow}(1-p)]^{1-p}}}\right\\}.$

For any r.v. $X$, such that $P(X>0)=1$, if

  * 1.

If $\alpha<0$, then

$\displaystyle p_{L,p}(X^{\alpha})$ $\displaystyle=$ $\displaystyle
1-F\left\\{\sqrt[\alpha]{\frac{1}{p}[F^{\leftarrow}(1-p)]^{\alpha}-\frac{1-p}{p}[F^{\leftarrow}(p)]^{\alpha}}\right\\},$
$\displaystyle p_{R,p}(X^{\alpha})$ $\displaystyle=$ $\displaystyle
F\left\\{\sqrt[\alpha]{\frac{1}{p}[F^{\leftarrow}(p)]^{\alpha}-\frac{1-p}{p}[F^{\leftarrow}(1-p)]^{\alpha}}\right\\}.$

  * 1.

If $\alpha>0$, then

$\displaystyle p_{L,p}(X^{\alpha})$ $\displaystyle=$ $\displaystyle
F\left\\{\sqrt[\alpha]{\frac{1}{p}[F^{\leftarrow}(p)]^{\alpha}-\frac{1-p}{p}[F^{\leftarrow}(1-p)]^{\alpha}}\right\\},$
$\displaystyle p_{R,p}(X^{\alpha})$ $\displaystyle=$ $\displaystyle
1-F\left\\{\sqrt[\alpha]{\frac{1}{p}[F^{\leftarrow}(1-p)]^{\alpha}-\frac{1-p}{p}[F^{\leftarrow}(p)]^{\alpha}}\right\\}.$

If

  * 1.

$a\in(0,1)$, then

$\displaystyle p_{L,p}(a^{X})$ $\displaystyle=$ $\displaystyle
1-F\left\\{log_{a}\left[\frac{1}{p}a^{F^{\leftarrow}(1-p)}-\frac{1-p}{p}a^{F^{\leftarrow}(p)}\right]\right\\},$
$\displaystyle p_{R,p}(a^{X})$ $\displaystyle=$ $\displaystyle
F\left\\{log_{a}\left[\frac{1}{p}a^{F^{\leftarrow}(p)}-\frac{1-p}{p}a^{F^{\leftarrow}(1-p)}\right]\right\\}.$

  * 1.

If $\alpha>1$, then

$\displaystyle p_{L,p}(a^{X})$ $\displaystyle=$ $\displaystyle
F\left\\{log_{a}\left[\frac{1}{p}a^{F^{\leftarrow}(p)}-\frac{1-p}{p}a^{F^{\leftarrow}(1-p)}\right]\right\\},$
$\displaystyle p_{R,p}(a^{X})$ $\displaystyle=$ $\displaystyle
1-F\left\\{log_{a}\left[\frac{1}{p}a^{F^{\leftarrow}(1-p)}-\frac{1-p}{p}a^{F^{\leftarrow}(p)}\right]\right\\}.$

###### Remark 1

Note that in Theorem 1, l), the expressions for $p_{R,p}(log_{a}(X))$ and
$p_{L,p}(log_{a}(X))$ do not depend on the exact value of $a$ but only on the
fact if $0<a<1$, or $a>1$. The last means that, according to this
classification of absolutely continuous probability distributions with respect
to the tails of their c.d.fs., if we decide to change these characteristics
and take a logarithm, the exact value of the basis of the logarithm is not
important for probabilities for outside values of the transformed
distribution. Only the fact that it is bigger or less than $1$ can influence
$p_{R,p}(log_{a}(X))$ and $p_{L,p}(log_{a}(X))$.

Corollary of f): Let $p\in(0,0.5]$ be fixed.

  * 1.

If $P(X>0)=1$, then $p_{L,p}(X)\leq p_{R,p}(\frac{1}{X}).$

  * 2.

If $P(X<0)=1$, then $p_{R,p}(X)\leq p_{L,p}(\frac{1}{X}).$

The next corollary corresponds to the well-known experience that taking a
logarithm with basis bigger than one of the data we decrease the chance to
observe right $p$-outside values and increase the chance to observe left
$p$-outside values. Together with Theorem 3, they show once again the
appropriateness of these characteristics when speaking about the tail of the
observed distribution.

Corollary of e): Let $p\in(0,0.5]$ be fixed. Suppose $P(X>0)=1$,

  * 1.

if $a>1$, then

$p_{L,p}(a^{X})\leq p_{L,p}(X)\leq p_{L,p}[log_{a}(X)],$ (12)
$p_{R,p}[log_{a}(X)]\leq p_{R,p}(X)\leq p_{R,p}(a^{X}).$ (13)

  * 1.

If $0<a<1$, then

$p_{L,p}(a^{X})\leq p_{R,p}(X),\quad p_{L,p}\left(\frac{1}{a^{X}}\right)\leq
p_{L,p}(X)\leq p_{R,p}[log_{a}(X)],$ (14) $p_{L,p}[log_{a}(X)]\leq
p_{R,p}(X)\leq p_{R,p}\left(\frac{1}{a^{X}}\right),\quad p_{L,p}(X)\leq
p_{R,p}(a^{X}).$ (15)

According to $p_{R,p}$ characteristics, taking powers bigger than 1 of the
data we increase the chance to observe right $p$-outside values, and decrease
the chance to observe left $p$-outside values in the observed distribution.

###### Theorem 3

For $p\in(0,0.5]$, and $P(X>0)=1$,

a)

$0<\alpha_{1}\leq\alpha_{2}$, then

$p_{L,p}(X^{\alpha_{2}})\leq p_{L,p}(X^{\alpha_{1}}),\quad
p_{R,p}(X^{\alpha_{1}})\leq p_{R,p}(X^{\alpha_{2}}).$ (16)

b)

If $\alpha>1$, and
$\frac{1}{1-p}\leq\left[\frac{F^{\leftarrow}(1-p)}{F^{\leftarrow}(p)}\right]^{\alpha}$,
then $p_{L,p}(X^{\alpha})=0$ and

$p_{R,p}(X^{1/\alpha})\leq p_{R,p}(X)\leq p_{R,p}(X^{\alpha}).$ (17)

c)

If $\alpha>1$, and
$\frac{1}{1-p}\geq\left[\frac{F^{\leftarrow}(1-p)}{F^{\leftarrow}(p)}\right]^{\alpha}$,
then $p_{R,p}(X^{\alpha})=0$ and

$p_{L,p}(X^{\alpha})\leq p_{L,p}(X)\leq p_{L,p}(X^{1/\alpha}).$ (18)

Application of these probabilities requires knowledge about their values for
different distributions. Therefore we have calculated some of their explicit
forms in the next section.

##  3 The most important particular cases

In order to choose the most appropriate class for modeling the tails of the
c.d.f. of the observed r.v. we can first calculate the probabilities for left
and right $p$-outside values, for as more as possible distributional types,
and then to compare these probabilities with corresponding estimators. This
approach is analogous to the comparison of the means in cases when we are
interested in the center of the distribution. Let us now present the exact
values of these characteristics in some of the most popular cases of
probability distributions used in practice for modeling heavy tails. Till the
end of this section, we assume that $p\in(0,0.5]$.

The dependencies of $p_{R,0.25}$ on the parameter $\alpha$, which
characterises the tail of the corresponding distribution in cases when $F$ is
$Gamma(\alpha,\beta)$, or Fr$\acute{e}$chet, Pareto, Stable, Weibull positive,
$H_{1}$, $H_{4}$, log-Pareto, Hill horror or Burr distributed are depicted on
Figure 2 and Figure 2, and could be seen also in Jordanova and Petkova (2018)
[36], and in the supplementary material of Soza et al. (2019) [56].

  * 1.

Exponential distribution. Let $\lambda>0$, and $X$ be Exponential with mean
$\frac{1}{\lambda}$.

It is well known that $\lambda$ is a scale parameter of the exponential
distribution, therefore due to Th. 1, c) without lost of generality (w.l.g.)
we can assume that $\lambda=1$ and this will not change the values of
$p_{L,p}(X)$ and $p_{R,p}(X)$. In this case,

$p_{L,p}(X)=P\left[X<log\frac{p^{\frac{1-p}{p}}}{(1-p)^{\frac{1}{p}}}\right]=\left\\{\begin{array}[]{ccc}0&,&p\in(0,p_{0}),\\\
p\frac{(1-p)^{\frac{1}{p}}}{p^{\frac{1}{p}}}&,&p\in[p_{0},\frac{1}{2})\end{array}\right.,$

where $p_{0}$ is the solution of the equation
$(1-p_{0})log(p_{0})=log(1-p_{0})$ and $p_{0}\approx 0.4096$.

$p_{R,p}(X)=P\left\\{X>log\left[\left(\frac{1-p}{p}\right)^{\frac{1}{p}}\frac{1}{1-p}\right]\right\\}=(1-p)\left(\frac{p}{1-p}\right)^{\frac{1}{p}}.$

In particular, for $p_{L,0.25}(X)=0$, the empirical right fence is
asymptotically unbiased and efficient estimator for the theoretical right
fence

$\lim_{i\to\infty}ER_{4i-1}(X,0.25)=R(X,0.25)=2log(2)+3log(3)=log(108)$
$\lim_{i\to\infty}DR_{4i-1}(X,0.25)=\lim_{i\to\infty}[16\psi^{\prime}(i)-\psi^{\prime}(4i)-15\psi^{\prime}(3i)]=0.$

where $\psi^{\prime}(i)=\frac{\partial^{2}log\Gamma(z)}{\partial z^{2}}$ is
the Polygamma function (for the last limit see Guo and Feng (2013) [28]), and
$p_{R,0.25}(X)=\frac{1}{108}=0.00925(925).$ See Jordanova and Petkova
(2017-2018) [36, 35].

Further on in this section (due to properties b) and c) Theorem 2, w.l.g. we
assume that $\mu=0$ and $\sigma=1$.

  * 1.

Generalized Pareto distribution (GPD). Consider $\mu\in R$ and $\sigma>0$.

$\displaystyle
F_{X}(x)=\left\\{\begin{array}[]{ccc}1-(1+\xi\frac{x-\mu}{\sigma})^{-1/\xi}&,&x>\mu,\xi>0\\\
1-(1+\xi\frac{x-\mu}{\sigma})^{-1/\xi}&,&\mu\leq
x\leq\mu-\frac{\sigma}{\xi},\xi<0\\\
1-e^{-\frac{x-\mu}{\sigma}}&,&x>\mu,\xi\to 0.\end{array}\right.$ (22)

We have already considered the case $\xi\to 0$. In that case, it is well known
that, the GPD coincides with Exponential distribution. So, here we assume that
$\xi\not=0$. Then the quantile function is
$F_{X}^{\leftarrow}(p)=\frac{1}{\xi}[(1-p)^{-\xi}-1].$ We replace it in the
formula for $L(X;p)$, then in the definition for $p_{L,p}(X)$, and obtain

$p_{L,p}(X)=P\left\\{X<\frac{1}{\xi}\left[\frac{1}{p}(1-p)^{-\xi}-1+p^{-\xi}-p^{-\xi-1}\right]\right\\}.$

In order to replace (22) in the last probability we need to consider
separately the following two cases:

  * $\cdot$

Case $\xi>0$. In this case
$\frac{1}{\xi}\left[\frac{1}{p}(1-p)^{-\xi}-1+p^{-\xi}-p^{-\xi-1}\right]<\frac{1}{\xi}$
$\left[\frac{1}{p}-1+\right.$ $\left.p^{-\xi}-p^{-\xi-1}\right]$. The last
expression is equal to
$\frac{1}{\xi}\left(1-\frac{1}{p}\right)\left(\frac{1}{p^{\xi}}-1\right)<0$,
therefore $p_{L,p}(X)=0$.

  * $\cdot$

Case $\xi<0$. In this case
$\frac{1}{\xi}\left[\frac{1}{p}(1-p)^{-\xi}-1+p^{-\xi}-p^{-\xi-1}\right]>0$,
therefore

$\displaystyle p_{L,p}(X)$ $\displaystyle=$ $\displaystyle
P\left\\{X<\frac{1}{\xi}\left[\frac{1}{p}(1-p)^{-\xi}-1+p^{-\xi}-p^{-\xi-1}\right]\right\\}$
$\displaystyle=$ $\displaystyle
1-(\frac{1}{p}(1-p)^{-\xi}+p^{-\xi}-p^{-\xi-1})^{-1/\xi}$

Analogously we replace the quantile function in the definition for $R(X;p)$,
then in $p_{R,p}(X)$, and obtain

$\displaystyle p_{R,p}(X)$ $\displaystyle=$ $\displaystyle
P\left\\{X>\frac{1}{\xi}[p^{-\xi}-1]+\frac{1-p}{p}\left[\frac{1}{\xi}(p^{-\xi}-1)-\frac{1}{\xi}[(1-p)^{-\xi}-1]\right]\right\\}$
$\displaystyle=$ $\displaystyle
P\left\\{X>\frac{1}{\xi}\left[\frac{1-p}{p}[1-(1-p)^{-\xi}]-\frac{1}{p}\left(1-p^{-\xi}\right)\right]\right\\}$

In order to calculate this expression we need to determine the sign of
$\frac{1}{\xi}\left[\frac{1-p}{p}[1-(1-p)^{-\xi}]-\frac{1}{p}\left(1-p^{-\xi}\right)\right]$.
Therefore again we consider two cases.

