# Maximum likelihood estimation for left-truncated log-logistic distributions
with a given truncation point

Markus Kreer  Feldbergschule, Oberhöchstadter Str. 20, 61440 Oberursel
(Taunus), Germany.  Ayşe Kızılersü  CSSM, Department of Physics, University of
Adelaide, 5005, Adelaide, Australia.  Jake Guscott  CSSM, Department of
Physics, University of Adelaide, 5005, Adelaide, Australia.  Lukas Christopher
Schmitz  Institut für Mathematik, Johannes Gutenberg-Universität Mainz,
Staudingerweg 9, 55128 Mainz, Germany.  Anthony W. Thomas  CSSM, Department of
Physics, University of Adelaide, 5005, Adelaide, Australia.

###### Abstract

The maximum likelihood estimation of the left-truncated log-logistic
distribution with a given truncation point is analyzed in detail from both
mathematical and numerical perspectives. These maximum likelihood equations
often do not possess a solution, even for small truncations. A simple
criterion is provided for the existence of a regular maximum likelihood
solution. In this case a profile likelihood function can be constructed and
the optimisation problem is reduced to one dimension. When the maximum
likelihood equations do not admit a solution for certain data samples, it is
shown that the Pareto distribution is the $L^{1}$-limit of the degenerated
left-truncated log-logistic distribution. Using this mathematical information,
a highly efficient Monte Carlo simulation is performed to obtain critical
values for some goodness-of-fit tests. The confidence tables and an
interpolation formula are provided and several applications to real world data
are presented.

##  1 Preliminaries

The log-logistic distribution, also known as the Fisk distribution, has been
popular in the econometric community since the early 1960s because of its
better description of income distributions ([1]), as compared with the Pareto
distribution. On the other hand, hydrologists in the late 1980s suggested that
the log-logistic distributions were useful for modelling Canadian
precipitation data ([2]), or flood frequencies in Scotland for annual flood
maxima ([3]). The log-logistic distribution is related to the logistic
distribution by a logarithmic transform. Practioners make use of log-logistic
distributions because of its easy calculability and closed form expressions
for both cummulative probability distribution (cdf) and probablity density
function (pdf) [4],[5]. For $x>0$ the pdf and cdf are given by

$\displaystyle f(x|\alpha,\beta)$ $\displaystyle=$
$\displaystyle\frac{\beta}{\alpha}\left(\frac{x}{\alpha}\right)^{\beta-1}\frac{1}{\left[1+(x/\alpha)^{\beta}\right]^{2}}$
$\displaystyle F(x|\alpha,\beta)$ $\displaystyle=$
$\displaystyle\frac{1}{1+(x/\alpha)^{-\beta}}$

where $\alpha>0$ is the scale parameter and $\beta>0$ the shape parameter. We
can recover the Pareto distribution for the tail by expanding the cdf for
“large” arguments, $x\gg\alpha$, because up to first order $F(x)\simeq
1-(x/\alpha)^{-\beta}$.

In the analysis of annual flood maxima, the log-logistic distribution in Ref.
[3] was modified by the introduction of a threshold parameter for practical
reasons: a flood maximum in a rainy country like Scotland should always be
above a certain threshold level. In this paper we pursue a different approach:
we keep the two-parameter distribution but introduce a fixed left-truncation
point $x_{L}>0$ instead. Thus we have now for $x>x_{L}$ the left-truncated
log-logistic pdf and cdf respectively as (see also Refs. [6, 7, 8] for
obtaining a truncated distribution from a complete distribution)

$\displaystyle f_{LT}(x|\alpha,\beta;x_{L})$ $\displaystyle=$
$\displaystyle\left(1+\left(\frac{x_{L}}{\alpha}\right)^{\beta}\right)\frac{\beta}{\alpha}\left(\frac{x}{\alpha}\right)^{\beta-1}\frac{1}{\left[1+\left(\frac{x}{\alpha}\right)^{\beta}\right]^{2}}\quad,$
(1) $\displaystyle F_{LT}(x|\alpha,\beta;x_{L})$ $\displaystyle=$
$\displaystyle\frac{\left(\frac{x}{\alpha}\right)^{\beta}-\left(\frac{x_{L}}{\alpha}\right)^{\beta}}{1+\left(\frac{x}{\alpha}\right)^{\beta}}\quad.$
(2)

The subscript “LT” stands for left-truncated.If a random variable $X$ is log-
logistically distributed with positive parameters $\alpha,\beta$ and left-
truncation $x_{L}$ we can denote $X\sim LL(\alpha,\beta;x_{L})$. From Eq. (2)
we see immediately how to generate a random variable $X\sim
LL(\alpha,\beta;x_{L})$ from a uniformly distributed random variable $U$ in
the interval $(0,1)$, namely

$\displaystyle
X=\left(\frac{\alpha^{\beta}U+x_{L}^{\beta}}{1-U}\right)^{1/\beta}=\alpha\left(\frac{U+\eta}{1-U}\right)^{1/\beta}\quad,$
(3)

where the $\eta=(x_{L}/\alpha)^{\beta}$.

When probability distributions are truncated some interesting effects can
happen: in [9] it is demonstrated that if a normal distribution is truncated,
there exists finite random samples for which the regular maximum likelihood
equations (MLE) do not possess a solution. Instead a new maximum likelihood
estimator as a limit case was obtained leading to a degenerated one-parameter
distribution, namely the exponential distribution“to fit” the sample data
appropriately. Similar effects were observed in the analysis of the left-
truncated Weibull distributions [10], where for certain samples the MLE do not
possess a solution. Indeed, in [8] it is found numerically that for a rather
large set of data samples the MLE for the left-truncated log-logistic
distribution do not possess a solution.

This was one of the main motivations for this paper, because it demonstrates a
clear need for a careful analysis of the left-truncated log-logistic
distribution from a rigorous mathematical point of view. To the best of our
knowledge there was no study to prove the existence of the maximum likelihood
estimator for a random sample drawn from the left-truncated log-logistic
distribution. Therefore any numerical studies without this proof, assuming the
existence of a solution, will end up in the worst case either not converging
at all, or converging to a degenerate solution in which the parameter
estimates can take values like zero or infinity.

Our paper is structured as follows: Section 2 contains the main theorems for
the existence of a non-trivial solution of the maximum likelihood equations
for the left-truncated log-logistic distribution. It also examines the
properties of a suitable profile likelihood function. These results will be
relevant for the efficient numerical implementation explained in section 3.
Moreover we also discuss in this section our findings for the critical values
for Kolmogorov-Smirnov (KS) and Anderson-Darling (AD) hypothesis tests based
on extensive Monte Carlos simulations at the supercomputer of Adelaide
University. As an illustration, in Section 4 we apply our technique to cancer
data and German precipitation data. All proofs are given in Section 5.

##  2 Mathematical results

###  2.1 Scaling property

We first provide a lemma dealing with a scaling property.

######  Lemma 1.

(Scaling Property) Let $X\sim LL(\alpha,\beta;x_{L})$ be a left-truncated log-
logistic random variable. Then for any $k>0$ we have $kX\sim
LL(k\alpha,\beta;kx_{L})$.

Proof:

By assumption $X>x_{L}$ with $X\sim LL(\alpha,\beta;x_{L})$. Thus $kX>kx_{L}$
and

$\displaystyle\text{prob}(kX<x\leavevmode\nobreak\ |\leavevmode\nobreak\
kX>kx_{L})$ $\displaystyle=$ $\displaystyle\text{prob}(X<x/k|X>x_{L})$
$\displaystyle=$
$\displaystyle\frac{\left(\frac{x/k}{\alpha}\right)^{\beta}-\left(\frac{x_{L}}{\alpha}\right)^{\beta}}{1+\left(\frac{x/k}{\alpha}\right)^{\beta}}=\frac{\left(\frac{x}{k\alpha}\right)^{\beta}-\left(\frac{kx_{L}}{k\alpha}\right)^{\beta}}{1+\left(\frac{x}{k\alpha}\right)^{\beta}}$

and the proof is finished. $\blacksquare$

As a consequence, for simplicity we will assume later on the left-truncation
point $x_{L}=1$. In other words we rescale the independent identically
distributed (i.i.d.) sample $X_{1},...,X_{N}$ with $k=1/x_{L}$, leading to
$X_{1}/x_{L},...,X_{N}/x_{L}$ (which is truncated at $1$). In any stage we can
go back to the original sample and original parameters by using a rescaling
factor $k=x_{L}$.

###  2.2 The first-order maximum likelihood equations

We introduce a new parametrization with $\lambda=\alpha^{\beta}$, and re-write
Eq. (1) as

$\displaystyle f_{LT}(x|\lambda,\beta)$ $\displaystyle=$
$\displaystyle\left(1+\frac{x_{L}^{\beta}}{\lambda}\right)\frac{\beta}{\lambda}x^{\beta-1}\frac{1}{\left[1+\frac{x^{\beta}}{\lambda}\right]^{2}}\quad.$
(4)

Note that with this notation we have $\eta=x_{L}^{\beta}/\lambda$.

Using Eq. (4) and a left-truncated log-logistic sample, denoted by
$X_{1},X_{2},...,X_{N}$ and all observations are bigger than $x_{L}>0$, the
log-likelihood function is given via by

$\displaystyle\ln{L_{LT}(\\{X_{i}\\}|\lambda,\beta;x_{L})}$ $\displaystyle=$
$\displaystyle
N\ln{\left(1+\frac{x_{L}^{\beta}}{\lambda}\right)}+N\ln{\beta}-N\ln{\lambda}$
(5)
$\displaystyle+(\beta-1)\sum\limits_{i=1}^{N}\ln{X_{i}}-2\sum\limits_{i=1}^{N}\ln{\left[1+\frac{X_{i}^{\beta}}{\lambda}\right]}\,.$

The MLE equations are obtained by differentiating Eq. (5) with respect to
$\lambda$ and $\beta$ respectively and putting the derivatives equals to zero.
From $\frac{\partial}{\partial\lambda}\ln{L_{LT}}=0$ we obtain for
$(\lambda,\beta)\in\mathbb{R^{+}}\times\mathbb{R^{+}}$

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{2}{N}\sum\limits_{i=1}^{N}\frac{\frac{X_{i}^{\beta}}{\lambda}}{1+\frac{X_{i}^{\beta}}{\lambda}}-1-\frac{\frac{x_{L}^{\beta}}{\lambda}}{1+\frac{x_{L}^{\beta}}{\lambda}}$
(6)

and from $\frac{\partial}{\partial\beta}\ln{L_{LT}}=0$ we obtain for
$(\lambda,\beta)\in\mathbb{R^{+}}\times\mathbb{R^{+}}$

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{\frac{x_{L}^{\beta}}{\lambda}\ln{x_{L}}}{1+\frac{x_{L}^{\beta}}{\lambda}}+\frac{1}{\beta}+\frac{1}{N}\sum\limits_{i=1}^{N}\ln{X_{i}}-\frac{2}{N}\sum\limits_{i=1}^{N}\frac{\frac{X_{i}^{\beta}}{\lambda}}{1+\frac{X_{i}^{\beta}}{\lambda}}\ln{X_{i}}\,.$
(7)

A solution of these MLE equations Eqs. (6)–(7) (if it exists) will be denoted
by $(\hat{\lambda},\hat{\beta})$. For finite untruncated samples, Ref. [11]
(respectively Ref. [12]) has shown that the MLE have a unique solution for the
log-logistic distribution (respectively the logistic distribution). These
proofs fail when a left-truncation point $x_{L}>0$ is introduced.

We will take $x_{L}=1$ from now on without loss of generality. We also define
for convenience the quantity $S=\sum\limits_{i=1}^{N}\ln{X_{i}}$ and introduce
a new objective function as in [11]

$\displaystyle\varphi(\lambda,\beta)$ $\displaystyle=$
$\displaystyle\ln{L_{LT}(\\{X_{i}\\}|\lambda,\beta;x_{L}=1)}+S\quad.$

The extrema of both the function $\varphi(\cdot,\cdot)$ and the log-likelihood
function are the same (and so are their MLE equations), because they differ by
a constant number, $S$.

###  2.3 Our main theorems

Our first theorem states the existence of a maximum for the objective function
under certain conditions.

######  Theorem 1.

(Existence) Consider the i.i.d. left-truncated sample ${X_{1},...,X_{N}>1}$,
for which at least two observations are different, and define the objective
function
$\varphi(\cdot,\cdot):\mathbb{R^{+}}\times\mathbb{R^{+}}\rightarrow\mathbb{R}$
by

$\displaystyle\varphi(\lambda,\beta)$ $\displaystyle=$ $\displaystyle
N\ln{\left(1+\frac{1}{\lambda}\right)}+N\ln{\beta}-N\ln{\lambda}+\beta S$ (8)
$\displaystyle-2\sum\limits_{i=1}^{N}\ln{\left[1+\frac{X_{i}^{\beta}}{\lambda}\right]}\,.$

Define $\beta_{C}>0$ as the unique solution of

$\displaystyle\frac{1}{2}=\frac{1}{N}\sum\limits_{i=1}^{N}\frac{1}{X_{i}^{\beta_{C}}}$
(9)

and $\beta_{0}>0$ by

$\displaystyle\frac{1}{\beta_{0}}=\frac{1}{N}\sum\limits_{i=1}^{N}\ln{X_{i}}\quad.$
(10)

Then the following holds true

  1. (1)

For $\beta_{0}>\beta_{C}$ the objective function $\varphi(\cdot,\cdot)$
posseses a global maximum
$(\hat{\lambda},\hat{\beta})\in\mathbb{R^{+}}\times\mathbb{R^{+}}$.

  2. (2)

For $\beta_{0}\leq\beta_{C}$ the objective function $\varphi(\cdot,\cdot)$
posseses a (local) maximum $(\hat{\lambda},\hat{\beta})=(0,\beta_{0})$ on the
boundary.

The proof is given in Section 5.

The next result is important for the numerical computation of the maxima and
leads to a profile likelihood function and provides a curve for the loci111
locus = a set of points that satisfy or are determined by some specific
condition of critical points.

######  Theorem 2.

(Loci of critical points)

Consider the i.i.d. left-truncated sample $X_{1},...,X_{N}>1$ for which at
least two observations are different. Then the following holds true:

  1. (1)

For fixed $\beta>\beta_{C}$ the equation

$\displaystyle\left.\frac{\partial}{\partial\lambda}\varphi(\lambda,\beta)\right|_{\lambda=\hat{\lambda}}=0$

has exactly one positive solution $\hat{\lambda}$ which depends on the fixed
parameter $\beta$. Furthermore we have the following inequalities

$\displaystyle\frac{\partial}{\partial\lambda}\varphi(\lambda,\beta)$
$\displaystyle>$ $\displaystyle 0\hskip 14.22636pt\text{for}\hskip
14.22636pt0<\lambda<\hat{\lambda}$
$\displaystyle\frac{\partial}{\partial\lambda}\varphi(\lambda,\beta)$
$\displaystyle<$ $\displaystyle 0\hskip 14.22636pt\text{for}\hskip
14.22636pt\lambda>\hat{\lambda}$

and thus

$\displaystyle\left.\frac{\partial^{2}}{\partial\lambda^{2}}\varphi(\lambda,\beta)\right|_{\lambda=\hat{\lambda}}<0\quad.$
(11)

  2. (2)

For fixed $0<\beta\leq\beta_{C}$ the equation

$\displaystyle\left.\frac{\partial}{\partial\lambda}\varphi(\lambda,\beta)\right|_{\lambda=\hat{\lambda}}=0\quad,$

has the only solution $\hat{\lambda}=0$.

Hence the non-negative continous function
$\Lambda(\cdot):\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}_{0}$, defined by

$\displaystyle\Lambda(\beta)=\left\\{\begin{array}[]{ll}0,&0<\beta\leq\beta_{C}\\\
\hat{\lambda},&\beta>\beta_{C}\end{array}\right.$ (14)

is the locus of all possible critical points of the objective function
$\varphi(\cdot,\cdot)$.

The proof is given in Section 5.

By inserting the function $\Lambda(\beta)$, as constructed in Eq. (14), into
the original objective function Eq. (8) we obtain the “profile likelihood
function”

$\displaystyle\tilde{\varphi}(\beta)=\varphi(\Lambda(\beta),\beta)$
$\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}N\ln{\beta}-\beta
S,&\beta\in(0,\beta_{C}]\\\ \\\
N\ln{\left(1+\frac{1}{\Lambda(\beta)}\right)}+N\ln{\beta}-N\ln{\Lambda(\beta)}&\\\
+\beta
S-2\sum\limits_{i=1}^{N}\ln{\left(1+\frac{X_{i}^{\beta}}{\Lambda(\beta)}\right)},&\beta\in(\beta_{C},\infty)\end{array}\right.$
(18)

Therefore, we have reduced the two-dimensional maximization problem for
$\varphi(\cdot,\cdot)$ on a finite region to a one-dimensional problem for
$\tilde{\varphi}(\cdot)$ on a finite interval. Figure 1 illustrates the
problem: the critical points (and therefore our maximum guaranteed by Theorem
1) are located inside the blue rectangle region and must be located on the red
curve. By condition Eq. (11), critical points can only be maxima or saddle
points. This simplification is important because it improves the speed of
finding the critical points in our numerical study dramatically.
Unfortunately, we were not able to prove uniqueness for the critical points,
as the necessary concavity arguments for the Hessian matrix seem intractable.
Numerically, though, we always found exactly one critical point for millions
of samples generated and this point was always a maximum.

Figure 1:  The function $\Lambda(\beta)$ for the profile likelihood.

A Corollary sumarises the situation of our two theorems.

######  Corollary 1.

Under the assumptions of Theorem 1. and 2. the following holds true

  1. (1)

If and only if $\beta_{0}>\beta_{C}$ the MLE equations Eq. (6)–(7) possess a
solution $(\hat{\lambda},\hat{\beta})\neq(0,0)$ (which might not be unique).
Critical points are obtained by constructing the function
$\Lambda(\cdot):\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}_{0}$ as defined for
all $\beta>0$ in Theorem 2, Eq. (14), and then solving Eq. (7), which now
reads

$\displaystyle
0=\frac{N}{\beta}+\sum\limits_{i=1}^{N}\ln{X_{i}}-2\sum\limits_{i=1}^{N}\frac{X_{i}^{\beta}\ln{X_{i}}}{\Lambda(\beta)+X_{i}^{\beta}}\quad,$
(20)

for the desired critical point $\hat{\beta}>\beta_{C}$.

  2. (2)

If and only if $\beta_{0}\leq\beta_{C}$ the MLE equations, Eq. (6)–(7), do not
possess a solution. The likelihood attains a maximum on the boundary
$(\hat{\lambda},\hat{\beta})=(0,\beta_{0})$ with a Pareto probabilty
distribution given by

$\displaystyle g(x|\beta_{0})=\frac{\beta_{0}}{x^{1+\beta_{0}}}\quad.$ (21)

The proof is given in Section 5.

Our final theorem demonstrates that the condition that at least two
observations in our sample be different is not only sufficient but also
necessary. We see that in this case no finite maximum of the MLE exists.

######  Theorem 3.

(Necessity of at least two different observations) Consider the i.i.d. left-
truncated sample $X_{1}=X_{2}=...=X_{N}>1$ for which all observations are
equal to $X_{1}$, say. Then the following holds true:

  1. (1)

$\beta_{0}=1/\ln{X_{1}}$ and $\beta_{C}=\ln{2}/\ln{X_{1}}$

  2. (2)

For fixed $\beta>\beta_{C}$ the function

$\displaystyle\Lambda(\beta)=X_{1}^{\beta}-2$

satisfies the ML equation for $\lambda$, Eq. (6).

  3. (3)

The profile likelihood function Eq. (LABEL:eq:profilelikelihood) reading here
as

$\displaystyle\tilde{\varphi}(\beta)=\varphi(\Lambda(\beta),\beta)=\left\\{\begin{array}[]{ll}N\ln{\beta}-N\beta\ln{X_{1}},&\beta\in(0,\beta_{C}]\\\
\\\ -N\ln{\left(X_{1}^{\beta}-1\right)}+N\ln{\beta}&\\\
+N\beta\ln{X_{1}}-2N\ln{2},&\beta\in(\beta_{C},\infty)\end{array}\right.$ (25)

is continous and strictly monotonically increasing in $\beta$. It therefore
does not possess a finite maximum.

The proof is given in Section 5.

This last Theorem, which can be also formulated in the same fashion for the
un-truncated log-logistic MLE, might explain the phenomenon observed in Ref.
[2], Table 1, p. 231: there the percentage of failures of the maximisation
algorithm for the likelihood is given for small samples. Indeed, when a sample
consists of e.g. $N=15$ data points which are all “nearly equal”, a direct
optimiziation by numerical methods is bound to fail if the parameter region
chosen is too “small”. The profile likelihood will be increasing in one
direction towards the boundaries of the “small” parameter region and the
Newton-Raphson solver of the MLE equations fails to converge. That the MLE
problem in this case possesses a solution nevertheless was established in Ref.
[11].