  * $\cdot$

Case $\xi>0$. Because of $p\in(0,0.5]$, we have $p^{-\xi}>(1-p)^{-\xi}$,
$p(1-p)^{-\xi}>p$ and

$(1-p)^{-\xi}+p<p^{-\xi}+p(1-p)^{-\xi}$
$1-(1-p)^{-\xi}-p[1-(1-p)^{-\xi}]>1-p^{-\xi}$
$(1-p)[1-(1-p)^{-\xi}]>1-p^{-\xi}$
$\frac{1-p}{p}[1-(1-p)^{-\xi}]>\frac{1}{p}\left(1-p^{-\xi}\right)$

Therefore

$\displaystyle p_{R,p}(X)$ $\displaystyle=$ $\displaystyle
P\left\\{X>\frac{1}{\xi}\left[\frac{1-p}{p}[1-(1-p)^{-\xi}]-\frac{1}{p}\left(1-p^{-\xi}\right)\right]\right\\}$
$\displaystyle=$
$\displaystyle\left\\{1+\frac{1-p}{p}[1-(1-p)^{-\xi}]-\frac{1}{p}\left(1-p^{-\xi}\right)\right\\}^{-1/\xi}$
$\displaystyle=$
$\displaystyle\left\\{\frac{p^{-\xi}}{p}+\left(1-\frac{1}{p}\right)(1-p)^{-\xi}\right\\}^{-1/\xi}$

In case $p=0.25$, and $\alpha=\frac{1}{\xi}$ this expression coincide with the
one in Jordanova and Petkova (2018)[36]

$p_{R,0.25}(X)=\frac{3}{4(4.3^{\frac{1}{\alpha}}-3)^{\alpha}}.$ (23)

Figure 1: The dependence of $p_{R,0.25}(X)$ on $\alpha$

Figure 2: The dependence of $p_{R,0.25}(X)$ on $\alpha$

  * $\cdot$

Case $\xi<0$. Because of $p\in(0,0.5)$ we have that
$\frac{1-p}{p}<\frac{1}{p}$, $1-(1-p)^{-\xi}<1-p^{-\xi}$ and
$\frac{1}{\xi}\left[\frac{1-p}{p}[1-(1-p)^{-\xi}]-\frac{1}{p}\left(1-p^{-\xi}\right)\right]>0.$

In case

$\frac{1}{\xi}\left[\frac{1-p}{p}[1-(1-p)^{-\xi}]-\frac{1}{p}\left(1-p^{-\xi}\right)\right]>-\frac{1}{\xi}$
$(1-p)[1-(1-p)^{-\xi}]-1+p^{-\xi}<-p$ $p^{-\xi}<(1-p)^{1-\xi}$
$\xi<\frac{log(1-p)}{log\left(\frac{1-p}{p}\right)}$

we have that $p_{R,p}(X)=0$.

If $0>\xi>\frac{log(1-p)}{log\left(\frac{1-p}{p}\right)}$,

$\displaystyle p_{R,p}(X)$ $\displaystyle=$ $\displaystyle
P\left\\{X>\frac{1}{\xi}\left[\frac{1-p}{p}[1-(1-p)^{-\xi}]-\frac{1}{p}\left(1-p^{-\xi}\right)\right]\right\\}$
$\displaystyle=$
$\displaystyle\left\\{1+\frac{1-p}{p}[1-(1-p)^{-\xi}]-\frac{1}{p}(1-p^{-\xi})\right\\}^{-\frac{1}{\xi}}$
$\displaystyle=$
$\displaystyle\left\\{(1-p)^{-\xi}\left(1-\frac{1}{p}\right)+\frac{1}{p}p^{-\xi}\right\\}^{-\frac{1}{\xi}}.$

When speaking about heavy tails we can not forget about Extreme value
distributions with respect to linear transformations. Therefore in the next
three points, we will consider them. At the beginning of the last century
Fisher and Tippet (1928)[23], Gnedenko (1943) [25], and Gumbel (1958)[27] have
shown that they appear as limiting distributions of maxima of i.i.d. r.vs.
after appropriate affine transformations.

  * 1.

Fr$\acute{e}$chet distribution. Let $\alpha>0$. W.l.g. we assume that in (2)
$c=1$.

Therefore for $p\in(0,1)$, $F^{\leftarrow}(p)=(-log(p))^{-\frac{1}{\alpha}}$.

$p_{L,p}(X)=P\left\\{X<(-log(p))^{-\frac{1}{\alpha}}-\frac{1-p}{p}[(-log(1-p))^{-\frac{1}{\alpha}}-(-log(p))^{-\frac{1}{\alpha}}]\right\\}$

and because of
$(-log(p))^{-\frac{1}{\alpha}}>\frac{1-p}{p}[(-log(1-p))^{-\frac{1}{\alpha}}-(-log(p))^{-\frac{1}{\alpha}}]$
$\Longleftrightarrow$

$-log(p)<(1-p)^{-\alpha}(-log(1-p))\Longleftrightarrow\alpha_{0}:=-\frac{log\left(\frac{log(p)}{log(1-p)}\right)}{log(1-p)}<\alpha$
$p_{L,p}(X)=exp\left\\{-\left[\frac{1}{p}(-log\,p)^{-1/\alpha}+\left(1-\frac{1}{p}\right)[-log\,(1-p)]^{-1/\alpha}\right]^{-\alpha}\right\\},$

when $\alpha>\alpha_{0}$, and $p_{L,p}(X)=0$, for $\alpha\in(0,\alpha_{0}]$.

Analogously

$\displaystyle p_{R,p}(X)$ $\displaystyle=$ $\displaystyle
P\left\\{X>(-log(1-p))^{-\frac{1}{\alpha}}+\frac{1-p}{p}[(-log(1-p))^{-\frac{1}{\alpha}}-(-log(p))^{-\frac{1}{\alpha}}]\right\\}$
$\displaystyle=$ $\displaystyle
P\left[X>\frac{1}{p}(-log(1-p))^{-\frac{1}{\alpha}}-\frac{1-p}{p}(-log(p))^{-\frac{1}{\alpha}}\right].$

Now we need to consider the expression after the inequality. As far as for all
$p\in(0,0.5)$ the following four expressions are equivalent

$(-log(1-p))^{-\frac{1}{\alpha}}>(1-p)(-log(p))^{-\frac{1}{\alpha}}$
$\frac{-log(1-p)}{-log(p)}<(1-p)^{-\alpha}$
$log\left[\frac{-log(1-p)}{-log(p)}\right]<-\alpha\,\,log(1-p)$
$-\frac{log\left[\frac{-log(1-p)}{-log(p)}\right]}{log(1-p)}<0<\alpha$

we have

$p_{R,p}(X)=exp\left\\{-\left(\frac{1}{p}(-log(1-p))^{-\frac{1}{\alpha}}-\frac{1-p}{p}(-log(p))^{-\frac{1}{\alpha}}\right)^{-\alpha}\right\\}.$

Figure 2 represents the dependence of $p_{R,p}(X)$ on $\alpha$ in case
$p=0.25$. The explicit formula for this case can be seen also in Jordanova and
Petkova (2018) [36].

  * 1.

Weibull negative distribution. Consider $\alpha>0$. W.l.g. $X\in
NegWeibull(\alpha,$ $1,$ $0)$. See (3). Therefore for $p\in(0,1)$,
$F^{\leftarrow}(p)=-(-log(p))^{\frac{1}{\alpha}}.$

Note that its positive version $-X$ coincides in distribution with a standard
Exponentially distributed r.v. raised to the power $\frac{1}{\alpha}$.
Therefore, for $\alpha>1$, according to our classification, this distribution
has heavier right tail than the exponential one and for $\alpha<1$ vice versa.
This can be seen also on Figure 2, where standard exponential distribution is
depicted as $\Gamma(1,1)$. More precisely

$\displaystyle p_{L,p}(X)$ $\displaystyle=$ $\displaystyle
P\left\\{X<-[-log(p)]^{\frac{1}{\alpha}}-\frac{1-p}{p}\left[[-log(p)]^{\frac{1}{\alpha}}-[-log(1-p)]^{\frac{1}{\alpha}}\right]\right\\}$
$\displaystyle=$ $\displaystyle
P\left\\{X<\frac{1-p}{p}[-log(1-p)]^{\frac{1}{\alpha}}-\frac{1}{p}[-log(p)]^{\frac{1}{\alpha}}\right\\}$

and because of for all $p\in(0,0.5]$ we have
$(1-p)[-log(1-p)]^{\frac{1}{\alpha}}<[-log(p)]^{\frac{1}{\alpha}}$, therefore

$p_{L,p}(X)=exp\left\\{-\left[\frac{1}{p}[-log(p)]^{\frac{1}{\alpha}}-\frac{1-p}{p}[-log(1-p)]^{\frac{1}{\alpha}}\right]^{\alpha}\right\\}$
(24)

Figure 2, depicts the dependence of $p_{L,0.25}(X)=p_{R,0.25}(-X)$, (i.e. $-X$
is Weibull positive) on $\alpha$.

Analogously

$\displaystyle p_{R,p}(X)$ $\displaystyle=$ $\displaystyle
P\left\\{X>-(-log(1-p))^{\frac{1}{\alpha}}+\frac{1-p}{p}[(-log(p))^{\frac{1}{\alpha}}-(-log(1-p))^{\frac{1}{\alpha}}]\right\\}$
$\displaystyle=$ $\displaystyle
P\left[X>\frac{1-p}{p}(-log(p))^{\frac{1}{\alpha}}-\frac{1}{p}(-log(1-p))^{\frac{1}{\alpha}}\right]$

As far as for all $p\in(0,0.5]$

$(-log(1-p))^{\frac{1}{\alpha}}<(1-p)(-log(p))^{\frac{1}{\alpha}}$
$\frac{-log(1-p)}{-log(p)}>(1-p)^{-\alpha}$
$log\left[\frac{-log(1-p)}{-log(p)}\right]>\alpha\,\,log(1-p)$
$\alpha_{1}:=\frac{log\left[\frac{-log(1-p)}{-log(p)}\right]}{log(1-p)}>\alpha$

we have

$p_{R,p}(X)=\left\\{\begin{array}[]{ccc}0&,&\alpha\in(0,\alpha_{1}]\\\
1-exp\left\\{-\left(\frac{1}{p}(-log(1-p))^{\frac{1}{\alpha}}-\frac{1-p}{p}(-log(p))^{\frac{1}{\alpha}}\right)^{\alpha}\right\\}&,&\alpha>\alpha_{1}\end{array}\right..$

  * 1.

Gumbell distribution. Let $\alpha>0$, $\mu\in R$ and $\sigma>0$

$F_{X}(x)=exp\left\\{-exp\left[-\frac{x-\mu}{\sigma}\right]\right\\},\quad
x\in R.$

W.l.g. $\mu=0$ and $\sigma=1$. $F^{\leftarrow}(p)=-log(-log(p))$,

$\displaystyle p_{L,p}(X)$ $\displaystyle=$ $\displaystyle
P\left\\{X<-log(-log(p))-\frac{1-p}{p}[log(-log(p))-log(-log(1-p))]\right\\}$
$\displaystyle=$ $\displaystyle
P\left\\{X<\frac{1-p}{p}log(-log(1-p))-\frac{1}{p}log(-log(p))\right\\}=p^{\left[\frac{log(p)}{log\,(1-p)}\right]^{\frac{1}{p}}}$
$\displaystyle p_{R,p}(X)$ $\displaystyle=$ $\displaystyle
P\left\\{X>-log(-log(1-p))+\frac{1-p}{p}[log(-log(p))-log(-log(1-p))]\right\\}$
$\displaystyle=$ $\displaystyle
P\left\\{X>\frac{1-p}{p}log(-log(p))-\frac{1}{p}log(-log(1-p))\right\\}=1-p^{\left[\frac{log(1-p)}{log\,(p)}\right]^{\frac{1}{p}}}$

Jordanova and Petkova (2018) [36] have calculated that $p_{L,0.25}(X)\approx
4.264\times 10^{-68}$ and $p_{R,0.25}(X)\approx 0.002568$.

  * 1.

Logistic distribution. Assume $\mu\in R$, $\sigma>0$ and

$F_{X}(x)=\frac{1}{1+exp\left(-\frac{x-\mu}{\sigma}\right)},\quad x\in R.$

W.l.g. $\mu=0$ and $\sigma=1$.
$F^{\leftarrow}(p)=log\left(\frac{p}{1-p}\right)$,

$p_{R,p}(X)=p_{L,p}(X)=P\left(X<\frac{2-p}{p}log\frac{p}{1-p}\right)=\left[1+\left(\frac{1-p}{p}\right)^{\frac{2-p}{p}}\right]^{-1}$

$\leq\left[1+3^{5}\right]^{-1}=p_{R,0.25}(X)\approx 0.000457,p\in(0,0.25).$

  * 1.

Log-logistic distribution. Assume $\mu\in R$, $\sigma>0$ and

$F_{X}(x)=\frac{1}{1+\left(\frac{x-\mu}{\sigma}\right)^{-\alpha}},\quad
x>\mu.$ (25)

W.l.g. $\mu=0$ and $\sigma=1$. As far as
$F^{\leftarrow}(p)=\sqrt[\alpha]{\frac{p}{1-p}}$, and for $p\in(0,0.5]$

$R(X,p)=\frac{1}{p}\sqrt[\alpha]{\frac{1-p}{p}}-\frac{1-p}{p}\sqrt[\alpha]{\frac{p}{1-p}}>0$

therefore by definition

$p_{R,p}(X)=\frac{1}{1+\left(\frac{1}{p}\sqrt[\alpha]{\frac{1-p}{p}}-\frac{1-p}{p}\sqrt[\alpha]{\frac{p}{1-p}}\right)^{\alpha}}.$
(26)

The plot of this function of $\alpha>0$ for $p=0.25$ can be seen on Figure 4.
We observe that for a fixed $\alpha>0$ its right tail is very similar to the
corresponding tails of Pareto and Fr$\acute{e}$chet distribution and heavier
than the Stable one. Analogously

$L(X,p)=\frac{1}{p}\sqrt[\alpha]{\frac{p}{1-p}}-\frac{1-p}{p}\sqrt[\alpha]{\frac{1-p}{p}},$

therefore for $0<\alpha<2[log_{1-p}(p)-1]$, and $p\in(0,0.5]$

$p_{L,p}(X)=1-\frac{1}{1+\left(\frac{1}{p}\sqrt[\alpha]{\frac{p}{1-p}}-\frac{1-p}{p}\sqrt[\alpha]{\frac{1-p}{p}}\right)^{\alpha}}.$

and $p_{L,p}(X)=0$ otherwise.

In the next two cases we assume that $\tau>0$ and $\alpha>0$. W.l.g. $\mu=0$,
because it is a location parameter, and $\sigma=1$ and $\delta=1$ because they
are scale parameters. See Burr (1942) [7] or Einmahl et al. (2008) [19].

  * 1.

Burr distribution. Let

$F_{X}(x)=\left\\{\begin{array}[]{ccc}0&,&x<\mu\\\
1-\left[\frac{\delta}{\delta+\left(\frac{x-\mu}{\sigma}\right)^{\tau}}\right]^{\alpha}&,&x\geq\mu\end{array}\right..$
(27)

For $p\in(0,1)$, the quantile function is

$F^{\leftarrow}(p)=\sqrt[\tau]{(1-p)^{-1/\alpha}-1}$

(see also Nair et al. (2013) [40]). Therefore in our context from (27) we have
that if

$\frac{(1-p)^{-\frac{1}{\alpha}}-1}{p^{-\frac{1}{\alpha}}-1}>(1-p)^{\tau},$

then

$\displaystyle p_{L,p}(X)$ $\displaystyle=$ $\displaystyle
P\left\\{X<\frac{1}{p}\sqrt[\tau]{(1-p)^{-1/\alpha}-1}-\frac{1-p}{p}\sqrt[\tau]{p^{-1/\alpha}-1}\right\\}$
$\displaystyle=$ $\displaystyle
1-\left\\{\frac{p^{\tau}}{p^{\tau}+\left[\sqrt[\tau]{(1-p)^{-1/\alpha}-1}-(1-p)\sqrt[\tau]{p^{-1/\alpha}-1}\right]^{\tau}}\right\\}^{\alpha}$

and $p_{L,p}(X)=0$ otherwise.

For $p\in(0,0.25)$ and $\tau>0$ the inequality

$\frac{p^{-\frac{1}{\alpha}}-1}{(1-p)^{-\frac{1}{\alpha}}-1}\geq
1\geq(1-p)^{\tau},$

the definition of $p_{R,p}(X)$, and (27) entail

$\displaystyle p_{R,p}(X)$ $\displaystyle=$ $\displaystyle
P\left[X>\frac{1}{p}\sqrt[\tau]{p^{-1/\alpha}-1}-\frac{1-p}{p}\sqrt[\tau]{(1-p)^{-1/\alpha}-1}\right]$
$\displaystyle=$
$\displaystyle\left[\frac{p^{\tau}}{p^{\tau}+\left(\sqrt[\tau]{p^{-1/\alpha}-1}-(1-p)\sqrt[\tau]{(1-p)^{-1/\alpha}-1}\right)^{\tau}}\right]^{\alpha}.$

The dependence of $p_{R,0.25}(X)$ on $\alpha$ and for different values of
$\tau$ is depicted on Figure 2. We see that when $\tau$ increases, the chance
to observe $p$-outside values in the considered distribution decreases. The
last means that for Burr distribution not only $\alpha$ but also $\tau$
influences the tail-behaviour.

  * 1.

Reverse Burr distribution.

$F_{X}(x)=\left\\{\begin{array}[]{ccc}1-\left[\frac{\delta}{\delta+\left(\frac{\mu-x}{\sigma}\right)^{-\tau}}\right]^{\alpha}&,&x\leq\mu\\\
1&,&x>\mu\end{array}\right..$ (28)

For $p\in(0,1)$, the corresponding quantile function is

$F^{\leftarrow}(p)=-[(1-p)^{-1/\alpha}-1]^{-1/\tau}.$

We are interested in the case when $p\in(0,0.5]$. It guarantees that

$(1-p)[p^{-1/\alpha}-1]^{-1/\tau}<[(1-p)^{-1/\alpha}-1]^{-1/\tau}.$

Therefore from (28) we have

$\displaystyle p_{L,p}(X)$ $\displaystyle=$ $\displaystyle
P\left\\{X<\frac{1-p}{p}[p^{-1/\alpha}-1]^{-1/\tau}-\frac{1}{p}[(1-p)^{-1/\alpha}-1]^{-1/\tau}\right\\}$
$\displaystyle=$ $\displaystyle
1-\left\\{\frac{1}{1+p^{\tau}\left[[(1-p)^{-1/\alpha}-1]^{-1/\tau}-(1-p)(p^{-1/\alpha}-1)^{-1/\tau}\right]^{-\tau}}\right\\}^{\alpha}.$

The dependence of $p_{R,0.25}(-X)=p_{L,0.25}(X)$ on $\alpha$ and for different
values of $\tau$ is depicted on Figure 4. We observe that for $\alpha>0.5$ the
tail behavior of the Reverse-Burr distribution is much more sensitive on
$\tau$ than on $\alpha$.

If $(1-p)[(1-p)^{-1/\alpha}-1]^{-1/\tau}<[p^{-1/\alpha}-1]^{-1/\tau},$

$\displaystyle p_{R,p}(X)$ $\displaystyle=$ $\displaystyle
P\left[X>\frac{1-p}{p}[(1-p)^{-1/\alpha}-1]^{-1/\tau}-\frac{1}{p}[p^{-1/\alpha}-1]^{-1/\tau}\right]$
$\displaystyle=$
$\displaystyle\left[\frac{1}{1+p^{\tau}\left((1-p)[(1-p)^{-1/\alpha}-1]^{-1/\tau}-[p^{-1/\alpha}-1]^{-1/\tau}\right)^{-\tau}}\right]^{\alpha}.$

and $p_{R,p}(X)=0$ otherwise.

Figure 3: The dependence of $p_{R,0.25}(-X)$ on parameter $\alpha$ in Reverse-
Burr case for $\tau=0.5,1,1.5,2$

Figure 4: The dependence of $p_{R,0.25}(X)$ on $\alpha$

  * 1.

Gompertz distribution. Assume $\mu\in R$, $\sigma>0$, $\alpha>0$, and

$F_{X}(x)=\left\\{\begin{array}[]{ccc}0&,&x\leq\mu\\\
1-exp\left\\{-\alpha\left[exp\left(\frac{x-\mu}{\sigma}\right)-1\right]\right\\}&,&x>\mu\end{array}\right..$
(29)

For $\mu=0$, $\sigma=1$ and $p\in(0,1)$,
$F^{\leftarrow}(p)=log\left[1-\frac{log(1-p)}{\alpha}\right].$

For
$1-\frac{1}{\alpha}log(1-p)\geq\left[1-\frac{1}{\alpha}log(p)\right]^{1-p}$ we
have that $p_{L,p}(X)$ is equal to

$\displaystyle
P\left\\{X<\frac{1}{p}log\left[1-\frac{log(1-p)}{\alpha}\right]-\frac{1-p}{p}log\left[1-\frac{log(p)}{\alpha}\right]\right\\}$
$\displaystyle=$ $\displaystyle
1-exp\left\\{-\alpha\left[exp\left(\frac{1}{p}log\left[1-\frac{log(1-p)}{\alpha}\right]-\frac{1-p}{p}log\left[1-\frac{log(p)}{\alpha}\right]\right)-1\right]\right\\}$
$\displaystyle=$ $\displaystyle
1-exp\left\\{-\alpha\left[\sqrt[p]{\frac{1-\alpha^{-1}log(1-p)}{[1-\alpha^{-1}log(p)]^{1-p}}}-1\right]\right\\}$

and $p_{L,p}(X)=0$ otherwise. In particular $p_{L,0.25}(X)=0$.

As far as for all $p\in(0,0.5]$
$\frac{log[1-\frac{1}{\alpha}log(p)]}{log[1-\frac{1}{\alpha}log(1-p)]}>1-p,$
therefore from (29) and the definition of $p_{R,p}(X)$ we have that it is
equal to

$\displaystyle
P\left\\{X>\frac{1}{p}log\left[1-\frac{log(p)}{\alpha}\right]-\frac{1-p}{p}log\left[1-\frac{log(1-p)}{\alpha}\right]\right\\}$
$\displaystyle=$ $\displaystyle
exp\left\\{-\alpha\left[exp\left[\frac{1}{p}log\left[1-\frac{log(p)}{\alpha}\right]-\frac{1-p}{p}log\left[1-\frac{log(1-p)}{\alpha}\right]\right]-1\right]\right\\}$
$\displaystyle=$ $\displaystyle
exp\left\\{-\alpha\left[\sqrt[p]{\frac{1-\alpha^{-1}log(p)}{[1-\alpha^{-1}log(1-p)]^{1-p}}}-1\right]\right\\}\approx
0.$

Note that or all fixed $\alpha>0$, and within the considered distributions
with $p_{R,p}(X)>0$ in this study, this distribution has smallest value of
$p_{R,p}(X)$.

The next two distributions that we consider belong to the class of so-called
p-max-stable laws. They can be generalized to a strictly increasing affine
transformation and $p_{L,p}$, and $p_{R,p}$ characteristics will not change.
Using power normalizations Pancheva (1985) [43] obtained them as limiting laws
of power transformed maximums. Falk et al. (2004) [22] describe domains of
attraction of these laws under power normalization. We have already seen in
Corollary 1 of e), Theorem 1 that these transformations increase the values of
$p_{R,p}$. Ravi and Saeb (2012) [49] have obtained their entropies.

  * 1.

$H_{1}$ type. (See Pancheva (1985) [43]). Let $\alpha>0$, and $X\in
H_{1}(\alpha)$, i.e.

$F_{X}(x)=\left\\{\begin{array}[]{ccc}0&,&x<1\\\
exp\left\\{-(log\,\,x)^{-\alpha}\right\\}&,&x\geq 1\end{array}\right..$ (30)

The quantile function of this distribution is
$F_{X}^{\leftarrow}(p)=exp\left\\{(-log\,\,p)^{-1/\alpha}\right\\}$. 222It can
be seen e.g. in the supplementary material of Soza et al. (2019) [56] who
consider the case $p=0.25$. Therefore

$p_{L,p}(X)=P\left\\{X<\frac{1}{p}exp\left\\{[-ln\,\,(p)]^{-1/\alpha}\right\\}-\frac{1-p}{p}exp\left\\{[-ln\,\,(1-p)]^{-1/\alpha}\right\\}\right\\}.$

and for
$exp\left\\{[-ln\,\,(p)]^{-1/\alpha}\right\\}-(1-p)exp\left\\{[-ln\,\,(1-p)]^{-1/\alpha}\right\\}>p$,
$p_{L,p}(X)$ is equal to

$exp\left\\{-\left\\{log\left[\frac{1}{p}exp\left\\{[-ln\,\,(p)]^{-1/\alpha}\right\\}-\frac{1-p}{p}exp\left\\{[-ln\,\,(1-p)]^{-1/\alpha}\right\\}\right]\right\\}^{-\alpha}\right\\},$

otherwise $p_{L,p}(X)=0$. The last mean that for large $\alpha>0$ it is
possible to observe also left outside values.

Now let us consider the right tail. As far as for all $\alpha>0$

$\frac{1}{p}exp\left\\{[-log\,\,(1-p)]^{-1/\alpha}\right\\}-\frac{1-p}{p}exp\left\\{[-log(p)]^{-1/\alpha}\right\\}>1$
$exp\left\\{[-log\,\,(1-p)]^{-1/\alpha}\right\\}-(1-p)exp\left\\{[-log(p)]^{-1/\alpha}\right\\}>p$
$exp\left\\{[-log\,\,(1-p)]^{-1/\alpha}\right\\}-exp\left\\{[-log(p)]^{-1/\alpha}\right\\}>0$

and

$0>p\left\\{1-exp\left\\{[-log(p)]^{-1/\alpha}\right\\}\right\\}$
$\displaystyle p_{R,p}(X)$ $\displaystyle=$ $\displaystyle
P\left\\{X>\frac{1}{p}exp\left\\{[-log\,\,(1-p)]^{-1/\alpha}\right\\}-\frac{1-p}{p}exp\left\\{[-log(p)]^{-1/\alpha}\right\\}\right\\}=$
(31) $\displaystyle=$ $\displaystyle
1-exp\left\\{-\left\\{log\left\\{\frac{1}{p}exp\left\\{[-log\,\,(1-p)]^{-1/\alpha}\right\\}\right.\right.\right.$
$\displaystyle-$
$\displaystyle\left.\left.\left.\frac{1-p}{p}exp\left\\{[-log(p)]^{-1/\alpha}\right\\}\right\\}\right\\}^{-\alpha}\right\\}$

Figure 2 shows their dependence on $\alpha$ in case $p=0.25$. This case is
considered in the supplementary material of Soza et al. (2019) [56].

  * 1.

$H_{4}$ type. Let $\alpha>0$,

$F_{X}(x)=\left\\{\begin{array}[]{ccc}exp\left\\{-[ln(-\,x)]^{\alpha}\right\\}&,&x<-1\\\
1&,&x\geq-1\end{array}\right..$

It is one of the limiting distributions of power-transformed maxima obtained
in Pancheva (1984) [43]. Ravi and Saeb (2012) [49] calculate its Shannon
entropy. $H_{4}$ type is known also as log-Weibull law, and it is one of the
p-max stable laws. The quantile function has the form
$F_{Y}^{\leftarrow}(p)=-exp\\{[-log(p)]^{1/\alpha}\\}$, $p\in(0,1).$ Therefore

$p_{L,p}(X)=P\left\\{X<\frac{1-p}{p}exp\\{[-log(1-p)]^{1/\alpha}\\}-\frac{1}{p}exp\\{[-log(p)]^{1/\alpha}\\}\right\\}.$

And for $exp\\{[-log(p)]^{1/\alpha}\\}-(1-p)exp\\{[-log(1-p)]^{1/\alpha}\\}>p$
we have that $p_{L,p}(X)$ is equal to

$exp\left\\{-\left\\{log\left\\{\frac{1}{p}exp\\{[-log(p)]^{1/\alpha}\\}-\frac{1-p}{p}exp\\{[-log(1-p)]^{1/\alpha}\\}\right\\}\right\\}^{\alpha}\right\\},$

and $p_{L,p}(X)=0$ otherwise.

When consider the right tail for all $\alpha>0$ as far as $p<1-p$

$\frac{1-p}{p}exp\\{[-log(p)]^{1/\alpha}\\}-\frac{1}{p}exp\\{[-log(1-p)]^{1/\alpha}\\}<-1$
$(1-p)exp\\{[-log(p)]^{1/\alpha}\\}-exp\\{[-log(1-p)]^{1/\alpha}\\}<-p$
$-(1-p)exp\\{[-log(p)]^{1/\alpha}\\}-\left\\{-exp\\{[-log(1-p)]^{1/\alpha}\\}\right\\}>$
$>-(1-p)exp\\{[-log(p)]^{1/\alpha}\\}-\left\\{-exp\\{[-log(p)]^{1/\alpha}\\}\right\\}$

The last expression is equal to $-(-p)exp\\{[-log(p)]^{1/\alpha}\\}>p$,
therefore $p_{R,p}(X)$ is equal to

$\displaystyle
P\left\\{X>\frac{1-p}{p}exp\\{[-log(p)]^{1/\alpha}\\}-\frac{1}{p}exp\\{[-log(1-p)]^{1/\alpha}\\}\right\\}$
$\displaystyle=$ $\displaystyle
1-exp\left\\{-\left\\{log\left\\{\frac{1}{p}exp\\{[-log(1-p)]^{1/\alpha}\\}-\frac{1-p}{p}exp\\{[-log(p)]^{1/\alpha}\\}\right\\}\right\\}^{\alpha}\right\\}$

As in the previous case for large values of $\alpha>0$ it is possible to
observe both left and right outside values. The dependance of $p_{R,0.25}(-X)$
on $\alpha$ is depicted on Figure 2. We call the distribution of $-X$,
”$H_{4}$ positive”, and we have denoted it by $H_{4}^{+}$.