##  3 Monte Carlo study and statistical hypothesis test

The mathematical results obtained in the previous section are now applied in
the framework of hypothesis testing. The determination of critical values for
a certain confidence level are necessary in the case that the distribution
parameters are estimated from the given sample, “in-sample testing”. Some of
us have shown in the case of left-truncated Weibull distribution that the
critical values will depend on sample size, the truncation point and also on
the parameters. This functional dependence will be investigated now in detail.
Extensive Monte Carlos studies have been performed in [8] to determine the
functional dependence. In this paper we present the results for the
Kolmogorov-Smirnov and Anderson-Darling hypothesis tests for various
confidence levels.

###  3.1 Kolmogorov-Smirnov and Anderson-Darling hypothesis test

If the MLE for a given sample of size $N$ yields non-trivial parameters
$(\hat{\lambda},\hat{\beta})$, then the Kolmogorov-Smirnov (KS) distance $D$
for the sample s, defined as being the maximum value of the absolute
difference between the empirical distribution function and the cummulative
distribution function $F(\cdot|\hat{\lambda},\hat{\beta})$ (see [13, 14]) is

$\displaystyle D=\max_{1\leq i\leq
N}{\left\\{\frac{i}{N}-F(X_{i}|\hat{\lambda},\hat{\beta}),\frac{i-1}{N}-F(X_{i}|\hat{\lambda},\hat{\beta})\right\\}}\,.$
(26)

We employ here as test statistic for the KS test the quantity $\sqrt{N}D$.

The Anderson-Darling (AD) test statistic $A^{2}$ is given by the following
expression ([15])

$\displaystyle
A^{2}=-N-\frac{1}{N}\sum\limits_{i=1}^{N}\\{2i-1\\}\left\\{\ln{[F(X_{i}|\hat{\lambda},\hat{\beta})]}+\ln{[1-F(X_{N-i+1}|\hat{\lambda},\hat{\beta})]}\right\\}\quad.$
(27)

If these positive test statistics are smaller than certain critical values
(depending on the confidence level) we cannot reject the statement that “the
sample data are distributed according to a truncated log-logistic distribution
with estimated parameters $(\hat{\lambda},\hat{\beta})$” for a given
confidence level.

###  3.2 Algorithmic implementation and results

The following Monte Carlo algorithm to determine critical numbers for the KS
and AD tests was performed in [8]. The maximum likelihood estimation in the
spirit of Corollary 1 is algorithmically very efficient and despite the fact
that we could not assert uniqueness, it turns out that numerical search for
critical points seems to yield always exactly one solution which satisfies the
properties of a maximum. The flow chart of the algorithm for a given sample
size is as follows

  1. 1.

For a fixed integer $N$ iterate the following steps 2 to 5, $C$ times ($C$
being an integer).

  2. 2.

Draw random sample of log-logistic numbers of size $N$ for given parameters
$\lambda,\beta$ for fixed truncation $x_{L}=1$ using Eq. (3) and
$\lambda=\alpha^{\beta}$.

  3. 3.

Calculate $\beta_{C}$ from Eq. (9) and $\beta_{0}$ from Eq. (10). If
$\beta_{0}\leq\beta_{C}$ then the critical point corresponding to a solution
to the maximum likelihood estimation problem is a Pareto distribution with
$\hat{\lambda}=0$ and $\hat{\beta}=\beta_{0}$ and finish at this step,
otherwise go to step 3.

  4. 4.

If $\beta_{0}>\beta_{C}$, then solve the equation Eq. (20) to obtain
$\hat{\beta}$ and $\hat{\lambda}=\Lambda(\hat{\beta})$ via Eq. (14)

  5. 5.

If $\hat{\lambda}>0$, compute KS and AD statistic like in Section 3.1, Eq.
(26) and Eq. (27) respecitvely and store values in lists and go back to 1.

  6. 6.

Sort list of KS and AD values in ascending order and find quantile in usual
way [8].

This has been repeated for various sample sizes.

The subsequent results have been obtained in [8]. Following the notation
therin and using our parameter $\eta$ from above, which is either zero in the
case of $x_{L}=0$ (untruncated case) or otherwise $\eta=1/\lambda$ with the
normalization of $x_{L}=1$, the quantity $p=F(x_{L}\leavevmode\nobreak\
|\leavevmode\nobreak\ \alpha,\beta)$ is introduced as a measure for the
“truncation percentage”, reading here as

$\displaystyle p=1-\frac{1}{1+\eta}=1-\frac{1}{1+\lambda^{-1}}\,.$

This truncation percentage tells us how many data have been “lost” by the
left-truncation and depends, with the normalisation $x_{L}=1$, only on the
scale parameter $\lambda$. Sample sizes have been chosen as
$N\in\\{30,50,100,200,500,1000,10000\\}$ and for sample size $N=10000$
altogether $C=100$ repetitions have been performed to keep $N\cdot C=1000000$.
Likewise, for $N=100$ $C=10000$ repetitions have been performed to get error
estimates for the obtained quantiles ([16, 17]). In Tables A – A we display
the critical values for the 95% confidence level for the KS and AD hypothesis
tests. Additional tables for other confidence levels are provided in Appendix
B. Extensive power testing has been done in Ref. [8]. We want to emphasize
here that our 95%-KS test is able to distinguish log-logistic distributions
from other fat-tailed distributions (such as Weibull, Mittag-Leffler and log-
normal) only for samples with sample size $N\geq 1000$, whereas the 95%-AD
test already performs well for sample size $N\geq 100$.

We summarise our results for various confidence levels in the following
interpolation formula for the test statistics $KS(\sqrt{N}D)$ and $AD(A^{2})$
respectively (see [8]):

$\displaystyle\text{Test
Statistics}=\frac{\theta_{1}\eta+\theta_{2}\eta^{1/2}+\theta_{3}}{\theta_{4}\eta^{1/2}+\theta_{5}+\eta}+\theta_{6}\left(\frac{\eta}{N}\right)^{1/2}+\theta_{7}\eta^{3/2}+\frac{\theta_{8}}{N^{1/2}}+\frac{\theta_{9}}{N}\quad.$
(28)

The values of the $\theta$’s are given in Table A in Appendix A. Note that we
take $\eta\simeq\hat{\eta}=1/\hat{\lambda}$ (with $x_{L}=1$).

Our critical values for $p=0$ may be compared with the corresponding critical
values given in Ref. [18] for the logistic distribution.

##  4 Applications to real world data

###  4.1 Bladder cancer data set from Lee and Wang (2003)

We apply our methods to the bladder cancer data set from Lee and Wang 2003
[19],Table 9.3, p. 231, which has been studied recently in Ref. [20] under the
hypothesis that the data are distributed according to a log-logistic
distribution. The original Table 9.3 contains a set of 137 remission times in
months from cancer patients. These remission times are subset of the data of a
bladder cancer study. As in [20], we have used only 128 data points from this
table and have taken out 9 censored observations (indicated by a “+” in the
original Table 9.3)

$\displaystyle
0.08,2.09,3.48,4.87,6.94,8.66,13.11,23.63,0.20,2.23,3.52,4.98,6.97,9.02,$
$\displaystyle
13.29,0.40,2.26,3.57,5.06,7.09,9.22,13.80,25.74,0.50,2.46,3.64,5.09,7.26,$
$\displaystyle
9.47,14.24,25.82,0.51,2.54,3.70,5.17,7.28,9.74,14.76,26.31,0.81,2.62,3.82,$
$\displaystyle
5.32,7.32,10.06,14.77,32.15,2.64,3.88,5.32,7.39,10.34,14.83,34.26,0.90,$
$\displaystyle
2.69,4.18,5.34,7.59,0.66,15.96,36.66,1.05,2.69,4.23,5.41,7.62,10.75,16.62,$
$\displaystyle
43.01,1.19,2.75,4.26,5.41,7.63,17.12,46.12,1.26,2.83,4.33,5.49,7.66,11.25,$
$\displaystyle
17.14,79.05,1.35,2.87,5.62,7.87,11.64,17.36,1.40,3.02,4.34,5.71,7.93,11.79,$
$\displaystyle
18.10,1.46,4.40,5.85,8.26,11.98,19.13,1.76,12.07,3.25,4.50,6.25,8.37,12.02,$
$\displaystyle
2.02,3.31,4.51,6.54,8.53,12.03,20.28,2.02,3.36,6.76,21.73,2.07,3.36,6.93,$
$\displaystyle 8.65,12.63,22.69$

Using our method we first estimate the parameters of the log-logistic
distribution and then perform KS and AD hypothesis testing using the
interpolation equation, Eq. (28), for the critical values. This was done for
the complete data set and various truncation points. Our results are displayed
in Table 1. Note that for the purpose of comparison we use the scale parameter
$\hat{\alpha}=\hat{\lambda}^{1/\hat{\beta}}$.

Table 1: Remission times for log-logistic distribution (95% confidence level) $x_{L}$ [m] | $N$ |  $\hat{\alpha}$ [m] | $\hat{\beta}$ | $\ln{L}$ |  KS ($\sqrt{N}D)$ |  AD $(A^{2})$ | KS | AD
---|---|---|---|---|---|---|---|---
0 | 128 | $5.97$ | $1.695$ | -410.89 | 0.4447 | 0.2684 | $\checkmark$ | $\checkmark$
0.25 | 126 | $6.11$ | $1.782$ | -402.20 | 0.4344 | 0.1657 | $\checkmark$ | $\checkmark$
1 | 120 | $6.32$ | $1.877$ | -379.28 | 0.4030 | 0.1253 | $\checkmark$ | $\checkmark$
6 | 64 | $8.63$ | $2.239$ | -206.00 | 0.5006 | 0.3086 | $\checkmark$ | $\checkmark$
12 | 31 | $8.36$ | $2.277$ | -103.85 | 0.4877 | 0.5129 | $\checkmark$ | $\checkmark$

We may compare our result without truncation with Ref. [20], which found
$\hat{\alpha}=6.08982$ months and $\hat{\beta}=1.725158$, with a log-
likelihood function value of $-411.4574$. These authors were using
“LLmodel.Optim() function in R with Newton-Raphson options […] as an iterative
process for maximizing the log-likelihood function”. Our likelihood value
being $-410.89$ is better and will lead to a better estimation of the
parameters $\hat{\alpha}$ and $\hat{\beta}$. Therefore our KS distance of
$(\sqrt{n}D)=0.4447$ is a more realistic value than theirs
$(\sqrt{n}D)=0.3629$.222Using their parameter values yields exactly this KS-
distance. Any difference is due to the R-optimizer.