Note that the function
$g(\alpha)=exp\\{[log(4)]^{1/\alpha}\\}-\frac{3}{4}exp\\{[log\frac{4}{3}]^{1/\alpha}\\}-\frac{1}{4}$
is decreasing in $\alpha$ and $g(2)\approx 1.713627$ therefore for
$\alpha\in(0,2]$

$\displaystyle p_{L,0.25}(X)$ $\displaystyle=$ $\displaystyle p_{R,0.25}(-X)$
$\displaystyle=$ $\displaystyle
exp\left\\{-\left\\{log\left\\{4exp\\{[log(4)]^{1/\alpha}\\}-3exp\\{[log(\frac{4}{3})]^{1/\alpha}\\}\right\\}\right\\}^{\alpha}\right\\}.$

See the supplementary material of Soza et al. [56].

  * 1.

$log-Par_{\alpha}$ type. Log-Pareto law with parameter $\alpha$ seems to be
introduced in Cormann and Reiss (2009) [11]. More precisely here we assume
that

$F_{X}(x)=\left\\{\begin{array}[]{ccc}0&,&x<e\\\ 1-(log(x))^{-\alpha}&,&x\geq
e\end{array}\right..$

This distribution belongs to $\Pi$ class considered e.g. in de Haan and
Ferreira (2006) [13] or Embrehts et al. (1997) [20]. Ravi and Saeb (2012) [49]
investigate their entropies. Due to Corollary 1 of Theorem 1 its tail is
heavier than the tail of Fr$\acute{e}$chet distribution. The quantile function
of this distribution is
$F_{X}^{\leftarrow}(p)=exp\left\\{(1-p)^{-1/\alpha}\right\\}$. Therefore for
$\frac{1}{p}exp\left\\{(1-p)^{-1/\alpha}\right\\}-\frac{1-p}{p}exp\left\\{p^{-1/\alpha}\right\\}>e$

$p_{L,p}(X)=1-\left\\{log\left[\frac{1}{p}exp\left[(1-p)^{-1/\alpha}\right]-\frac{1-p}{p}exp\left(p^{-1/\alpha}\right)\right]\right\\}^{-\alpha},$

otherwise $p_{L,p}(X)=0$.

When consider the right tail for all $\alpha>0$ as far as $p<1-p$

$exp\left(p^{-1/\alpha}\right)+p\,exp\left[(1-p)^{-1/\alpha}\right]\geq
exp\left[(1-p)^{-1/\alpha}\right]+ep$
$exp\left(p^{-1/\alpha}\right)-(1-p)exp\left[(1-p)^{-1/\alpha}\right]\geq ep$
$\frac{1}{p}exp\left(p^{-1/\alpha}\right)-\frac{1-p}{p}exp\left[(1-p)^{-1/\alpha}\right]\geq
e$

therefore from the definition of $p_{R,p}(X)$ we obtain

$\displaystyle p_{R,p}(X)$ $\displaystyle=$ $\displaystyle
P\left\\{X>\frac{1}{p}exp\left(p^{-1/\alpha}\right)-\frac{1-p}{p}exp\left[(1-p)^{-1/\alpha}\right]\right\\}$
$\displaystyle=$
$\displaystyle\left\\{log\left\\{\frac{1}{p}exp\left(p^{-1/\alpha}\right)-\frac{1-p}{p}exp\left[(1-p)^{-1/\alpha}\right]\right\\}\right\\}^{-\alpha}.$

The dependence of $p_{R,0.25}(X)$ on $\alpha$ could be seen on Figure 2. We
observe that its tail behaviour almost coincide with log-Fr$\acute{e}$chet,
i.e. $H_{1}$, and within the considered distributions, for fixed $\alpha$
according to $p_{R,0.25}(X)$ the last two distributions have highest
probabilities to observe extreme outside values. Having in mind that without
transformations the tails of Fr$\acute{e}$chet and Pareto distributions are
heavy-tailed Cl. Neves et al. (2008) [41], Corman and Reiss (2009) [11] or
Falk (2004) [22] call them ”super heavy-tailed”.

Further on in this section, we consider distributions which quantile function
have no explicit form. Therefore we use R software (2018) [48] in order to
obtain obtain $F^{\leftarrow}(0.25)$ and $F^{\leftarrow}(0.75)$. Then we come
back to the well-known formulas for c.d.f. and obtain $p_{R,0.25}(X)$
characteristics.

  * 1.

Normal distribution. Assume $\mu\in R$, $\sigma^{2}>0$ and $X\in
N(\mu,\sigma^{2})$.

W.l.g. $\mu=0$ and $\sigma^{2}=1$. Due to the symmetry of this distribution
with respect to (w.r.t.) Oy, for all $p\in(0,0.5)$ we have that
$p_{L,p}(X)=p_{R,p}(X)$. In particular $p_{L,0.25}(X)=p_{R,0.25}(X)\approx
0.000001171$.

  * 1.

$t$-distribution. Assume $n\in N$ and $X\in t(n)$.

The symmetry of the p.d.fs. of these distributions w.r.t. $Oy$ implies
$p_{L,p}(X)=p_{R,p}(X)$, for all $p\in(0,0.5)$. The values of these
characteristics for $n=1,2,...,10$ are presented in Table 1, and could be seen
also in Jordanova and Petkova [36], and in the supplementary material of Soza
et al. [56].

n | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
$p_{L,0.25}(X)=p_{R,0.25}(X)$ | 0.0452 | 0.0146 | 0.0064 | 0.0033 | 0.0019
n | 6 | 7 | 8 | 9 | 10
$p_{L,0.25}(X)=p_{R,0.25}(X)$ | 0.0012 | 0.0008 | 0.0006 | 0.0004 | 0.0003
Table 1: The dependence of $p_{L,0.25}(X)=p_{R,0.25}(X)$ on $n$. $X\in t(n)$.

  * 1.

Gamma distribution. Assume $\alpha>0$, $\beta>0$ and $X\in
Gamma(\alpha,\beta)$ which means that

$F_{X}(x)=\left\\{\begin{array}[]{ccc}0&,&x<0\\\
\int_{0}^{x}\frac{\beta^{\alpha}}{\Gamma(\alpha)}y^{\alpha-1}e^{-\beta
y}dy&,&x\geq 0\end{array}\right..$

W.l.g. we assume that $\beta=1$. The plot of the dependence of $p_{R,0.25}(X)$
on $\alpha$ is depicted on Figure 2.

  * 1.

Hill-horror distribution. For $\alpha>0$ Embrechts et al. (1997) [20] define
it via its quantile function

$F_{X}^{\leftarrow}(p)=\frac{-log(1-p)}{\sqrt[\alpha]{1-p}},\quad p\in(0,1).$
(32)

Then $p_{L,0.25}(X)=0$. For the values of $p_{R,0.25}(X)$ see Figure 2. We
observe that this distribution has one of the heaviest right tails within the
considered probability types.

Following this approach, we can find explicit values or plots of $p_{L,p}$ and
$p_{R,p}$ characteristics for many other distributions and in this way to
compare their tails. For example log-Positive Weibull, log-Gumbel, Invence-
Gamma, Log-Gamma, Beta prime, or powers bigger than one, of these and other
distributions.

##  4 Properties of the estimators

In the previous section, we have considered some particular cases of
probability laws and we have shown which parameter governs the heaviness of
the tail of the corresponding distribution according to our classification.
For the distributions with regularly varying tails, it coincides with the very
well-known index of regular variation. In this section, we obtain different
asymptotic properties of the estimators of the corresponding parameters of
heaviness of the tails. The general formula for the joint distribution of
order statistics is very well known. Together with the formula for their
conditional distributions they could be found e.g. in Nevzorov (2001) [42] or
in Arnold et al. (1992) [2]. The following lemma is their immediate corollary.
Its first part summarises the same results in the terms of equality in
distributions. In vi) we have expressed the bivariate vector of order
statistics as a bivariate function of independent r.vs. This allows as to make
the same with the fences in the next property. Finally two explicit formulae
for the probability mass functions of the numbers of left and right
$p$-outside values in a sample of independent observations are presented. Due
to their complicated forms further on the section proceeds with asymptotic
results.

###### Lemma 1

If $\xi\in Beta(i,j)$, $i\in N$, $j\in N$, $F(z)=P(X<z)$ is some c.d.f. of a
r.v. $X$ and $c>0$ is a constant, then

  1. i)

$-\frac{log\,(\xi)}{c}\stackrel{{\scriptstyle d}}{{=}}\epsilon_{(j,i+j-1)},$
where $\epsilon_{(j,i+j-1)}$ is the $j$-th order statistics in a sample of
$i+j-1$ independent observations on i.i.d. Exponential r.vs. with parameter
$c>0$.

  2. ii)

$\frac{1}{c\xi^{\alpha}}\stackrel{{\scriptstyle d}}{{=}}\nu_{(j,i+j-1)}$,
where $\nu_{(j,i+j-1)}$ is the $j$-th order statistics in a sample of $i+j-1$
independent observations on i.i.d. Pareto distributed r.vs. with parameters
$\frac{1}{\alpha}>0$ and $\delta=\frac{1}{c}$.

  3. iii)

$\frac{-1}{c\,\,log(\xi)}\stackrel{{\scriptstyle d}}{{=}}\phi_{(j,i+j-1)}$,
where $\phi_{(j,i+j-1)}$ is the $j$-th order statistics in a sample of $i+j-1$
independent observations on i.i.d. Fr$\acute{e}$chet distributed r.vs. with
parameter $\alpha=1$ and scale parameter $c$.

  4. iv)

$F^{\leftarrow}(\xi)\stackrel{{\scriptstyle d}}{{=}}\kappa_{(i,i+j-1)},$ and
$(1-F)^{\leftarrow}(\xi)\stackrel{{\scriptstyle d}}{{=}}\kappa_{(j,i+j-1)}$,
where $\kappa_{(s,i+j-1)}$ is the $s$-th order statistic of a sample of
$i+j-1$ independent observations on a r.v. with absolutely continuous c.d.f.
$F$, $s=i,j$.

  5. v)

For $1\leq i<j$

$(X_{(j,i+j-1)}|X_{(i,i+j-1)}=x)\stackrel{{\scriptstyle
d}}{{=}}\tilde{\tilde{\theta}}_{(j-i,j-1)}\stackrel{{\scriptstyle
d}}{{=}}G_{x}^{\leftarrow}(\xi^{*}),$
$(X_{(i,i+j-1)}|X_{(j,i+j-1)}=x)\stackrel{{\scriptstyle
d}}{{=}}\tilde{\theta}_{(i,j-1)}\stackrel{{\scriptstyle
d}}{{=}}T_{x}^{\leftarrow}(\xi^{**}),$

where $\tilde{\tilde{\theta}}_{(j-i,j-1)}$ is the $j-i$-th order statistics in
a sample of $j-1$ independent observations on i.i.d. r.vs. with $X_{LT}$
c.d.f. $G_{x}(y)=\frac{F(y)-F(x)}{1-F(x)}$, $y>x$, $\tilde{\theta}_{(i,j-1)}$
is the $i$-th order statistics in a sample of $j-1$ independent observations
on i.i.d. r.vs. with $X_{RT}$ c.d.f. $T_{x}(y)=\frac{F(y)}{F(x)}$, $y<x$,
$\xi^{*}\in Beta(j-i,i)$, and $\xi^{**}\in Beta(i,j-i)$.

  6. vi)

Assume $\xi$ and $\xi^{*}$ are independent, and $\xi^{*}\in Beta(j-i,i)$.
Denote by $\xi^{**}=1-\xi^{*}$, and $\xi^{***}=1-\xi$.333Then
$1-\xi^{*}=:\xi^{**}\in Beta(i,j-i)$ and $1-\xi=:\xi^{***}\in Beta(j,i)$. Then
for $n=i+j-1$, $i,j\in N$, $1\leq i<j$,

    1. (a)

$(X_{(j,i+j-1)},X_{(i,i+j-1)})$

$\displaystyle\stackrel{{\scriptstyle d}}{{=}}$
$\displaystyle\\{F^{\leftarrow}(1-\xi^{**}\xi^{***}),F^{\leftarrow}(1-\xi^{***})\\}$
$\displaystyle\stackrel{{\scriptstyle d}}{{=}}$
$\displaystyle\left\\{(1-F)^{\leftarrow}(\xi^{**}\xi^{***}),(1-F)^{\leftarrow}(\xi^{***})\right\\}$
$\displaystyle\stackrel{{\scriptstyle d}}{{=}}$
$\displaystyle\left\\{\left(\frac{1}{1-F}\right)^{\leftarrow}\left(\frac{1}{\xi^{**}\xi^{***}}\right),\left(\frac{1}{1-F}\right)^{\leftarrow}\left(\frac{1}{\xi^{***}}\right)\right\\}$

Moreover $\xi^{**}\xi^{***}\stackrel{{\scriptstyle d}}{{=}}\xi$.

    2. (b)

The empirical $\frac{i}{i+j}$-right fences

$\displaystyle R_{n}(X,\frac{i}{i+j})$ $\displaystyle=$
$\displaystyle\left(1+\frac{j}{i}\right)X_{j,i+j-1}-\frac{j}{i}X_{i,i+j-1}$
$\displaystyle\stackrel{{\scriptstyle d}}{{=}}$
$\displaystyle\left(1+\frac{j}{i}\right)(1-F)^{\leftarrow}(\xi^{**}\xi^{***})-\frac{j}{i}(1-F)^{\leftarrow}(\xi^{***}).$

    3. (c)

The empirical $\frac{i}{i+j}$-left fences

$\displaystyle L_{n}(X,\frac{i}{i+j})$ $\displaystyle=$
$\displaystyle\left(1+\frac{j}{i}\right)X_{i,i+j-1}-\frac{j}{i}X_{j,i+j-1}$
$\displaystyle\stackrel{{\scriptstyle d}}{{=}}$
$\displaystyle\left(1+\frac{j}{i}\right)(1-F)^{\leftarrow}(\xi^{***})-\frac{j}{i}(1-F)^{\leftarrow}(\xi^{**}\xi^{***}).$

  7. vii)

For $k=0,1,...,i+j$,

$P(n_{R}(\frac{i}{i+j},i+j-1)=k)$

$\displaystyle=$ $\displaystyle\frac{(i+j-1)!}{k!(i+j-k-1)!}$ .
$\displaystyle\int_{0}^{1}\int_{0}^{1}\left\\{\frac{1-F\left[\left(1+\frac{j}{i}\right)(1-F)^{\leftarrow}(xy)-\frac{j}{i}(1-F)^{\leftarrow}(x)\right]}{F\left[\left(1+\frac{j}{i}\right)(1-F)^{\leftarrow}(xy)-\frac{j}{i}(1-F)^{\leftarrow}(x)\right]}\right\\}^{k}$
.
$\displaystyle\frac{x^{j-1}(1-x)^{i-1}}{B(i,j)}\frac{y^{i-1}(1-y)^{j-i-1}}{B(j-i,i)}$
.
$\displaystyle\left\\{F\left[\left(1+\frac{j}{i}\right)(1-F)^{\leftarrow}(xy)-\frac{j}{i}(1-F)^{\leftarrow}(x)\right]\right\\}^{i+j-1}dydx$

$P(n_{L}(\frac{i}{i+j},i+j-1)=k)$

$\displaystyle=$ $\displaystyle\frac{(i+j-1)!}{k!(i+j-k-1)!}$ .
$\displaystyle\int_{0}^{1}\int_{0}^{1}\left\\{\frac{F\left[\left(1+\frac{j}{i}\right)(1-F)^{\leftarrow}(x)-\frac{j}{i}(1-F)^{\leftarrow}(xy)\right]}{1-F\left[\left(1+\frac{j}{i}\right)(1-F)^{\leftarrow}(x)-\frac{j}{i}(1-F)^{\leftarrow}(xy)\right]}\right\\}^{k}$
.
$\displaystyle\frac{x^{j-1}(1-x)^{i-1}}{B(i,j)}\frac{y^{i-1}(1-y)^{j-i-1}}{B(j-i,i)}$
.
$\displaystyle\left\\{1-F\left[\left(1+\frac{j}{i}\right)(1-F)^{\leftarrow}(x)-\frac{j}{i}(1-F)^{\leftarrow}(xy)\right]\right\\}^{i+j-1}dydx$

###### Remark 2

As an additional result, we can use the above theorem to obtain different
univariate and bivariate distributions and new relations between them. This
approach is analogous to the one applied by Eugene et al. (2002) [21] or
Cordeiro et al. (2012) [1] among others, who consider Generalized-Beta
generalized distributions.

###### Remark 3

Using the general formula for the moments of order statistics, for
$m=1,2,...,n$, and $r\in R$

$E[X_{(m,n)}^{r}]=\frac{n!}{(m-1)!(n-m)!}\int_{-\infty}^{\infty}x^{r}[F(x)]^{m-1}[1-F(x)]^{n-m}dF(x),$

which could be seen e.g. in the books of Arnold et al. (1992)[2] or Nevzorov
(2001) [42], we can easily obtain the general formulae for the mean and the
variance of $L_{n}(X,\frac{i}{i+j})$ and $R_{n}(X,\frac{i}{i+j})$ in cases
when they exist. For example in Pareto (1) case, for $i=k$, $j=sk$,
$s=2,3,...,$ $L_{k(s+1)-1}(X,\frac{1}{1+s})$ and
$R_{k(s+1)-1}(X,\frac{1}{1+s})$, are asymptotically unbiased estimators
correspondingly for $L(F,\frac{1}{s+1})$ and $R(F,\frac{1}{s+1})$. More
precisely

$\lim_{k\to\infty}EL_{k(s+1)-1}(X,\frac{1}{s+1})=L(F,\frac{1}{s+1}),\quad and$
$\lim_{k\to\infty}ER_{k(s+1)-1}(X,\frac{1}{s+1})=R(F,\frac{1}{s+1}).$

The following result is an immediate corollary of the definition of
convergence in probability, quantile transform, a.s. convergence of empirical
quantiles to the corresponding theoretical one, and Slutsky’s theorem about
continuous functions. See e.g. Embrechts et al. (1997) [20].

###### Theorem 4

Given a sample of independent observations, for any fixed $s=2,3,...$

$\displaystyle L_{(s+1)k-1}\left(F,\frac{1}{s+1}\right)$
$\displaystyle\stackrel{{\scriptstyle P}}{{\to}}$ $\displaystyle
L(F,\frac{1}{s+1}),\quad k\to\infty,$ (35) $\displaystyle
R_{(s+1)k-1}\left(F,\frac{1}{s+1}\right)$
$\displaystyle\stackrel{{\scriptstyle P}}{{\to}}$ $\displaystyle
R(F,\frac{1}{s+1}),\quad k\to\infty.$ (36)

Cadwell (1953) [9] finds the distribution of quasi-ranges in samples from a
normal population. He gives us the idea about the next result. Rider (1959)
[50] obtains their exact distribution in case of samples from an exponential
population. Sarhan et al. (1963) [53] propose simplified estimates in this
case. The asymptotic normality of the appropriately normalized univariate
distributions of the central order statistics is investigated in Smirnov
(1949) [54]. Note that in the next theorem, because of the special choice of
the numbers of order statistics, $p$ and $n$ his conditions $\frac{k}{n}\to
p\in(0,1)$ and $\sqrt{n}(\frac{k}{n}-p)\to\mu\in(-\infty,\infty)$ are
satisfied. Moreover, in our case $\mu=0$. The theorem about the joint
distribution of the central order statistics of i.i.d. observations, could be
seen e.g. in Nair (2013) [40], p.330, or Arnold et al. (1992) [2], p. 226,
among others. The multivariate delta method is a very powerful technique for
obtaining confidence intervals in such cases. It can be seen e.g. in Sobel
(1982) [55]. In the next theorem we use these results and obtain the limiting
distribution of the fences of central order statistics.

###### Theorem 5

Consider a sample of $n=(s+1)k-1$, $s=1,2,3,...$ observations on a r.v. $X$
with c.d.f. $F$ and p.d.f. $f=F^{\prime}$. Suppose that there exists
$c_{F,s}:=f\left[F^{\leftarrow}(\frac{1}{s+1})\right]\in(0,\infty)$ and
$d_{F,s}=f\left[F^{\leftarrow}(\frac{s}{s+1})\right]\in(0,\infty)$. Then

$\lim_{k\to\infty}EL_{n}\left(F,\frac{1}{s+1}\right)=L(F,\frac{1}{s+1})$
$\lim_{k\to\infty}ER_{n}\left(F,\frac{1}{s+1}\right)=R(F,\frac{1}{s+1})$

and for $k\to\infty$

$\sqrt{(s+1)k-1}\left|L_{n}\left(F,\frac{1}{s+1}\right)-L(F,\frac{1}{s+1})\right|\stackrel{{\scriptstyle
d}}{{\to}}N\left(0;V_{L,F,s}\right),$ (37)
$\sqrt{(s+1)k-1}\left|R_{n}\left(F,\frac{1}{s+1}\right)-R(F,\frac{1}{s+1})\right|\stackrel{{\scriptstyle
d}}{{\to}}N\left(0;V_{R,F,s}\right),$ (38)

where
$V_{L,F,s}=\frac{s}{(s+1)^{2}}\left[\frac{(s+1)^{2}}{c_{F,s}^{2}}-\frac{2(s+1)}{c_{F,s}d_{F,s}}+\frac{s^{2}}{d_{F,s}^{2}}\right]$
and
$V_{R,F,s}=\frac{s}{(s+1)^{2}}\left[\frac{s^{2}}{c_{F,s}^{2}}-\frac{2(s+1)}{c_{F,s}d_{F,s}}+\frac{(s+1)^{2}}{d_{F,s}^{2}}\right].$

These allows as to compute the asymptotic confidence intervals of these
estimators for $k\to\infty$, $n=(s+1)k-1$, and $s=1,2,3,...$, fixed. Denote
$L_{F,s,k}=L_{(s+1)k-1}\left(F,\frac{1}{s+1}\right)$ and
$R_{F,s,k}=R_{(s+1)k-1}\left(F,\frac{1}{s+1}\right)$. If the conditions of
Theorem 5 are satisfied, then given $\alpha\in(0,1)$,

$L_{F,s,k}-z_{\alpha}\sqrt{\frac{V_{L,F,s}}{(s+1)k-1}}\leq
L(F,\frac{1}{s+1})\leq L_{F,s,k}+z_{\alpha}\sqrt{\frac{V_{L,F,s}}{(s+1)k-1}},$
(39) $R_{F,s,k}-z_{\alpha}\sqrt{\frac{V_{R,F,s}}{(s+1)k-1}}\leq
R(F,\frac{1}{s+1})\leq R_{F,s,k}+z_{\alpha}\sqrt{\frac{V_{R,F,s}}{(s+1)k-1}}.$
(40)

The next theorem explains why different probabilities for outside values can
be useful for estimating the tail behaviour of the observed distribution. For
a fixed $s=2,3,...$ and $k\to\infty$, we apply the approach of Dembinska
(2012) [17], for the bivariate case, and obtain that
$\frac{n_{L}\left(\frac{1}{s+1};(s+1)k-1\right)}{(s+1)k-1}$ and
$\frac{n_{R}\left(\frac{1}{s+1};(s+1)k-1\right)}{(s+1)k-1}$ are strongly
consistent estimators correspondingly of $p_{L,\frac{1}{s+1}}(X)$ and
$p_{R,\frac{1}{s+1}}(X)$. Moreover Dembinska (2017) [16] shows that this
approach works not only for i.i.d., but also also for strictly stationary and
ergodic sequences.

###### Theorem 6

Let $s=2,3,...$ be fixed. Assume
$f[F^{\leftarrow}(\frac{1}{s+1})]\in(0,\infty)$ and
$f[F^{\leftarrow}(\frac{s}{s+1})]\in(0,\infty)$.

  1. 1.

If $f[F^{\leftarrow}(\frac{1}{s+1})]\in(0,\infty)$, then

$\frac{n_{L}\left(\frac{1}{s+1};(s+1)k-1\right)}{(s+1)k-1}{\mathop{\to}\limits_{k\to\infty}^{a.s.}}p_{L,\frac{1}{s+1}}(X).$

  2. 2.

If $f[F^{\leftarrow}(\frac{1}{s+1})]\in(0,\infty)$, then

$\frac{n_{R}\left(\frac{1}{s+1};(s+1)k-1\right)}{(s+1)k-1}{\mathop{\to}\limits_{k\to\infty}^{a.s.}}p_{R,\frac{1}{s+1}}(X).$

##  5 Simulation study

In this section we assume that
$\mathbf{X}_{1},\mathbf{X}_{2},...,\mathbf{X}_{n}$ are independent
realizations of $\mathbf{X}$, with c.d.f. $F$. Jordanova and Petkova
(2017-2018) [35, 36] assume that $F$ has regularly varying tail. More
precisely they consider only the cases when there exists $\alpha>0$, such that
for all $x>0$,

$\lim_{t\to\infty}\frac{1-F(xt)}{1-F(t)}=x^{-\alpha}.$ (41)

The number $-\alpha$ is called ”index of regular variation of the tail of
c.d.f.”, see e.g. de Haan and Ferreira (2006) [13] or Resnick (1987) [51].
Using the explicit form of the corresponding probabilities for extreme
outliers according the definition in Devore(2015) [18],($0.25$-outliers)
Jordanova and Petkova (2017-2018) [35, 36] obtain distribution sensitive
estimators of the unknown parameter $\alpha$ which governs the tail of the
considered distributional type. The algorithm consists of the following three
main steps.

  1. 1.

Using the results from the previous two sections the explorer chooses the most
appropriate probability type (let us call it $T$) for modeling the tail of the
distribution of the observed r.v.

  2. 2.

Using the formula for $p_{R,0.25}$ in case $T$ one expresses the unknown
parameter $\alpha$.

  3. 3.

Replace the theoretical characteristics in the previous step with the
corresponding estimators and obtain a new estimator for the parameter which
governs the tail behavior.

In their work Jordanova and Petkova (2017-2018) [35, 36] compare the obtained
in this way estimators in Pareto, Fr$\acute{e}$chet, and Hill-Horror case with
Hill, t-Hill, Pickands, and Deckers-Einmahl-de Haan estimators and depict the
results via a simulation study. Here we consider two more cases: Log-Logistic
case (25) and $H_{1}$ case (30). Although in the last case the right tail of
the c.d.f. is not regularly varying the next study shows that the approach
still gives very good results.

In any of the following five examples using the functions implemented in R
(2018), [48] we have simulated $m=1000$ samples of $n$ independent
observations separately on $X$. Then for any fixed $n=10,11,...,500$ and for
any fixed sample we have computed the estimators
$\hat{p}_{R}(0.25,n)=\frac{n_{R}(0.25,n)}{n},$ $F_{n}^{\leftarrow}(0.25)$,
$F_{n}^{\leftarrow}(0.75)$, and $\hat{\alpha}_{\bullet,n}$. Here
$n_{R}(0.25,n)$ is the numbers of right extreme outside values in the
considered sample of $n$ independent observations, and $\bullet$ means one of
the abbreviations $Par$, $Fr$, $HH$, $H_{1}$ or $LL$ explained below. Finally
we have fixed one of the last estimators and we have averaged the
corresponding values of $\hat{\alpha}_{\bullet,n}$ over the considered $n$.
The next Figures 6-14 depict the dependence of these values, together with the
corresponding asymptotic normal 95% confidence intervals, on the real type of
the observed r.v., and on the sample size, for $\alpha=0.5$, or $1$. We have
chosen only the cases when $\alpha$ is small because our observations show
that for a fixed sample size the more the outside values, the heavier of the
tail of the c.d.f. is and the better the corresponding estimator is.

Let us depict this approach with some examples. In any of them we suppose that
$F_{n}^{\leftarrow}\left(\frac{3}{4}\right)>1$ and $\hat{p}_{R}(0.25,n)>0$.