We used four different truncation points and estimated the parameters and
performed the hypothesis testing. For all truncation points we cannot reject
the hypothesis of the data being log-logistic distributed. For small
truncation points (up to 5% of the data) the parameter estimates remain fairly
stable. Even for larger truncation points ( 75% of the data) our hypothesis
cannot be rejected.

###  4.2 Annual precipitation for Berlin and Toronto

In the literature the log-logistic distributions is often used to describe
precipitation data, for example Refs. [2] and [21]. For reason of homogeneity,
[2] focused on precipitation data of various Canadian cities. We have chosen
as an example the annual precipitation for the city-state of Berlin, because
of its small area the data are likely to be i.i.d. The data is available from
the German met office [22]. The annual precipitation data consists of 141 data
points starting in 1881 and extending to 2021. Also, to compare our method
with Ref. [2], we analyse the annual precipitation data for Toronto from 1
July to 30 June, as given in Ref. [23], from 1937 until 2021.

For Berlin our results are given in Table 2 and for Toronto in Table 3. Note
that in our analysis the shape parameters, $\hat{\beta}$, for both cities are
similar. We compare our results with [2], where for Toronto
$\hat{\alpha}=789.8$ mm and $\hat{\beta}=14.4$ were given. Whereas the
estimates of the scale parameter $\hat{\alpha}$ are in good agreement, the
values of $\hat{\beta}$ are different, but of similar order of magnitude. It
is interesting to note that for Toronto a truncation is needed to pass the
Anderson-Darling hypothesis test.

Table 2: Berlin precipitation data for log-logistic distribution (95% confidence level). $x_{L}$ [mm] | $N$ |  $\hat{\alpha}$ [mm] | $\hat{\beta}$ | $\ln{L}$ |  KS $(\sqrt{N}D)$ |  AD $(A^{2})$ | KS | AD
---|---|---|---|---|---|---|---|---
0 | 141 | $564.0$ | $11.8$ | -825.30 | 0.6255 | 0.4943 | $\checkmark$ | $\checkmark$
300 | 141 | $563.9$ | $11.8$ | -825.22 | 0.6286 | 0.5001 | $\checkmark$ | $\checkmark$
400 | 137 | $565.9$ | $12.1$ | -791.86 | 0.5908 | 0.4773 | $\checkmark$ | $\checkmark$
500 | 113 | $574.5$ | $13.6$ | -616.60 | 0.5319 | 0.2926 | $\checkmark$ | $\checkmark$
Table 3: Toronto precipitation data for log-logistic distribution (95% confidence level). $x_{L}$ [mm] | $N$ |  $\hat{\alpha}$ [mm] | $\hat{\beta}$ | $\ln{L}$ |  KS $(\sqrt{N}D)$ |  AD $(A^{2})$ | KS | AD
---|---|---|---|---|---|---|---|---
0 | 85 | $771.5$ | $11.6$ | -526.58 | 0.5549 | 0.6757 | $\checkmark$ | ✗
300 | 85 | $771.5$ | $11.6$ | -526.58 | 0.5551 | 0.6768 | $\checkmark$ | ✗
400 | 84 | $774.1$ | $12.3$ | -514.65 | 0.5456 | 0.6626 | $\checkmark$ | $\checkmark$
500 | 83 | $775.9$ | $12.7$ | -504.75 | 0.5803 | 0.5240 | $\checkmark$ | $\checkmark$

###  4.3 Outlook

The log-logistic distribution is heavier tailed than the log-normal
distribution and enjoys a closeness to the Pareto distribution. For modelling
the distribution of wealth and income a left-truncated version of the log-
logistic distribution [1] is a sensible approach, because often very low
incomes (below a certain threshold) in many economies are earned in a shadow
economy. Therefore they are not captured in official government statistics. We
have shown that the log-logistic hypothesis can explain certain cancer
survival data very well and accurately, even passing KS and AD hypothesis
tests. We can also confirm that in general the log-logistic distribution
describes precipitation data regardless of their region fairly well. It is
noteworthy that even in recent daily precipitation data there are days missing
resulting in a too “small” annual precipitation value if the missing data were
replaced by zero. While the effect of this might be neglegible for higher
annual precipitation it will have an effect for lower annual precipitation. A
remedy for the statistical analysis could be the introduction of a left-
truncation point to overcome this problem: we have seen that in this case both
KS and AD tests are passed.

In recent literature it was suggested that one might investigate various
modifications of the log-logistic distributions, such as, e.g., the Burr XII
family of distributions [25]-[27].

Also in the case of mixed distributions where one mixture component is light
tailed and the other is heavier tailed, such as log-logistic, our method will
allow us to estimate the parameters of the log-logistic component with a good
degree of accuracy. Thus, one has a tool at hand to decide whether or not a
two-component mixture contains a log-logistic component.

##  5 Proofs for theorems from Chapter 2

###  5.1 Proof of Theorem 1.

We first begin with a lemma dealing with “small” $\lambda$ for the objective
function given in Eq. (8). Keep in mind that we have chosen $x_{L}=1$.

######  Lemma 2.

For any $\beta>0$, there exists some $\lambda_{0}>0$ such that for
$\lambda\in[0,\lambda_{0})$ we have

$\displaystyle\varphi(\lambda,\beta)=\varphi(0,\beta)+N\left\\{-\frac{2}{N}\sum\limits_{i=1}^{N}\frac{1}{X_{i}^{\beta}}+1\right\\}\lambda+{\cal{O}}(\lambda^{2})$
(29)

where

$\displaystyle\varphi(0,\beta)=\lim_{\lambda\rightarrow
0+}\varphi(\lambda,\beta)=\sum\limits_{i=1}^{N}\left[-\beta\ln{X_{i}}+\ln{\beta}\right]\quad.$
(30)

Proof:

For fixed $\beta>0$ we obtain by differentiating Eq. (8) with respect to
$\lambda$

$\displaystyle\frac{\partial}{\partial\lambda}\leavevmode\nobreak\
\varphi(\lambda,\beta)$ $\displaystyle=$
$\displaystyle-\frac{N}{\lambda}\frac{\frac{1}{\lambda}}{1+\frac{1}{\lambda}}-\frac{N}{\lambda}+\frac{2}{\lambda}\sum\limits_{i=1}^{N}\frac{\frac{X_{i}^{\beta}}{\lambda}}{1+\frac{X_{i}^{\beta}}{\lambda}}$
(31) $\displaystyle=$
$\displaystyle\frac{N}{\lambda}\left\\{\frac{2}{N}\sum\limits_{i=1}^{N}\frac{1}{1+\frac{\lambda}{X_{i}^{\beta}}}-1-\frac{1}{1+\lambda}\right\\}$
$\displaystyle=$
$\displaystyle\frac{N}{\lambda}\left\\{-\frac{2}{N}\sum\limits_{i=1}^{N}\frac{1}{X_{i}^{\beta}}\leavevmode\nobreak\
\lambda+\lambda+{\cal{O}}(\lambda^{2})\right\\}\,,$

where for $\lambda\in(0,\lambda_{0})$ with $0<\lambda_{0}<1$ the geometric
series is absolutely convergent. Thus integrating Eq. (31) over
$\lambda^{\prime}$ from $0$ to $\lambda$ we get

$\displaystyle\varphi(\lambda,\beta)-\varphi(0,\beta)$ $\displaystyle=$
$\displaystyle\lim_{\varepsilon\rightarrow
0}\int\limits_{\varepsilon}^{\lambda}d\lambda^{\prime}\leavevmode\nobreak\
N\left\\{-\frac{2}{N}\sum\limits_{i=1}^{N}\frac{1}{X_{i}^{\beta}}+1+{\cal{O}}(\lambda^{\prime})\right\\}$
$\displaystyle=$ $\displaystyle
N\lambda\left\\{-\frac{2}{N}\sum\limits_{i=1}^{N}\frac{1}{X_{i}^{\beta}}+1\right\\}+{\cal{O}}(\lambda^{2})\,.$

The computation of $\varphi(0,\beta)=\lim_{\varepsilon\rightarrow
0}\varphi(\varepsilon,\beta)$ is straight forward. $\blacksquare$

In the next lemma we derive an anologous result like in [11] about the
behaviour of the objective function $\varphi(\lambda,\beta)$ in a certain
rectangle in the first quadrant of the $(\lambda,\beta)$-coordinate plane. The
situation is depicted in Fig. 2.

######  Lemma 3.

Consider an i.i.d. left-truncated sample $X_{1},...,X_{N}$ all bigger than one
and for which at least two observations are different. For any $M>0$ we have
the following inequalities valid for $\varphi(\lambda,\beta)$ defined in Eq.
(8)

  1. (1)

There exists $\beta_{1}>0$ such that for $\beta\in(0,\beta_{1})$ we have
$\varphi(\lambda,\beta)<-M$, independent of $\lambda$

  2. (2)

There exists $\beta_{2}>\beta_{1}$ such that for $\beta>\beta_{2}$ we have
$\varphi(\lambda,\beta)<-M$, independent of $\lambda$

  3. (3)

For $\beta\in[\beta_{1},\beta_{2}]$ there exists $\lambda_{1}>0$ such that for
$\lambda>\lambda_{1}$ we have also $\varphi(\lambda,\beta)<-M$

Proof:

(1): We write Eq. (8) as

$\displaystyle\varphi(\lambda,\beta)$ $\displaystyle=$ $\displaystyle
N\ln{\left(1+\frac{1}{\lambda}\right)}+N\ln{\beta}-N\ln{\lambda}+\beta S$ (32)
$\displaystyle-2\sum\limits_{i=1}^{N}\ln{\left[1+\frac{X_{i}^{\beta}}{\lambda}\right]}$
$\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{N}\ln{\frac{1+\frac{1}{\lambda}}{1+\frac{X_{i}^{\beta}}{\lambda}}}+N\ln{\beta}+\beta
S+\sum\limits_{i=1}^{N}\ln{\frac{\frac{1}{\lambda}}{1+\frac{X_{i}^{\beta}}{\lambda}}}$
$\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{N}\ln{\frac{\lambda+1}{\lambda+X_{i}^{\beta}}}+N\ln{\beta}+\beta
S+\sum\limits_{i=1}^{N}\ln{\frac{1}{\lambda+X_{i}^{\beta}}}$ $\displaystyle<$
$\displaystyle N\ln{\beta}+\beta S$

because $\lambda+X_{i}^{\beta}>\lambda+1>1$. From the bound Eq. (51) we see
that $\beta_{1}>0$ can be chosen independent of $\lambda\geq 0$ and the first
statement is proven.

(2): see proof in Appendix C

(3): For $\lambda>1$ and $\beta\in[\beta_{1},\beta_{2}]$ we have

$\displaystyle\varphi(\lambda,\beta)\leq
N\ln{\left(1+1\right)}+N\ln{\beta_{2}}-N\ln{\lambda}+\beta_{2}S$ (33)

because the last term in Eq. (8) is always negative and the argument of the
logarithm is always bigger than 1. From inequality Eq. (53) we see the
existance of a sufficiently large $\lambda_{1}>1$ such that the third
statement follows. $\blacksquare$

Now we have all the ingredients for the proof of Theorem 1.

Proof of Theorem 1.:

We start the proof by looking at Fig. 2.

Figure 2: The closed intervall $[\beta_{1},\beta_{2}]$ for the maximum of
$\varphi(\lambda,\beta)$.

For statement (1) apply Lemma 3 to conclude that outside the shaded area our
objective function $\varphi(\lambda,\beta)<-M$ for some arbitrarily large
$M>0$, i.e. our objective function gets arbitrarily small. Thus a global
maximum of the twice differentiable objective function $\varphi(\cdot,\cdot)$
is either inside the shaded region
$[0,\lambda_{1}]\times[\beta_{1},\beta_{2}]$ or on the boundary with
$\lambda=0$, somewhere in the closed interval $[\beta_{1},\beta_{2}]$. One
easily computes the only critical point and thus a candidate for a maximum of
the objective function on this boundary-line to be $\beta_{0}$, as given in
Eq. (10). By condition $\beta_{0}>\beta_{C}$ and Lemma 2 this can be excluded:
there exists some small $\lambda_{0}>0$ such that for
$\lambda\in(0,\lambda_{0})$

$\displaystyle\varphi(\lambda,\beta_{0})>\varphi(0,\beta_{0})$

and statement (1) is proven.

For statement (2) we use the following argument: by continuity there exists
$\epsilon>0$ (first line) and by Lemma 2 a positive $\lambda_{0}$ (second
line) such that

$\displaystyle\varphi(0,\beta_{0})$ $\displaystyle=$
$\displaystyle\max{\left\\{\varphi(0,\beta)|\beta\in[\beta_{0}-\epsilon,\beta_{0}+\epsilon]\right\\}}$
$\displaystyle\geq$
$\displaystyle\max{\left\\{\varphi(\lambda,\beta)|\beta\in[\beta_{0}-\epsilon,\beta_{0}+\epsilon],\lambda\in[0,\lambda_{0}]\right\\}}$

and thus $\beta_{0}$ is local maximum. $\blacksquare$

###  5.2 Proof of Theorem 2.

For $\lambda>0$ we start from the rewritten Eq. (6)

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{2}{N}\sum\limits_{i=1}^{N}\frac{X_{i}^{\beta}}{\lambda+X_{i}^{\beta}}-1-\frac{1}{\lambda+1}$
(34)

and investigate this equation Eq. (34) in greater detail. We firstly note the
following

######  Lemma 4.

Consider a left-truncated log-logistic sample $X_{1},...,X_{N}$ all bigger
than $1$. Define the function $q:\mathbb{R}_{+}\rightarrow(0,1)$ by

$\displaystyle
q\left(\beta|(X_{1},...,X_{N})\right)\equiv\frac{1}{N}\sum\limits_{i=1}^{N}\left(\frac{1}{X_{i}}\right)^{\beta}$

Then the following holds true

  1. (1)

$q(\cdot)$ is a monotonically decreasing function.

  2. (2)

The equation $q\left(\beta|(X_{1},...,X_{N})\right)=\frac{1}{2}$ has a unique
positive solution $\beta_{C}$.

  3. (3)

Equation Eq. (34) has for $\beta\in(0,\beta_{C}]$ only the trivial solution
$\hat{\lambda}=\Lambda(\beta)=0$ and for $\beta>\beta_{C}$ in addition exactly
one positive solution $\hat{\lambda}=\Lambda(\beta)>0$.

  4. (4)

For any fixed $\beta>0$ all the roots $\hat{\lambda}=\hat{\lambda}(\beta)$ of
the MLE equation Eq (34) depend continously on the sample $(X_{1},...,X_{N})$.

  5. (5)

For a fixed sample $(X_{1},...,X_{N})$ all the roots $\hat{\lambda}$ of the
MLE equation Eq (34) depend continously on $\beta\in(0,+\infty)$. In addition
we have $\lim_{\beta\rightarrow\beta_{C}+}\Lambda(\beta)=0$, where
$\Lambda(\beta)$ is defined in part (3) of the Lemma.

Proof:

(1) Note that $0<1/X_{i}<1...$ and therefore $(1/X_{i})^{\beta}$ is
monotonically decreasing and the first statement follows.

(2) Next note that because of

$\displaystyle\lim_{\beta\rightarrow 0+}q\left(\beta|(X_{1},...,X_{N})\right)$
$\displaystyle=$ $\displaystyle 1$
$\displaystyle\lim_{\beta\rightarrow\infty}q\left(\beta|(X_{1},...,X_{N})\right)$
$\displaystyle=$ $\displaystyle 0$

and the monotonicity of $q(\cdot)$ the second statment follows immediately.

(3) To prove the third statement we start from the likelihood equation, Eq.
(34), defining for fixed $\beta>0$ for convenience $Y_{i}=X_{i}^{\beta}$,

$\displaystyle
1+\frac{1}{\lambda+1}=\frac{2}{N}\sum\limits_{i=1}^{N}\frac{Y_{i}}{\lambda+Y_{i}}\,.$
(35)

Next define by the right-hand side of Eq. (35), the function

$\displaystyle
k(\lambda)=\frac{2}{N}\sum\limits_{i=1}^{N}\frac{Y_{i}}{\lambda+Y_{i}}$ (36)

which is monotonically decreasing in $\lambda\in[0,\infty)$, because

$\displaystyle
k^{\prime}(\lambda)=-\frac{2}{N}\sum\limits_{i=1}^{N}\frac{Y_{i}}{(\lambda+Y_{i})^{2}}<0\,.$
(37)

Afterwards define a function by the left-hand side of Eq. (35) as

$\displaystyle\ell(\lambda)=1+\frac{1}{\lambda+1}$ (38)

which is also monotonically decreasing in $\lambda\in[0,\infty)$, because

$\displaystyle\ell^{\prime}(\lambda)=-\frac{1}{(\lambda+1)^{2}}<0\,.$ (39)

Note that we have

$\displaystyle\lim_{\lambda\rightarrow 0+}k(\lambda)$ $\displaystyle=$
$\displaystyle 2$ $\displaystyle\lim_{\lambda\rightarrow+\infty}k(\lambda)$
$\displaystyle=$ $\displaystyle 0$

and also

$\displaystyle\lim_{\lambda\rightarrow 0+}\ell(\lambda)$ $\displaystyle=$
$\displaystyle 2$ $\displaystyle\lim_{\lambda\rightarrow+\infty}\ell(\lambda)$
$\displaystyle=$ $\displaystyle 1\quad.$

Thus the trivial solution $\hat{\lambda}=0$ is always a solution. We next
consider three cases:

Case I: $k^{\prime}(0)<\ell^{\prime}(0)$

In this case according to Eq. (37) and Eq. (39) the function $k(\cdot)$ falls
faster than the function $\ell(\cdot)$, at least for argument $\lambda$ near
0. If this is true for all $\lambda>0$, i.e.
$k^{\prime}(\lambda)<\ell^{\prime}(\lambda)$, then clearly there will not be
any intersection and $\hat{\lambda}=0$ is the only solution of Eq. (34). If
the latter is not true, then the situation will be as depicted in Fig. 3:

Figure 3: Case I: The functions $k(\lambda)$ (red) and $\ell(\lambda)$ (blue).

In this case $k(\cdot)$ initially falls quicker than $\ell(\cdot)$ and then
somehow slows down to intersect at a point $\lambda_{1}>0$ and possibly a
second time at a point $\lambda_{2}>\lambda_{1}$. Thus we have for
$\lambda_{1}>0$ that the function graph of $k(\cdot)$ crosses the function
graph of $\ell(\cdot)$ from below, meaning that the slopes satisfy the
following inequality

$\displaystyle k^{\prime}(\lambda_{1})>\ell^{\prime}(\lambda_{1})\,.$

By the intermediate value theorem there will exist a positive number
$\lambda^{*}\in(0,\lambda_{1})$ such that

$\displaystyle k^{\prime}(\lambda^{*})=\ell^{\prime}(\lambda^{*})\,.$ (40)

We shall show that this will lead to a contradiction as follows. We write the
Eq. (40) in detail as

$\displaystyle-\frac{2}{N}\sum\limits_{i=1}^{N}\frac{Y_{i}}{(\lambda^{*}+Y_{i})^{2}}=-\frac{1}{(\lambda^{*}+1)^{2}}$

or rearranging

$\displaystyle\frac{1}{N}\sum\limits_{i=1}^{N}{Y_{i}}\frac{(\lambda^{*}+1)^{2}}{(\lambda^{*}+Y_{i})^{2}}=\frac{1}{2}\quad.$
(41)

Define the function $h(\cdot):\mathbb{R_{+}}\rightarrow\mathbb{R_{+}}$ by

$\displaystyle
h(\lambda)=\frac{1}{N}\sum\limits_{i=1}^{N}{Y_{i}}\frac{(\lambda+1)^{2}}{(\lambda+Y_{i})^{2}}\quad.$
(42)

Thus Eq. (41) is equivalent to $h(\lambda^{*})=\frac{1}{2}$. Note that

$\displaystyle
h^{\prime}(\lambda)=\frac{1}{N}\sum\limits_{i=1}^{N}{Y_{i}}\frac{2(\lambda+1)(\lambda+Y_{i})-2(\lambda+1)^{2}}{(\lambda+Y_{i})^{3}}>0$
(43)

because all terms are positive. In particular, the numerator is positive
because $Y_{i}>1$. Note that we have the following limit

$\displaystyle\lim_{\lambda\rightarrow\infty}h(\lambda)$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum\limits_{i=1}^{N}{Y_{i}}>1$
$\displaystyle\lim_{\lambda\rightarrow 0+}h(\lambda)$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum\limits_{i=1}^{N}\frac{1}{Y_{i}}<1\,.$

Hence for the equation $h(\lambda)=\frac{1}{2}$ to have a solution in
$(0,\lambda_{1})$ we need to require that the second limit is smaller than
$\frac{1}{2}$, i.e.