###### Example 1

Assime $\mathbf{X}\in Par(\alpha,\delta)$. See (1). Having a sample of $n$
independent observations on $\mathbf{X}$, analogously to the generalized
method of moments, and following the above algorithm Jordanova and Petkova
(2018) [36] obtain

$\hat{\alpha}_{Par,n}=-\frac{log\,\,\hat{p}_{R}(0.25,n)}{log\left\\{F_{n}^{\leftarrow}\left(\frac{3}{4}\right)+3\left[F_{n}^{\leftarrow}\left(\frac{3}{4}\right)-F_{n}^{\leftarrow}\left(\frac{1}{4}\right)\right]\right\\}}.$
(42)

###### Example 2

If $\mathbf{X}\in Fr(\alpha,c)$, see (2), Jordanova and Petkova (2018) [36]
propose

$\hat{\alpha}_{Fr,n}=-\frac{log\\{-log[1-\hat{p}_{R}(0.25,n)]\\}}{log\left\\{F_{n}^{\leftarrow}\left(\frac{3}{4}\right)+3\left[F_{n}^{\leftarrow}\left(\frac{3}{4}\right)-F_{n}^{\leftarrow}\left(\frac{1}{4}\right)\right]\right\\}}.$
(43)

###### Example 3

Let $\mathbf{X}$ be Hill-Horror distributed. This distribution is usually
defined via its quantile function (32). Given
$F_{n}^{\leftarrow}\left(\frac{3}{4}\right)+3\left[F_{n}^{\leftarrow}\left(\frac{3}{4}\right)-F_{n}^{\leftarrow}\left(\frac{3}{4}\right)\right]\not=-log\,\hat{p}_{R}(0.25,n)$
Jordanova and Petkova (2018) [36] use

$\hat{\alpha}_{HH,n}=\frac{log\,\hat{p}_{R}(0.25,n)}{log\,\left\\{\frac{-log\,\hat{p}_{R}(0.25,n)}{F_{n}^{\leftarrow}\left(\frac{3}{4}\right)+3\left[F_{n}^{\leftarrow}\left(\frac{3}{4}\right)-F_{n}^{\leftarrow}\left(\frac{1}{4}\right)\right]}\right\\}}.$
(44)

The next two estimators seems to be new. They show that this approach can be
applied in much wider than the regularly varying case.

###### Example 4

Let $\mathbf{X}\in H_{1}$, see (30). It is difficult to solve (31) with
respect to $\alpha$, therefore we solve the equation

$p_{R,0.25}(X)=1-exp\\{-[log\,R(X,0.25)]^{-\alpha}\\}.$

When express $\alpha$ and replace the theoretical characteristics with the
corresponding empirical one we obtain the estimator

$\hat{\alpha}_{H_{1},n}=-\frac{log\\{-log[1-\hat{p}_{R}(0.25,n)]\\}}{log\left\\{log\left\\{F_{n}^{\leftarrow}\left(\frac{3}{4}\right)+3\left[F_{n}^{\leftarrow}\left(\frac{3}{4}\right)-F_{n}^{\leftarrow}\left(\frac{1}{4}\right)\right]\right\\}\right\\}}.$
(45)

###### Example 5

Suppose $\mathbf{X}$ follow Log-logistic probability law (25). The equation
(26) have no explicit solution for $\alpha$, therefore we solve the equation

$p_{R,0.25}(X)=\frac{1}{1+R(X,0.25)^{\alpha}}.$

Then we replace the theoretical characteristics with the corresponding
empirical one we obtain the estimator

$\hat{\alpha}_{LL,n}=\frac{log\left[\frac{1}{\hat{p}_{R}(0.25,n)}-1\right]}{log\left\\{F_{n}^{\leftarrow}\left(\frac{3}{4}\right)+3\left[F_{n}^{\leftarrow}\left(\frac{3}{4}\right)-F_{n}^{\leftarrow}\left(\frac{1}{4}\right)\right]\right\\}}.$
(46)

Figures 6-14 depict the dependence of these estimators, together with their
empirical 95% confidence intervals on the sample size, probability law of the
simulated r.v., and the estimated parameter $\alpha$. The names of the
estimators in these figures are abbreviated as follows:
$\hat{\alpha}_{Par,n}=aParn$, $\hat{\alpha}_{Fr,n}=aFrn$,
$\hat{\alpha}_{HH,n}=aHHn$, $\hat{\alpha}_{LL,n}=aLLn$, and
$\hat{\alpha}_{H_{1},n}=aH1n$.

Figure 5: Pareto case, $\alpha=0.5$: Dependence of $\hat{\alpha}_{Par,n}$,
$\hat{\alpha}_{Fr,n}$, $\hat{\alpha}_{HH,n}$, $\hat{\alpha}_{H_{1},n}$,
$\hat{\alpha}_{LL,n}$ on the sample size

Figure 6: Pareto case, $\alpha=1$: Dependence of $\hat{\alpha}_{Par,n}$,
$\hat{\alpha}_{Fr,n}$, $\hat{\alpha}_{HH,n}$, $\hat{\alpha}_{H_{1},n}$,
$\hat{\alpha}_{LL,n}$ on the sample size

Figure 7: Fr$\acute{e}$chet case, $\alpha=0.5$: Dependence of
$\hat{\alpha}_{Par,n}$, $\hat{\alpha}_{Fr,n}$, $\hat{\alpha}_{HH,n}$,
$\hat{\alpha}_{H_{1},n}$, $\hat{\alpha}_{LL,n}$ on the sample size

Figure 8: Fr$\acute{e}$chet case, $\alpha=1$: Dependence of
$\hat{\alpha}_{Par,n}$, $\hat{\alpha}_{Fr,n}$, $\hat{\alpha}_{HH,n}$,
$\hat{\alpha}_{H_{1},n}$, $\hat{\alpha}_{LL,n}$ on the sample size

Figure 9: Hill-Horror case, $\alpha=0.5$: Dependence of
$\hat{\alpha}_{Par,n}$, $\hat{\alpha}_{Fr,n}$, $\hat{\alpha}_{HH,n}$,
$\hat{\alpha}_{H_{1},n}$, $\hat{\alpha}_{LL,n}$ on the sample size

Figure 10: Hill-Horror case, $\alpha=1$: Dependence of $\hat{\alpha}_{Par,n}$,
$\hat{\alpha}_{Fr,n}$, $\hat{\alpha}_{HH,n}$, $\hat{\alpha}_{H_{1},n}$,
$\hat{\alpha}_{LL,n}$ on the sample size

Figure 11: $H_{1}$ case, $\alpha=0.5$: Dependence of $\hat{\alpha}_{Par,n}$,
$\hat{\alpha}_{Fr,n}$, $\hat{\alpha}_{HH,n}$, $\hat{\alpha}_{H_{1},n}$,
$\hat{\alpha}_{LL,n}$ on the sample size

Figure 12: $H_{1}$ case, $\alpha=1$: Dependence of $\hat{\alpha}_{Par,n}$,
$\hat{\alpha}_{Fr,n}$, $\hat{\alpha}_{HH,n}$, $\hat{\alpha}_{H_{1},n}$,
$\hat{\alpha}_{LL,n}$ on the sample size

Figure 13: Log-Logistic case, $\alpha=0.5$: Dependence of
$\hat{\alpha}_{Par,n}$, $\hat{\alpha}_{Fr,n}$, $\hat{\alpha}_{HH,n}$,
$\hat{\alpha}_{H_{1},n}$, $\hat{\alpha}_{LL,n}$ on the sample size

Figure 14: Log-Logistic case, $\alpha=1$: Dependence of
$\hat{\alpha}_{Par,n}$, $\hat{\alpha}_{Fr,n}$, $\hat{\alpha}_{HH,n}$,
$\hat{\alpha}_{H_{1},n}$, $\hat{\alpha}_{LL,n}$ on the sample size

The above simulation study shows that within the considered set of
distributions given a small sample of observations the considered estimators
outperform the properties of the well-known estimators proposed by Hill (1975)
[30], Pickands (1975)[47] and Deckers-Einmahl-de Haan Dekkers (1989) [15].
Within the right probability type, the rate of convergence of any of them
increases when the sample size increases and $\alpha>0$ decreases. However,
according to our investigation, these estimators are too distribution
sensitive. The biggest their advantage is that they are applicable for
relatively small samples.

##  6 Conclusive remarks

To the best knowledge of the author, a universal numerical characteristic of
the tail of the c.d.f., which is invariant within distributional type (with
respect to increasing affine transformation) is still not known. Here we show
that probabilities of the events an observation to be $p$-outside value can be
very useful in this sense. They can be used for making a reasonable
classification of the tails of probability distributions. They outperform the
role e.g. of the excess in characterizing the tail of the observed
distribution because they do not depend on the moments of the observed r.v.
and could be applied also in cases when moments do not exist. Their estimators
are appropriate for usage in preliminary statistical analysis in presence of
corresponding outside values. They can help the practitioners to find the most
appropriate classes of probability laws for modeling the tails of the
distribution of the observed r.v. Within that family the parameter which
influences the tails needs further estimation. According to our simulation
study, the proposed algorithm for making estimators gives better results when
$\alpha>0$ decreases. The fast rate of convergence allows one to apply these
estimators also for relatively small samples. However, the main disadvantage
of all these estimators is that they are distribution sensitive. The last
means that their good properties may disappear if the distributional type is
not correctly determined.

##  7 Acknowledgements

The author would like to thank Prof. Milan Stehlik for bringing her to the
question about statistical modeling of extremes given small samples.

##  8 Apendix

Proof of Theorem 1: a) By definition of $L(X,p)$ and formula for derivative of
the inverse function $\frac{\partial F^{\leftarrow}(p)}{\partial
p}=\frac{1}{f(F^{\leftarrow}(p))}$ we obtain

$\frac{\partial L(X,p)}{\partial
p}=\frac{1}{pf[F^{\leftarrow}(p)]}+\frac{F^{\leftarrow}(1-p)-F^{\leftarrow}(p)}{p^{2}}+\frac{1-p}{pf[F^{\leftarrow}(1-p)]}.$

$f(x)$ is a density function of the r.v. $X$, therefore it is non-negative.
The difference $F^{\leftarrow}(1-p)-F^{\leftarrow}(p)\geq 0$ because
$p\in(0;0.5]$. Therefore $\frac{\partial L(X,p)}{\partial p}\geq 0$.

b) By definition of $R(X,p)$ and the same formula for derivative of the
inverse function we have that

$\frac{\partial R(X,p)}{\partial
p}=\frac{F^{\leftarrow}(p)-F^{\leftarrow}(1-p)}{p^{2}}-\frac{1}{pf[F^{\leftarrow}(p)]}-\frac{1-p}{pf[F^{\leftarrow}(1-p)]}.$

Now the difference $F^{\leftarrow}(1-p)-F^{\leftarrow}(p)\leq 0$ because
$p\in(0;0.5]$. Therefore $\frac{\partial R(X,p)}{\partial p}\leq 0$.

c) follows by a), b), and monotonicity of probability measures. Q.A.D.

Proof of Theorem 2: b) For $c\in R$ from the definition of the quantile
function we have that $F_{X+c}^{\leftarrow}(p)=F_{X}^{\leftarrow}(p)+c$.
Therefore

$\displaystyle p_{R,p}(X+c)$ $\displaystyle=$ $\displaystyle
P(X+c>\frac{1}{p}F_{X+c}^{\leftarrow}(1-p)-\frac{1-p}{p}F_{X+c}^{\leftarrow}(p))$
$\displaystyle=$ $\displaystyle
P(X+c>\frac{1}{p}F_{X}^{\leftarrow}(1-p)+\frac{c}{p}-\frac{1-p}{p}F_{X}^{\leftarrow}(p)-\frac{c}{p}(1-p))$
$\displaystyle=$ $\displaystyle p_{R,p}(X).$

c) For $c>0$ again from the definition of the quantile function
$F_{cX}^{\leftarrow}(p)=cF_{X}^{\leftarrow}(p)$. Therefore

$\displaystyle p_{R,p}(cX)$ $\displaystyle=$ $\displaystyle
P(cX>\frac{1}{p}F_{cX}^{\leftarrow}(1-p)-\frac{1-p}{p}F_{cX}^{\leftarrow}(p))$
$\displaystyle=$ $\displaystyle
P(cX>\frac{c}{p}F_{X}^{\leftarrow}(1-p)-c\frac{1-p}{p}F_{X}^{\leftarrow}(p))=p_{R,p}(X).$

d) In this case $c<0$, therefore $F_{cX}^{\leftarrow}(p)=cF^{\leftarrow}(1-p)$
and

$\displaystyle p_{R,p}(cX)$ $\displaystyle=$ $\displaystyle
P(cX>\frac{1}{p}F_{cX}^{\leftarrow}(1-p)-\frac{1-p}{p}F_{cX}^{\leftarrow}(p))$
$\displaystyle=$ $\displaystyle
P(cX>\frac{c}{p}F_{X}^{\leftarrow}(p)-c\frac{1-p}{p}F_{X}^{\leftarrow}(1-p))$
$\displaystyle=$ $\displaystyle
P(X<\frac{1}{p}F_{X}^{\leftarrow}(p)-\frac{1-p}{p}F_{X}^{\leftarrow}(1-p))=p_{L,p}(X).$

e) Because of $g$ is a strictly increasing and continuous function we have
that $F_{g(X)}^{\leftarrow}(p)=g(F^{\leftarrow}(p))$ and
$F_{g(X)}^{\leftarrow}(1-p)=g(F^{\leftarrow}(1-p))$. Therefore

$\displaystyle p_{R,p}(g(X))$ $\displaystyle=$ $\displaystyle
P\left[g(X)>\frac{1}{p}F_{g(X)}^{\leftarrow}(1-p)-\frac{1-p}{p}F_{g(X)}^{\leftarrow}(p)\right]$
$\displaystyle=$ $\displaystyle
P\left\\{g(X)>\frac{1}{p}g[F^{\leftarrow}(1-p)]-\frac{1-p}{p}g[F^{\leftarrow}(p)]\right\\}$
$\displaystyle\leq$ $\displaystyle
P\left\\{g(X)>g[\frac{1}{p}F^{\leftarrow}(1-p)-\frac{1-p}{p}F^{\leftarrow}(p)]\right\\}$
$\displaystyle=$ $\displaystyle
P\left[X>\frac{1}{p}F^{\leftarrow}(1-p)-\frac{1-p}{p}F^{\leftarrow}(p)\right]=p_{R,p}(X)$

These, together with the definition of $p_{R,p}(X)$ and monotonicity of
probability measures entail (6).