$\displaystyle\frac{1}{N}\sum\limits_{i=1}^{N}\frac{1}{Y_{i}}<\frac{1}{2}\iff
k^{\prime}(0)>\ell^{\prime}(0)\,,$

which is a contradiction. Thus in Case I we have as the only real solution
$\hat{\lambda}=\lambda_{0}=0$. Note that Case I corresponds to
$\beta<\beta_{C}$.

Case II: $k^{\prime}(0)>\ell^{\prime}(0)$

Here the function $k(\cdot)$ decreases slower than the function $\ell(\cdot)$
for $\lambda$ near 0 and only here can the graph of $k(\cdot)$ intersect the
graph of $\ell(\cdot)$ from above at $\lambda_{1}>0$ before tending to zero as
$\lambda$ gets larger, as depicted in Fig. 4:

Figure 4: Case II: the functions $k(\lambda)$ (red) and $\ell(\lambda)$ (blue)
.

We need to show now that this intersection is unique. We note that at
$\lambda_{1}$ the graph of $k(\cdot)$ crosses the graph of $\ell(\cdot)$ from
above, i.e.

$\displaystyle k^{\prime}(\lambda_{1})<\ell^{\prime}(\lambda_{1})\quad.$ (44)

However, if there were a second crossing point, $\lambda_{2}>\lambda_{1}$, the
graph of $k(\cdot)$ would cross the graph of $\ell(\cdot)$ from below, i.e.

$\displaystyle k^{\prime}(\lambda_{2})>\ell^{\prime}(\lambda_{2})\,.$

As in Case I we conclude by the intermediate value theorem that there exists
$\lambda^{*}\in(\lambda_{1},\lambda_{2})$ such that

$\displaystyle k^{\prime}(\lambda^{*})=\ell^{\prime}(\lambda^{*})$

which is equivalent to

$\displaystyle\frac{1}{N}\sum\limits_{i=1}^{N}{Y_{i}}\frac{(\lambda^{*}+1)^{2}}{(\lambda^{*}+Y_{i})^{2}}=\frac{1}{2}$
(45)

and this in turn is similar to Eq. (41). We use the function $h(\cdot)$, as
defined in Eq. (42), to obtain the following limits

$\displaystyle\lim_{\lambda\rightarrow+\infty}h(\lambda)$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum\limits_{i=1}^{N}{Y_{i}}>1$
$\displaystyle\lim_{\lambda\rightarrow\lambda_{1}}h(\lambda)$ $\displaystyle>$
$\displaystyle\frac{1}{2}$

and Eq. (45) has no solution. The last limit follows from Eq. (44) and the
definition of $h(\cdot)$ from Eq. (45)

$\displaystyle-\frac{2}{N}\sum\limits_{i=1}^{N}\frac{Y_{i}}{(\lambda_{1}+Y_{i})^{2}}<-\frac{1}{(\lambda_{1}+1)^{2}}\iff
k^{\prime}(\lambda_{1})<\ell^{\prime}(\lambda_{1})\,.$

Thus there exists only in this case one positive root $\lambda_{1}$ to Eq.
(34). Case II corresponds to $\beta>\beta_{C}$.

Case III: $k^{\prime}(0)=\ell^{\prime}(0)$

Here one needs to look at the next higher derivatives, i.e.
$k^{\prime\prime}(0)>\ell^{\prime\prime}(0)$, which reads here as
$1<\frac{2}{N}\sum\limits_{i=1}^{N}\frac{1}{Y_{i}^{2}}$ or

$\displaystyle\frac{1}{2}<\frac{1}{N}\sum\limits_{i=1}^{N}\left(\frac{1}{Y_{i}}\right)^{2}=q(2\beta)\,.$

This cannot be true if

$\displaystyle\frac{1}{2}=\frac{1}{N}\sum\limits_{i=1}^{N}\left(\frac{1}{Y_{i}}\right)=q(\beta)\,,$

because of the properties of the above defined function $q(\cdot)$. Note that
Case III corresponds to $\beta=\beta_{C}$.Thus the third statement has been
proven.

(4) For the forth statement rewrite the MLE equation Eq. (34) as in Eq. (35)
and multiply both sides of the equation with
$(\lambda+1)\prod_{i=1}^{N}(\lambda+Y_{i})$. Thus after rearranging one
obtains a polynomial of $(N+1)$-st order in the variable $\lambda$

$\displaystyle\lambda^{N+1}+b_{N}(Y_{1},...,Y_{N})\lambda^{N}+...+b_{0}(Y_{1},...,Y_{N})=0$
(46)

where the coefficients $b_{i}(Y_{1},...Y_{N})$ are certain continous
polynomial functions in $(Y_{1},...Y_{N})$ tediously obtained in the process
of re-arranging terms and sorting by the powers of $\lambda$. From Appendix A
in Ref. [28], the roots $\hat{\lambda}$ of some algebraic equation (i.e. a
polynomial equation) depend continously on the coefficients and the statement
is proven.

(5) The last statement is shown in a similar manner to the previous statement
and because all $X_{i}^{\beta}$ are continous functions in $\beta$
respectively, the desired statement is proven because for $\beta\leq\beta_{C}$
the only real root is $\hat{\lambda}=0$. $\blacksquare$

The next Lemma provides some additional information about the function
$\Lambda(\cdot)$ defined implicitly by Eq. (34) via Lemma 4. (3).

######  Lemma 5.

Consider a left-truncated log-logistic sample $X_{1},...,X_{N}$ all bigger
than $1$. Then the non-zero solution of the MLE equation Eq. (34)
$\Lambda(\cdot):(\beta_{C},\infty)\rightarrow\mathbb{R^{+}}$ has the following
properties:

  1. (1)

For fixed $\beta$ $\Lambda(\beta)$ is a root of Eq. (34) with multiplicity 1.

  2. (2)

$\Lambda(\cdot)$ is an analytic function in the range of $(\beta_{C},\infty)$.

Proof:

Recall that $x_{L}=1$. From Eq. (5) respectively Eq. (8) we obtain for fixed
$\beta\in(\beta_{C},+\infty)$ by direct computation

$\displaystyle\left.\frac{\partial}{\partial\lambda}\leavevmode\nobreak\
\varphi(\lambda,\beta)\right|_{\Lambda(\beta)}$ $\displaystyle=$
$\displaystyle\frac{N}{\Lambda(\beta)}\left[\frac{2}{N}\sum\limits_{i=1}^{N}\frac{X_{i}^{\beta}}{\lambda+X_{i}^{\beta}}-1-\frac{1}{\lambda+1}\right]_{\Lambda(\beta)}=0$
$\displaystyle\left.\frac{\partial^{2}}{\partial\lambda^{2}}\leavevmode\nobreak\
\varphi(\lambda,\beta)\right|_{\Lambda(\beta)}$ $\displaystyle=$
$\displaystyle-\frac{N}{\Lambda(\beta)}\left[\frac{2}{N}\sum\limits_{i=1}^{N}\frac{X_{i}^{\beta}}{(\lambda+X_{i}^{\beta})^{2}}-\frac{1}{(\lambda+1)^{2}}\right]_{\Lambda(\beta)}$

where the first derivative is equal to zero for $\lambda=\Lambda(\beta)$ by
construction. Next we define a polynomial function
$p(\cdot):\mathbb{R}^{+}_{0}\rightarrow\mathbb{R}$ by

$\displaystyle p(\lambda)$ $\displaystyle\equiv$
$\displaystyle\left[\frac{2}{N}\sum\limits_{i=1}^{N}\frac{X_{i}^{\beta}}{\lambda+X_{i}^{\beta}}-\frac{\lambda+2}{\lambda+1}\right]\,(\lambda+1)\prod_{j=1}^{N}(\lambda+X_{j}^{\beta})$
(47) $\displaystyle=$
$\displaystyle\frac{2}{N}\sum\limits_{i=1}^{N}X_{i}^{\beta}(\lambda+1)\prod_{1\leq
j\leq N\atop j\neq
i}(\lambda+X_{j}^{\beta})-(\lambda+2)\prod_{j=1}^{N}(\lambda+X_{j}^{\beta})$

The first claim of the Lemma states that $\Lambda(\beta)$ is a root of
$p(\cdot)$, Eq. (47), but not of $p^{\prime}(\cdot)$. By construction we have
indeed $p(\Lambda(\beta))=0$. We show that the assumption
$p^{\prime}(\Lambda(\beta))=0$ leads to a contradiction. Compute the
derivative in Eq. (47)

$\displaystyle p^{\prime}(\lambda)$ $\displaystyle=$
$\displaystyle\frac{2}{N}\sum\limits_{i=1}^{N}X_{i}^{\beta}\prod_{1\leq j\leq
N\atop j\neq
i}(\lambda+X_{j}^{\beta})+\frac{2}{N}\sum\limits_{i=1}^{N}X_{i}^{\beta}(\lambda+1)\sum\limits_{k=1}^{N}\prod_{1\leq
j\leq N\atop j\neq i,j\neq k}(\lambda+X_{j}^{\beta})$
$\displaystyle-\prod_{j=1}^{N}(\lambda+X_{j}^{\beta})-(\lambda+2)\sum\limits_{k=1}^{N}\prod_{1\leq
j\leq N\atop j\neq k}(\lambda+X_{j}^{\beta})\quad.$

Then write down
$p^{\prime}(\Lambda(\beta))/\prod_{j}(\Lambda(\beta)+X_{j}^{\beta})$, which by
our asumption $p^{\prime}(\Lambda(\beta))=0$ vanishes

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{2}{N}\sum\limits_{i=1}^{N}\frac{X_{i}^{\beta}}{\Lambda(\beta)+X_{i}^{\beta}}+\frac{2}{N}(\Lambda(\beta)+1)\sum\limits_{i=1}^{N}\frac{X_{i}^{\beta}}{\Lambda(\beta)+X_{i}^{\beta}}\sum\limits_{k=1}^{N}\frac{1}{\Lambda(\beta)+X_{k}^{\beta}}$
$\displaystyle-1-(\Lambda(\beta)+2)\sum\limits_{k=1}^{N}\frac{1}{\Lambda(\beta)+X_{k}^{\beta}}$
$\displaystyle=$
$\displaystyle\frac{2}{N}\sum\limits_{i=1}^{N}\frac{X_{i}^{\beta}}{\Lambda(\beta)+X_{i}^{\beta}}\left[1+(\Lambda(\beta)+1)\sum\limits_{k=1}^{N}\frac{1}{\Lambda(\beta)+X_{k}^{\beta}}\right]$
$\displaystyle-1-(\Lambda(\beta)+2)\sum\limits_{k=1}^{N}\frac{1}{\Lambda(\beta)+X_{k}^{\beta}}$
$\displaystyle=$
$\displaystyle\frac{\Lambda(\beta)+2}{\Lambda(\beta)+1}\left[1+(\Lambda(\beta)+1)\sum\limits_{k=1}^{N}\frac{1}{\Lambda(\beta)+X_{k}^{\beta}}\right]$
$\displaystyle-1-(\Lambda(\beta)+2)\sum\limits_{k=1}^{N}\frac{1}{\Lambda(\beta)+X_{k}^{\beta}}$
$\displaystyle=$
$\displaystyle\frac{\Lambda(\beta)+2}{\Lambda(\beta)+1}-1=\frac{1}{\Lambda(\beta)+1}$

which is a contradiction, because $\Lambda(\beta)>0$ and thus the first claim
is proven. Hence for $\beta\in(\beta_{C},+\infty)$ fixed and any critical
point $\Lambda(\beta)$ of $\varphi(\cdot,\beta)$ we have always

$\displaystyle\left.\frac{\partial^{2}}{\partial\lambda^{2}}\varphi(\lambda,\beta)\right|_{\Lambda(\beta)}$
$\displaystyle\neq$ $\displaystyle 0$

We know that for a maximum the second derivative is negative, so we have
always

$\displaystyle\left.\frac{\partial^{2}}{\partial\lambda^{2}}\varphi(\lambda,\beta)\right|_{\Lambda(\beta)}$
$\displaystyle<$ $\displaystyle 0$

from which we conclude that for all $\beta>\beta_{C}$

$\displaystyle\left[\frac{2}{N}\sum\limits_{i=1}^{N}\frac{X_{i}^{\beta}}{(\lambda+X_{i}^{\beta})^{2}}-\frac{1}{(\lambda+1)^{2}}\right]_{\Lambda(\beta)}>0$
(48)

For the second claim we note that by its definition as root of multiplicity 1
the function $\Lambda(\cdot)$ is analytic for argument $\beta>\beta_{C}$ by
simple application of the implicit function theorem (e.g. [28])

and the proof is finished. $\blacksquare$

Now we have all the ingredients for the proof of Theorem 2.

Proof of Theorem 2.:

Lemma 4. (3) contains already the main statements concerning the unique
construction of the function
$\Lambda(\cdot):\mathbb{R}_{0}^{+}\rightarrow\mathbb{R}^{+}_{0}$ . The
behaviour of the derivatives
$\frac{\partial}{\partial\lambda}\varphi(\lambda,\beta)$ for
$\lambda\neq\Lambda(\beta)$ follows from the easily obtained asymptotics for
$\lambda\rightarrow\infty$:

$\displaystyle\frac{\partial}{\partial\lambda}\varphi(\lambda,\beta)=-\frac{N}{\lambda}\left\\{1++{\cal{O}}\left(\frac{1}{\lambda}\right)\right\\}$

and Lemma 2 for small $\lambda\geq 0$ and the fact that
$\frac{\partial}{\partial\lambda}\varphi(\lambda,\beta)$ has only one
zero.$\blacksquare$

The next Lemma treats the situation where the log-logistic distribution
degenerates when $\lambda\rightarrow 0+$. We formulate it for the convencience
of general $x_{L}>0$.

######  Lemma 6.

Consider the left-truncated log-logistic distribution given in Eq. (1) with
$x_{L}>0$. In the limit $\lambda\rightarrow 0+$ respectively
$\alpha\rightarrow 0+$ the following holds true

  1. (1)

There is $L^{1}$-convergence in the sense of Lebesgue of

$\displaystyle
f_{LT}(\cdot|\alpha,\beta)\xrightarrow{\text{$L^{1}$}}g(\cdot|\beta)\hskip
28.45274pt\text{as}\hskip 14.22636pt\alpha\rightarrow 0+$

where

$\displaystyle
g(x|\beta)=\left(\frac{\beta}{x_{L}}\right)\left(\frac{x}{x_{L}}\right)^{-(1+\beta)}$
(49)

is the Pareto probability distribution density.

  2. (2)

While the first ML equation for $\alpha$ respectively $\lambda$, Eq. (6),
becomes redunant, the second ML equation for $\beta$, Eq. (7) simplifies to

$\displaystyle\frac{1}{\beta}=\frac{1}{N}\sum\limits_{i=1}^{N}\ln{\left(\frac{X_{i}}{x_{L}}\right)}$
(50)

Proof:

(1) To prove statement (1) we first we show pointwise convergence for some
$x>x_{L}$ fixed, namly if we chose any arbitrary positive sequence
$(\alpha_{n})_{n\in\mathbb{N}}\rightarrow 0+$ as $n\rightarrow\infty$

$\displaystyle\left[1+\left(\frac{x_{L}}{\alpha_{n}}\right)^{\beta}\right]\frac{\beta}{\alpha_{n}}\left(\frac{x}{\alpha_{n}}\right)^{\beta-1}\frac{1}{\left[1+\left(\frac{x}{\alpha_{n}}\right)^{\beta}\right]^{2}}$
$\displaystyle=$
$\displaystyle\left(\frac{x_{L}}{\alpha_{n}}\right)^{\beta}\frac{\beta}{\alpha_{n}}\left(\frac{x}{\alpha_{n}}\right)^{\beta-1}\left(\frac{x}{\alpha_{n}}\right)^{-2\beta}+\hskip
7.11317pt\mathcal{O}(\alpha_{n}^{2\beta})$
$\displaystyle\xrightarrow{n\rightarrow\infty}$
$\displaystyle\left(\frac{\beta}{x_{L}}\right)\left(\frac{x}{x_{L}}\right)^{-\beta-1}=g(x|\beta)\quad.$

Next we need to show that the functions $f_{LT}(\cdot|\alpha_{n},\beta)$ can
be bounded by a positive integrable function. We have decided for the
following bound

$\displaystyle 0<f_{LT}(x|\alpha_{n},\beta)<c\cdot g(x|\beta)$

where the constant $c>0$ needs to be chosen in a suitable way. To this end we
start from the following

$\displaystyle\left[1+\left(\frac{x_{L}}{\alpha_{n}}\right)^{\beta}\right]\frac{\beta}{\alpha_{n}}\left(\frac{x}{\alpha_{n}}\right)^{\beta-1}\frac{1}{\left[1+\left(\frac{x}{\alpha_{n}}\right)^{\beta}\right]^{2}}$
$\displaystyle<$ $\displaystyle
c\left(\frac{\beta}{x_{L}}\right)\left(\frac{x}{x_{L}}\right)^{-(1+\beta)}$

which can be reduced to

$\displaystyle\frac{\alpha_{n}^{\beta}+x_{L}^{\beta}}{x_{L}^{\beta}}\left(\frac{x^{\beta}}{\alpha_{n}^{\beta}+x^{\beta}}\right)^{2}<c\,.$

Now there exists $n_{0}>0$ such that
$(\alpha_{n}^{\beta}+x_{L}^{\beta})/x_{L}^{\beta}\leq 2$ for $n>n_{0}$.
Because $x^{\beta}/(\alpha_{n}^{\beta}+x^{\beta})<1$ we may chose

$\displaystyle c=\max{\left\\{2,\max_{0\leq n\leq
n_{0}}\left(1+\frac{\alpha_{n}^{\beta}}{x_{L}^{\beta}}\right)\right\\}}$

The desired convergence result in $L^{1}$-Lebesgue sense follows from the
Lebesgue dominated convergence theorem.

(2) To prove statement (2) note that in Eq. (7) as $\alpha_{n}\rightarrow 0+$
the leading term on the right-hand side is
$1/\beta+\ln{\left(x_{L}/\alpha_{n}\right)}$ whereas the leading term on the
left-hand side is
$\sum\limits_{i=1}^{N}\ln{\left(X_{i}/\alpha_{n}\right)}\cdot 1$ and re-
arranging terms and performing the limit yields the desired result.
$\blacksquare$

###  5.3 Proof of Corollary 1.

Proof:

Part (1) is an immediate consequence of our Theorems. The proof of statement
(2) is provided by the Lemma 6. $\blacksquare$

###  5.4 Proof of Theorem 3.

Proof:

(1): The expressions follow immediately from Theorem 1. Note that $X_{1}>1$
throughout.

(2): Because $\lambda=\Lambda(\beta)>0$ for $\beta>\beta_{C}$ we can write Eq.
(6) as

$\displaystyle
0=2\frac{X_{1}^{\beta}}{\lambda+X_{1}^{\beta}}-1-\frac{1}{\lambda+1}$

Now inserting $\lambda=\Lambda(\beta)=X_{1}^{\beta}-2$ on the right-hand side
yields indeed

$\displaystyle
2\frac{X_{1}^{\beta}}{2(X_{1}^{\beta}-1)}-1-\frac{1}{X_{1}^{\beta}-1}=\frac{X_{1}^{\beta}}{X_{1}^{\beta}-1}-\frac{X_{1}^{\beta}-1}{X_{1}^{\beta}-1}-\frac{1}{X_{1}^{\beta}-1}=0\,.$

ad (3): We see continuity by straight-forward computation

$\displaystyle\lim_{\beta\rightarrow\beta_{C}-}\tilde{\varphi}(\beta)=\lim_{\beta\rightarrow\beta_{C}+}\tilde{\varphi}(\beta)=N\ln{\beta_{C}}-N\ln{2}\quad.$

For monotonicity we differentiate the profile likelihood with respect to
$\beta$. For $\beta\in(0,\beta_{C})$ we obtain

$\displaystyle\tilde{\varphi}^{\prime}(\beta)=N\left(\frac{1}{\beta}-\ln{X_{1}}\right)>0$

because $\beta<\beta_{C}=\ln{2}/\ln{X_{1}}<1/\ln{X_{1}}$. Finally for
$\beta\in(\beta_{C},+\infty)$ we obtain after some easy computation