(7) is a corollary of (10), applied for $\tilde{g}(x)=-g(x)$.

f) The equalities $F^{\leftarrow}_{g(X)}(1-p)=g(F^{\leftarrow}_{X}(p))$ and
$F^{\leftarrow}_{g(X)}(p)=g(F^{\leftarrow}_{X}(1-p))$ entail

$\displaystyle p_{R,p}(g(X))$ $\displaystyle=$ $\displaystyle
P\left[g(X)>\frac{1}{p}F_{g(X)}^{\leftarrow}(1-p)-\frac{1-p}{p}F_{g(X)}^{\leftarrow}(p)\right]$
$\displaystyle=$ $\displaystyle
P\left\\{g(X)>\frac{1}{p}g[F^{\leftarrow}(p)]-\frac{1-p}{p}g[F^{\leftarrow}(1-p)]\right\\}$
$\displaystyle\geq$ $\displaystyle
P\left\\{g(X)>g[\frac{1}{p}F^{\leftarrow}(p)-\frac{1-p}{p}F^{\leftarrow}(1-p)]\right\\}$
$\displaystyle=$ $\displaystyle
P\left[X<\frac{1}{p}F^{\leftarrow}(p)-\frac{1-p}{p}F^{\leftarrow}(1-p)\right]=p_{L,p}(X).$

Now the definitions of $p_{R,p}(X)$, and $p_{L,p}(X)$, and the monotonicity of
probability measures entail (10).

(11) follows by (6), when replace the function $g(x)$ with
$\tilde{g}(x)=-g(x)$ and take into account that
$p_{R,p}(-g(X))=p_{L,p}(g(X))$.

g) As far as $F^{\leftarrow}_{X_{(n,n)}}(p)=F^{\leftarrow}_{X}(\sqrt[n]{p})$

$\displaystyle p_{R,p}(X_{(n,n)})$ $\displaystyle=$ $\displaystyle
P\left\\{X>\frac{1}{p}F^{\leftarrow}_{X}(\sqrt[n]{1-p})-\frac{1-p}{p}F^{\leftarrow}_{X}(\sqrt[n]{p})\right\\}$
$\displaystyle=$ $\displaystyle
1-F_{X}^{n}\left[\frac{1}{p}F^{\leftarrow}_{X}(\sqrt[n]{1-p})-\frac{1-p}{p}F^{\leftarrow}_{X}(\sqrt[n]{p})\right]$
$\displaystyle p_{L,p}(X_{(n,n)})$ $\displaystyle=$ $\displaystyle
P\left\\{X<\frac{1}{p}F^{\leftarrow}_{X}(\sqrt[n]{p})-\frac{1-p}{p}F^{\leftarrow}_{X}(\sqrt[n]{1-p})\right\\}$
$\displaystyle=$ $\displaystyle
F_{X}^{n}\left[\frac{1}{p}F^{\leftarrow}_{X}(\sqrt[n]{p})-\frac{1-p}{p}F^{\leftarrow}_{X}(\sqrt[n]{1-p})\right]$

h) Using $F^{\leftarrow}_{X_{(1,n)}}(p)=F^{\leftarrow}_{X}(1-\sqrt[n]{1-p})$
we obtain

$\displaystyle p_{R,p}(X_{(1,n)})$ $\displaystyle=$ $\displaystyle
P\left\\{X>\frac{1}{p}F^{\leftarrow}_{X}(1-\sqrt[n]{p})-\frac{1-p}{p}F^{\leftarrow}_{X}(1-\sqrt[n]{1-p})\right\\}$
$\displaystyle=$
$\displaystyle\left\\{1-F_{X}\left[\frac{1}{p}F^{\leftarrow}_{X}(1-\sqrt[n]{p})-\frac{1-p}{p}F^{\leftarrow}_{X}(1-\sqrt[n]{1-p})\right]\right\\}^{n}$
$\displaystyle p_{L,p}(X_{(1,n)})$ $\displaystyle=$ $\displaystyle
P\left\\{X<\frac{1}{p}F^{\leftarrow}_{X}(1-\sqrt[n]{1-p})-\frac{1-p}{p}F^{\leftarrow}_{X}(1-\sqrt[n]{p})\right\\}$
$\displaystyle=$ $\displaystyle
1-\left\\{1-F_{X}\left[\frac{1}{p}F^{\leftarrow}_{X}(1-\sqrt[n]{1-p})-\frac{1-p}{p}F^{\leftarrow}_{X}(1-\sqrt[n]{p})\right]\right\\}^{n}.$

i) Consider $t>0$. The relation between the quantile function of the
exceedances and the c.d.f. of $F$ (see e.g. in Nair et al. (2013) [40])
entails

$\displaystyle p_{R,p}(X-t|X>t)$ $\displaystyle=$ $\displaystyle
P\left\\{X-t>\frac{1}{p}F^{\leftarrow}_{X-t|X>t}(1-p)-\frac{1-p}{p}F^{\leftarrow}_{X-t|X>t}(p)|X>t\right\\}$
$\displaystyle=$ $\displaystyle
P\left\\{X-t>\frac{1}{p}F^{\leftarrow}_{X}[1-p+pF_{X}(p)]-\frac{t}{p}\right.$
$\displaystyle-$
$\displaystyle\frac{1-p}{p}F_{X}^{\leftarrow}[p+(1-p)F_{X}(t)]+\left.\frac{t(1-p)}{p}|X>t\right\\}$
$\displaystyle=$
$\displaystyle\frac{P\left\\{X>\frac{1}{p}F^{\leftarrow}_{X}[1-p+pF_{X}(p)]-\frac{1-p}{p}F^{\leftarrow}_{X}[p+(1-p)F_{X}(t)]\right\\}}{P(X>t)}$
$\displaystyle=$
$\displaystyle\frac{P\left\\{X>\frac{1}{p}F^{\leftarrow}_{X|X>t}(1-p)-\frac{1-p}{p}F^{\leftarrow}_{X|X>t}(p)\right\\}}{P(X>t)}$
$\displaystyle=$ $\displaystyle
P\left\\{X>\frac{1}{p}F^{\leftarrow}_{X|X>t}(1-p)-\frac{1-p}{p}F^{\leftarrow}_{X|X>t}(p)|X>t\right\\}=p_{R,p}(X|X>t).$

Analogously for $p_{L,p}(X-t|X>t)$. 444This property can be obtained also as a
corollary of b).

k) We obtain this property when replace the relations between the quantile
functions of left-, right-, and double-truncated r.vs., $F_{X}^{\leftarrow}$
and $F_{X}$, i.e.

$\displaystyle F_{X_{LT}}^{\leftarrow}(p)$ $\displaystyle=$ $\displaystyle
F_{X}^{\leftarrow}(p+(1-p)F_{X}(l))$ $\displaystyle
F_{X_{RT}}^{\leftarrow}(p)$ $\displaystyle=$ $\displaystyle
F_{X}^{\leftarrow}(pF_{X}(u))$ $\displaystyle F_{X_{DT}}^{\leftarrow}(p)$
$\displaystyle=$ $\displaystyle F_{X}^{\leftarrow}(pF_{X}(u)+(1-p)F_{X}(l))$

in the definitions of $L(X,p)$ and $R(X,p)$. The above equalities is not
difficult to calculate, and could be found e.g. in Nair et al. (2013) [40].

Finally we use the definitions of $p_{L,p}(X)$ and $p_{R,p}(X)$.

l) Assume $p\in(0,0.5]$.

\- Case $a>1$. In this case
$F_{log_{a}(X)}^{\leftarrow}(p)=log_{a}[F^{\leftarrow}(p)]$, and the function
$a^{x}$ is increasing in $x$, therefore

$\displaystyle p_{L,p}[log_{a}(X)]$ $\displaystyle=$ $\displaystyle
P\left[log_{a}(X)<\frac{1}{p}F_{log_{a}(X)}^{\leftarrow}(p)-\frac{1-p}{p}F_{log_{a}(X)}^{\leftarrow}(1-p)\right]$
$\displaystyle=$ $\displaystyle
P\left\\{log_{a}(X)<\frac{1}{p}log_{a}[F^{\leftarrow}(p)]-\frac{1-p}{p}log_{a}[F^{\leftarrow}(1-p)]\right\\}$
$\displaystyle=$ $\displaystyle
F\left\\{\sqrt[p]{\frac{F^{\leftarrow}(p)}{[F^{\leftarrow}(1-p)]^{1-p}}}\right\\}.$
$\displaystyle p_{R,p}[log_{a}(X)]$ $\displaystyle=$ $\displaystyle
P\left[log_{a}(X)>\frac{1}{p}F_{log_{a}(X)}^{\leftarrow}(1-p)-\frac{1-p}{p}F_{log_{a}(X)}^{\leftarrow}(p)\right]$
$\displaystyle=$ $\displaystyle
P\left\\{log_{a}(X)>\frac{1}{p}log_{a}[F^{\leftarrow}(1-p)]-\frac{1-p}{p}log_{a}[F^{\leftarrow}(p)]\right\\}$
$\displaystyle=$ $\displaystyle
1-F\left\\{\sqrt[p]{\frac{F^{\leftarrow}(1-p)}{[F^{\leftarrow}(p)]^{1-p}}}\right\\}.$

\- Case $0<a<1$. here we use the fact that $1/a>1$, therefore our computations
in the previous case imply

$\displaystyle p_{L,p}[log_{a}(X)]$ $\displaystyle=$ $\displaystyle
p_{L,p}[-log_{1/a}(X)]=p_{R,p}[log_{1/a}(X)]$ $\displaystyle=$ $\displaystyle
1-F\left\\{\sqrt[p]{\frac{F^{\leftarrow}(1-p)}{[F^{\leftarrow}(p)]^{1-p}}}\right\\}.$
$\displaystyle p_{R,p}[log_{a}(X)]$ $\displaystyle=$ $\displaystyle
p_{R,p}[-log_{1/a}(X)]=p_{L,p}[log_{1/a}(X)]$ $\displaystyle=$ $\displaystyle
F\left\\{\sqrt[p]{\frac{F^{\leftarrow}(p)}{[F^{\leftarrow}(1-p)]^{1-p}}}\right\\}.$

m) Case $\alpha>0$. In this case
$F_{X^{\alpha}}^{\leftarrow}(p)=[F^{\leftarrow}(p)]^{\alpha}$, and the
function $x^{\alpha}$ is increasing in $x$, therefore we apply Theorem 1, e)
(4) and (5) and obtain the desired result.

Case $\alpha<0$. Now
$F_{X^{\alpha}}^{\leftarrow}(p)=[F^{\leftarrow}(1-p)]^{\alpha}$, and the
function $x^{\alpha}$ is decreasing in $x$, therefore we apply Theorem 1, f)
(8) and (9) complete the proof.

n) In case $\alpha\in(0,1)$ we take into account that
$F_{a^{X}}^{\leftarrow}(p)=a^{[}F^{\leftarrow}(1-p)]$, and the functions
$a^{x}$ and $log_{a}(x)$ are decreasing in $x$, then we apply Theorem 1, f)
(8) and (9) and finish the proof of this case.

Analogously, if $\alpha>1$ then
$F_{a^{X}}^{\leftarrow}(p)=a^{[}F^{\leftarrow}(p)]$, and the function
$x^{\alpha}$ is increasing in $x$. Therefore we apply Theorem 1, e) and after
some algebra complete the proof.

Q.A.D.

Proof of Corollary of f): Assume $p\in(0,0.5]$. Consider the case $P(X>0)=1$.
Theorem 1, f), applied for $g(x)=x^{-1}$ entails

$p_{R,p}\left(\frac{1}{X}\right)=F\left[\frac{pF^{\leftarrow}(p)F^{\leftarrow}(1-p)}{F^{\leftarrow}(1-p)-(1-p)F^{\leftarrow}(p)}\right].$

By the definition for $p_{L,p}(X)$ and monotonicity of probability measures in
order to prove that $p_{L,p}(X)\leq p_{R,p}(X^{-1})$ we need to show that

$\frac{1}{p}F^{\leftarrow}(p)-\frac{1-p}{p}F^{\leftarrow}(1-p)\leq\frac{pF^{\leftarrow}(p)F^{\leftarrow}(1-p)}{F^{\leftarrow}(1-p)-(1-p)F^{\leftarrow}(p)}$
(47)

The last inequality is equivalent to

$1-(1-p)\frac{F^{\leftarrow}(1-p)}{F^{\leftarrow}(p)}\leq
p^{2}\frac{F^{\leftarrow}(1-p)}{F^{\leftarrow}(p)}\left[\frac{F^{\leftarrow}(1-p)}{F^{\leftarrow}(p)}-1+p\right]^{-1}.$

For $z:=\frac{F^{\leftarrow}(1-p)}{F^{\leftarrow}(p)}\geq 1$, $z-1+p\geq 0$,
therefore $(z-1+p)[1-(1-p)z]\leq p^{2}z$, which completes the proof of this
part.

The assertion for the case $P(X<0)=1$ follows by the fact that

$p_{L,p}\left(\frac{1}{X}\right)=p_{R,p}\left(\frac{1}{-X}\right).$

Q.A.D.

Proof of Corollary of e): Assume $p\in(0,0.5]$.

\- Case $a>1$. Because of
$F_{log_{a}(X)}^{\leftarrow}(p)=log_{a}[F^{\leftarrow}(p)]$, in order to use
Theorem 2, e) for $g(x)=log_{a}(x)$, which is increasing in $x$ we need to
prove that

$\frac{1}{p}log_{a}[F^{\leftarrow}(1-p)]-\frac{1-p}{p}log_{a}[F^{\leftarrow}(p)]\geq
log_{a}\left[\frac{1}{p}F^{\leftarrow}(1-p)-\frac{1-p}{p}F^{\leftarrow}(p)\right].$

The function $a^{x}$ is also increasing in $x$, therefore the above inequality
is equivalent to

$\sqrt[p]{\frac{F^{\leftarrow}(1-p)}{F^{\leftarrow}(p)}}\geq\frac{1}{p}\frac{F^{\leftarrow}(1-p)}{F^{\leftarrow}(p)}-\frac{1}{p}+1.$
(48)

The function $\sqrt[p]{z}-\frac{1}{p}z+\frac{1}{p}-1$ is increasing in $z\geq
1$, and for $p\in(0,0.5]$, $\frac{F^{\leftarrow}(1-p)}{F^{\leftarrow}(p)}\geq
1$. Therefore, for $z\geq 1$, $\sqrt[p]{z}-\frac{1}{p}z+\frac{1}{p}-1\geq 0$
proves (48) and completes the proof of (13).

By Theorem 2, f), applied for $g(x)=log_{a}(x)$, (because now $log_{a}(x)$ is
also increasing in $x$) in order to compare $p_{L,p}[log_{a}(X)]$ and
$p_{L,p}(X)$ in (12) we have to show that

$\sqrt[p]{\frac{F^{\leftarrow}(p)}{[F^{\leftarrow}(1-p)]^{1-p}}}\geq\frac{1}{p}F^{\leftarrow}(p)-\frac{1-p}{p}F^{\leftarrow}(1-p).$
(49)

Which is the same as

$\sqrt[p]{\frac{F^{\leftarrow}(p)}{F^{\leftarrow}(1-p)}}\geq\frac{1}{p}\frac{F^{\leftarrow}(p)}{F^{\leftarrow}(1-p)}-\frac{1}{p}+1.$

The function $\sqrt[p]{z}-\frac{1}{p}z+\frac{1}{p}-1$ is decreasing in
$z\in(0,1)$, and because of by assumption $P(X>0)=1$, for $p\in(0,0.5]$,
$0<\frac{F^{\leftarrow}(1-p)}{F^{\leftarrow}(p)}\leq 1$. Therefore, for $z\geq
1$, $\sqrt[p]{z}-\frac{1}{p}z+\frac{1}{p}-1\geq 0$ proves (49) and (12).

\- Case $0<a<1$. Because of $1/a\geq 1$, by the previous case, and the
definitions for $p_{R,p}$ and $p_{L,p}$, entail

$p_{R,p}(log_{a}(X))=p_{R,p}(-log_{1/a}(X))=p_{L,p}(log_{1/a}(X))\geq
p_{L,p}(X)$

and complete the proof of (14).

The fact that $1/a\geq 1$ and (13) entail

$p_{L,p}(log_{a}(X))=p_{L,p}(-log_{1/a}(X))=p_{R,p}(log_{1/a}(X))\leq
p_{R,p}(X)\leq p_{R,p}(\frac{1}{a^{X}}),$

and this proves inequalities in (15). Q.A.D.

Proof of Theorem 3: a) Consider $p\in(0,0.5]$, $0\leq\alpha_{1}\leq\alpha_{2}$
and $P$ almost sure positive r.v. $X$, i.e. $P(X>0)=1$. According to
monotonicity of probability measures, the definition of $p_{R,p}$, and the
equalities
$F_{X^{\alpha_{i}}}^{\leftarrow}(p)=[F^{\leftarrow}(p)]^{\alpha_{i}}$, $i=1,2$
we need to show that

$\sqrt[\alpha_{1}]{\frac{1}{p}[F^{\leftarrow}(1-p)]^{\alpha_{1}}-\frac{1-p}{p}[F^{\leftarrow}(p)]^{\alpha_{1}}}\geq\sqrt[\alpha_{2}]{\frac{1}{p}[F^{\leftarrow}(1-p)]^{\alpha_{2}}-\frac{1-p}{p}[F^{\leftarrow}(p)]^{\alpha_{2}}}.$
(50)

It is the same as

$\sqrt[\alpha_{1}]{\frac{1}{p}\left[\frac{F^{\leftarrow}(1-p)}{F^{\leftarrow}(p)}\right]^{\alpha_{1}}-\frac{1}{p}+1}\geq\sqrt[\alpha_{2}]{\frac{1}{p}\left[\frac{F^{\leftarrow}(1-p)}{F^{\leftarrow}(p)}\right]^{\alpha_{2}}-\frac{1}{p}+1}.$

Denote by $z:=\frac{F^{\leftarrow}(1-p)}{F^{\leftarrow}(p)}\geq 1$. The last
inequality is true, because of for any fixed $z\geq 1$ and $p\in(0;0.5]$ the
function

$h(\alpha)=\sqrt[\alpha]{\frac{1}{p}z^{\alpha}-\frac{1}{p}+1}$

is decreasing in $\alpha>0$. Therefore (50) is also true, and the proof of
(16) is completed.

b) The r.v. $X$ is almost sure positive, so we have the same for $X^{\alpha}$.
Therefore, by the definition of $p_{L,p}(X^{\alpha})$ it is enough to show
that, in this case $L(X^{\alpha},p)\leq 0$. Now we use the equality
$F^{\leftarrow}_{X^{\alpha}}(p)=[F^{\leftarrow}_{X}(p)]^{\alpha}$ in the
definition of $L(X^{\alpha},p)$ and obtain that given the condition in c), the
value of

$L(X^{\alpha},p)=\frac{1}{p}\left\\{[F^{\leftarrow}(p)]^{\alpha}-(1-p)[F^{\leftarrow}(1-p)]^{\alpha}\right\\}\leq
0.$

c) It is enough to prove the second inequality in (18). It is equivalent to

$P\left[X<\frac{1}{p}F^{\leftarrow}(p)-\frac{1-p}{p}F^{\leftarrow}(1-p)\right]\geq
P\left[X^{\alpha}<\frac{1}{p}F_{X^{\alpha}}^{\leftarrow}(p)-\frac{1-p}{p}F_{X^{\alpha}}^{\leftarrow}(1-p)\right],$

and using the equality
$F_{X^{\alpha}}^{\leftarrow}(p)=[F^{\leftarrow}(p)]^{\alpha}$ it is the same
as

$P\left[X<\frac{1}{p}F^{\leftarrow}(p)-\frac{1-p}{p}F^{\leftarrow}(1-p)\right]\geq
P\left[X<\sqrt[\alpha]{\frac{1}{p}[F^{\leftarrow}(p)]^{\alpha}-\frac{1-p}{p}[F^{\leftarrow}(1-p)]^{\alpha}}\right].$

The monotonicity of probability measures entails that the above inequality
would be true if

$\frac{1}{p}F^{\leftarrow}(p)-\frac{1-p}{p}F^{\leftarrow}(1-p)\geq\sqrt[\alpha]{\frac{1}{p}[F^{\leftarrow}(p)]^{\alpha}-\frac{1-p}{p}[F^{\leftarrow}(1-p)]^{\alpha}},\,\,i.e.$
$\left[\frac{1}{p}F^{\leftarrow}(p)-\frac{1-p}{p}F^{\leftarrow}(1-p)\right]^{\alpha}\geq\frac{1}{p}[F^{\leftarrow}(p)]^{\alpha}-\frac{1-p}{p}[F^{\leftarrow}(1-p)]^{\alpha},\,\,that\,\,is$
$\left[\frac{1}{p}-\frac{1-p}{p}\frac{F^{\leftarrow}(1-p)}{F^{\leftarrow}(p)}\right]^{\alpha}\geq\frac{1}{p}-\frac{1-p}{p}\left[\frac{F^{\leftarrow}(1-p)}{F^{\leftarrow}(p)}\right]^{\alpha}.$
(51)

Again denote by $z:=\frac{F^{\leftarrow}(1-p)}{F^{\leftarrow}(p)}\geq 1$, and
consider the function

$t(z):=\frac{1}{p}-\frac{1-p}{p}z^{\alpha}-\left[\frac{1}{p}-\frac{1-p}{p}z\right]^{\alpha}.$

Given $\alpha>1$, it is decreasing for $z\geq 1$ and $t(1)=0$. Therefore
$t(z)\leq 0$, for $z\geq 1$. This entails (51) and completes the proof of
(18). Q.A.D.

Proof of Theorem 5: Let us fix $s\in N$. From the theorem about the joint
asymptotic normality of the order statistics, because the limit exists we have
that for any subsequence $\\{n:n=(s+1)k-1,s\in N\\}$,

$\sqrt{(s+1)k-1}\left(\begin{array}[]{c}X_{(k,(s+1)k-1)}-F^{\leftarrow}(\frac{1}{s+1})\\\
X_{(ki,(s+1)k-1)}-F^{\leftarrow}(\frac{i}{s+1})\\\
\end{array}\right)\stackrel{{\scriptstyle
d}}{{\to}}N\left[\left(\begin{array}[]{c}0\\\ 0\\\
\end{array}\right);D\right],\quad k\to\infty,$

where the asymptotic covariance matrix of this bivariate distribution is

$D=\frac{1}{(s+1)^{2}}\left(\begin{array}[]{cc}\frac{s}{f^{2}\left[F^{\leftarrow}(\frac{1}{s+1})\right]}&\frac{1}{f\left[F^{\leftarrow}(\frac{1}{s+1})\right]f\left[F^{\leftarrow}(\frac{s}{s+1})\right]}\\\
\frac{1}{f\left[F^{\leftarrow}(\frac{1}{s+1})\right]f\left[F^{\leftarrow}(\frac{s}{s+1})\right]}&\frac{s}{f^{2}\left[F^{\leftarrow}(\frac{s}{s+1})\right]}\\\
\end{array}\right)$

and the asymptotic correlation between these two order statistics is
$\frac{1}{i}$.

1.) Consider the function $g(x,y)=(s+1)x-sy$. For $x>0$ and $y>0$ it is
continuously differentiable. The asymptotic mean is

$\displaystyle\lim_{k\to\infty}EL_{n}\left(X,\frac{1}{s+1}\right)$
$\displaystyle=$ $\displaystyle
g\left[F^{\leftarrow}(\frac{1}{s+1}),F^{\leftarrow}(\frac{s}{s+1})\right]$
$\displaystyle=$
$\displaystyle(s+1)F^{\leftarrow}(\frac{1}{s+1})-iF^{\leftarrow}(\frac{s}{s+1})=L\left(X,\frac{1}{s+1}\right).$

The Jacobian of the transformation is

$I_{L}:=\left[\frac{\partial g(x,y)}{\partial x},\frac{\partial
g(x,y)}{\partial y}\right]=\left(s+1,-s\right).$

Now we apply the Multivariate Delta method Sobel (1982) [55], and obtain that
the asymptotic variance of $L_{n}\left(X,\frac{1}{s+1}\right)$ is

$\displaystyle V_{L}:$ $\displaystyle=$ $\displaystyle I_{L}\times D\times
I_{L}^{\prime}=\left(s+1,-s\right)\left|{}_{x=F^{\leftarrow}(\frac{1}{s+1}),y=F^{\leftarrow}(\frac{s}{s+1})}\right.$
$\displaystyle\times$
$\displaystyle\frac{1}{(s+1)^{2}}\left(\begin{array}[]{cc}\frac{s}{f^{2}\left[F^{\leftarrow}(\frac{1}{s+1})\right]}&\frac{1}{f\left[F^{\leftarrow}(\frac{1}{s+1})\right]f\left[F^{\leftarrow}(\frac{s}{s+1})\right]}\\\
\frac{1}{f\left[F^{\leftarrow}(\frac{1}{s+1})\right]f\left[F^{\leftarrow}(\frac{s}{s+1})\right]}&\frac{i}{f^{2}\left[F^{\leftarrow}(\frac{s}{s+1})\right]}\\\
\end{array}\right)$ $\displaystyle\times$
$\displaystyle\left(\begin{array}[]{c}s+1\\\ -s\\\
\end{array}\right)\left|{}_{x=F^{\leftarrow}(\frac{1}{s+1}),y=F^{\leftarrow}(\frac{s}{s+1})}\right.$
$\displaystyle=$ $\displaystyle\frac{1}{(s+1)^{2}}$ $\displaystyle\times$
$\displaystyle\left[\left(s+1,-s\right)\left(\begin{array}[]{cc}\frac{i}{f^{2}(x)}&\frac{1}{f(x)f(y)}\\\
\frac{1}{f(x)f(y)}&\frac{i}{f^{2}(y)}\\\
\end{array}\right)\left(\begin{array}[]{c}s+1\\\ -s\\\
\end{array}\right)\right]\left|{}_{x=F^{\leftarrow}(\frac{1}{s+1}),y=F^{\leftarrow}(\frac{s}{s+1})}\right.$
$\displaystyle=$
$\displaystyle\frac{1}{(s+1)^{2}}\left[\frac{s(s+1)^{2}}{f^{2}(x)}-\frac{2s(s+1)}{f(x)f(y)}+\frac{s^{3}}{f^{2}(y)}\right]\left|{}_{x=F^{\leftarrow}(\frac{1}{s+1}),y=F^{\leftarrow}(\frac{s}{s+1})}\right.$
$\displaystyle=$
$\displaystyle\frac{s}{(s+1)^{2}}\left[\frac{(s+1)^{2}}{c_{F,s}^{2}}-\frac{2(s+1)}{c_{F,s}d_{F,s}}+\frac{s^{2}}{d_{F,s}^{2}}\right].$

2.) Analogously, because of

$\displaystyle R_{n}\left(F,\frac{1}{s+1}\right)$ $\displaystyle=$
$\displaystyle(s+1)X_{(ks,(s+1)k-1)}-iX_{(k,(s+1)k-1)},\,\,and$ $\displaystyle
R\left(X,\frac{1}{s+1}\right)$ $\displaystyle=$
$\displaystyle(s+1)F^{\leftarrow}\left(\frac{s}{s+1}\right)+sF^{\leftarrow}\left(\frac{1}{s+1}\right),$

we consider function $g_{R}(x,y)=-sx+(s+1)y$.

For $x>0$ and $y>0$ it is continuously differentiable.

The asymptotic mean is

$\displaystyle\lim_{k\to\infty}ER_{n}\left(F,\frac{1}{s+1}\right)$
$\displaystyle=$ $\displaystyle
g_{R}\left[F^{\leftarrow}(\frac{1}{s+1}),F^{\leftarrow}(\frac{s}{s+1})\right]$
$\displaystyle=$ $\displaystyle-
sF^{\leftarrow}(\frac{1}{s+1})+(s+1)F^{\leftarrow}(\frac{s}{s+1})=R\left(X,\frac{1}{s+1}\right).$

In order to obtain the asymptotic variance of
$R_{n}\left(X,\frac{1}{s+1}\right)$ we calculate the Jacobian of the
transformation. It is

$I_{R}:=\left[\frac{\partial g_{R}(x,y)}{\partial x},\frac{\partial
g_{R}(x,y)}{\partial y}\right]=\left(s+1,-s\right).$

Now we apply the Multivariate Delta method (see e.g. Sobel (1982) [55]),
calculate $I_{R}\times D\times I_{R}^{\prime}$ and obtain the asymptotic
variance of $R_{n}\left(F,\frac{1}{s+1}\right)$ is

$V_{R}:=I_{R}\times D\times
I_{R}^{\prime}=\frac{s}{(s+1)^{2}}\left[\frac{s^{2}}{c_{F,s}^{2}}-\frac{2(s+1)}{c_{F,s}d_{F,s}}+\frac{(s+1)^{2}}{d_{F,s}^{2}}\right].$

Q.A.D.

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