$\displaystyle\tilde{\varphi}^{\prime}(\beta)=N\left(\frac{1}{\beta}-\frac{\ln{X_{1}}}{X_{1}^{\beta}-1}\right)=\left(\frac{1}{\beta}-\frac{1}{\beta_{C}}\frac{\ln{2}}{X_{1}^{\beta}-1}\right)>0$

where the last inequality follows from the inequality

$\displaystyle\beta_{C}(X_{1}^{\beta}-1)>\beta\ln{2}\,.$

To see this define two functions $F(\beta)=\beta_{C}(X_{1}^{\beta}-1)$ and
$G(\beta)=\beta\ln{2}$. Obviously we have
$F(\beta_{C})=\beta_{C}>G(\beta_{C})=\beta_{C}\ln{2}$. With the derivatives
$F^{\prime}(\beta)=\beta_{C}X_{1}^{\beta}\ln{X_{1}}=X_{1}^{\beta}\ln{2}>G^{\prime}(\beta)=\ln{2}$
we obtain the desired result by

$\displaystyle
F(\beta)-G(\beta)=F(\beta_{C})-G(\beta_{C})+\int\limits_{\beta_{C}}^{\beta}db\leavevmode\nobreak\
\left(F^{\prime}(b)-G^{\prime}(b)\right)>0$

because all terms on the right-hand side are positive. $\blacksquare$

##  Appendix A Tables for $95\%$ confidence levels

left-truncated log-logistic KS($\sqrt{N}D$) significance level 95% p
$\sqrt{\eta}$ 30 50 100 200 500 1000 10000 0 0 0.7661(06) 0.7774(07)
0.7860(07) 0.7916(08) 0.7976(07) 0.8002(07) 0.8039(07) 0.0323 0.18 0.7602(06)
0.7711(07) 0.7799(07) 0.7871(07) 0.7915(06) 0.7942(08) 0.7981(06) 0.1 0.33
0.7640(07) 0.7746(07) 0.7840(07) 0.7898(07) 0.7952(07) 0.7982(07) 0.8017(07)
0.2 0.5 0.7705(06) 0.7809(06) 0.7912(07) 0.7972(06) 0.8017(07) 0.8048(08)
0.8095(08) 0.3 0.65 0.7761(07) 0.7869(07) 0.7975(08) 0.8032(07) 0.8090(06)
0.8117(07) 0.8148(07) 0.4 0.82 0.7795(07) 0.7919(07) 0.8024(08) 0.8094(08)
0.8144(07) 0.8171(08) 0.8211(07) 0.5 1 0.7832(07) 0.7952(08) 0.8071(07)
0.8138(07) 0.8192(07) 0.8210(07) 0.8253(07) 0.6 1.22 0.7843(06) 0.7981(07)
0.8098(07) 0.8164(07) 0.8224(07) 0.8248(07) 0.8292(07) 0.7 1.53 0.7865(07)
0.7991(07) 0.8124(08) 0.8205(07) 0.8258(07) 0.8287(08) 0.8326(07) 0.8 2
0.7880(07) 0.8015(07) 0.8146(08) 0.8215(07) 0.8292(07) 0.8319(07) 0.8352(08)
0.8605 2.48 0.7885(07) 0.8015(08) 0.8144(08) 0.8227(07) 0.8301(07) 0.8328(08)
0.8375(07) 0.9 3 0.7884(07) 0.8028(07) 0.8157(08) 0.8242(08) 0.8310(07)
0.8339(08) 0.8393(07) left-truncated log-logistic AD($A^{2}$) significance
level 95%  p $\sqrt{\eta}$ 30 50 100 200 500 1000 10000 0 0 0.6594(12)
0.6614(14) 0.6632(12) 0.6634(15) 0.6647(13) 0.6653(14) 0.6652(13) 0.0323 0.18
0.6580(12) 0.6589(14) 0.6572(12) 0.6598(13) 0.6592(12) 0.6595(13) 0.6589(13)
0.1 0.33 0.6695(13) 0.6689(14) 0.6707(13) 0.6699(12) 0.6699(13) 0.6705(13)
0.6698(13) 0.2 0.5 0.6857(14) 0.6845(14) 0.6851(13) 0.6855(13) 0.6849(14)
0.6865(14) 0.6869(13) 0.3 0.65 0.6972(12) 0.6987(13) 0.6995(14) 0.6995(15)
0.7006(13) 0.6996(14) 0.6996(15) 0.4 0.82 0.7055(14) 0.7091(15) 0.7108(16)
0.7113(13) 0.7110(15) 0.7114(15) 0.7127(15) 0.5 1 0.7133(14) 0.7166(15)
0.7203(14) 0.7224(15) 0.7217(13) 0.7224(15) 0.7222(16) 0.6 1.22 0.7161(14)
0.7235(16) 0.7280(15) 0.7296(14) 0.7319(14) 0.7312(14) 0.7317(15) 0.7 1.53
0.7219(14) 0.7270(15) 0.7340(15) 0.7376(15) 0.7392(15) 0.7400(15) 0.7388(14)
0.8 2 0.7231(16) 0.7316(16) 0.7394(15) 0.7403(15) 0.7460(16) 0.7479(15)
0.7468(15) 0.8605 2.48 0.7256(15) 0.7314(16) 0.7400(13) 0.7440(16) 0.7496(16)
0.7513(18) 0.7523(15) 0.9 3 0.7252(17) 0.7345(16) 0.7399(17) 0.7469(16)
0.7507(16) 0.7525(14) 0.7541(16) Interpolation formula Eq.(28): coefficients
for different significance levels s.l. $\theta_{1}$ $\theta_{2}$ $\theta_{3}$
$\theta_{4}$ $\theta_{5}$ $\theta_{6}$ $\theta_{7}$ $\theta_{8}$ $\theta_{9}$
AD ($A^{2}$) 85% 0.5644 -0.0026 0.1307 0.0406 0.2612 -0.0432 0.0001 0.0350
-0.1715 AD ($A^{2}$) 90% 0.6390 -0.0005 0.1540 0.0361 0.2750 -0.0497 0.0000
0.0398 -0.2066 AD ($A^{2}$) 95% 0.7669 -0.0189 0.1927 0.0094 0.2914 -0.0641
0.0001 0.0494 -0.2364 AD ($A^{2}$) 99% 1.0714 -0.0589 0.2933 0.0301 0.3272
-0.0943 0.0002 0.0598 -0.1787 KS ($\sqrt{n}D$) 85% 0.7421 -0.0492 0.1565
-0.0517 0.2210 -0.0238 0.0000 0.1492 -0.2115 KS ($\sqrt{n}D$) 90% 0.7821
-0.0818 0.1742 -0.0917 0.2336 -0.0298 0.0001 0.1500 -0.2723 KS ($\sqrt{n}D$)
95% 0.8443 -0.1204 0.1987 -0.1318 0.2470 -0.0298 0.0001 0.1417 -0.3753 KS
($\sqrt{n}D$) 99% 0.9711 -0.1729 0.2672 -0.1687 0.2895 -0.0392 0.0001 0.1416
-0.5325

##  Appendix B Tables for other confidence levels

left-truncated log-logistic KS($\sqrt{N}D$) significance level 85% p
$\sqrt{\eta}$ 30 50 100 200 500 1000 10000 0 0 0.6740(04) 0.6830(05)
0.6907(05) 0.6961(05) 0.7012(04) 0.7036(04) 0.7072(05) 0.0323 0.18 0.6687(05)
0.6775(04) 0.6854(04) 0.6913(05) 0.6961(04) 0.6982(05) 0.7019(04) 0.1 0.33
0.6719(04) 0.6806(05) 0.6891(04) 0.6941(05) 0.6993(04) 0.7022(04) 0.7053(05)
0.2 0.5 0.6774(04) 0.6864(04) 0.6943(04) 0.7000(05) 0.7047(05) 0.7075(05)
0.7117(05) 0.3 0.65 0.6818(04) 0.6909(05) 0.6997(05) 0.7056(05) 0.7104(04)
0.7130(04) 0.7162(05) 0.4 0.82 0.6846(04) 0.6948(05) 0.7040(06) 0.7098(05)
0.7145(05) 0.7171(05) 0.7208(05) 0.5 1 0.6875(05) 0.6974(05) 0.7078(04)
0.7139(05) 0.7184(05) 0.7207(05) 0.7246(05) 0.6 1.22 0.6889(05) 0.7000(04)
0.7098(05) 0.7163(04) 0.7218(05) 0.7239(06) 0.7278(04) 0.7 1.53 0.6904(05)
0.7011(05) 0.7119(05) 0.7189(05) 0.7240(05) 0.7269(05) 0.7306(05) 0.8 2
0.6916(05) 0.7025(05) 0.7139(04) 0.7200(05) 0.7268(05) 0.7294(05) 0.7331(05)
0.8605 2.48 0.6920(04) 0.7032(04) 0.7140(05) 0.7211(05) 0.7275(04) 0.7305(05)
0.7344(05) 0.9 3 0.6922(05) 0.7038(04) 0.7140(05) 0.7220(05) 0.7281(05)
0.7309(05) 0.7357(05) left-truncated log-logistic KS($\sqrt{N}D$) significance
level 90% p $\sqrt{\eta}$ 30 50 100 200 500 1000 10000 0 0 0.7101(05)
0.7199(06) 0.7280(05) 0.7334(06) 0.7388(05) 0.7412(05) 0.7450(06) 0.0323 0.18
0.7047(05) 0.7139(05) 0.7225(05) 0.7286(05) 0.7334(05) 0.7356(06) 0.7394(05)
0.1 0.33 0.7079(05) 0.7174(05) 0.7262(05) 0.7316(06) 0.7368(05) 0.7398(05)
0.7429(06) 0.2 0.5 0.7139(05) 0.7236(06) 0.7322(05) 0.7381(05) 0.7428(06)
0.7457(06) 0.7496(05) 0.3 0.65 0.7186(05) 0.7285(05) 0.7380(06) 0.7441(06)
0.7490(05) 0.7514(05) 0.7548(05) 0.4 0.82 0.7219(05) 0.7331(05) 0.7425(06)
0.7486(06) 0.7540(06) 0.7564(06) 0.7600(06) 0.5 1 0.7251(05) 0.7357(05)
0.7468(05) 0.7531(06) 0.7579(05) 0.7604(06) 0.7639(06) 0.6 1.22 0.7266(05)
0.7385(05) 0.7489(06) 0.7554(06) 0.7611(06) 0.7638(06) 0.7674(05) 0.7 1.53
0.7282(05) 0.7397(06) 0.7511(06) 0.7586(06) 0.7639(06) 0.7669(06) 0.7705(05)
0.8 2 0.7295(06) 0.7410(05) 0.7533(05) 0.7599(06) 0.7670(05) 0.7697(05)
0.7734(05) 0.8605 2.48 0.7300(05) 0.7417(05) 0.7532(05) 0.7609(06) 0.7676(06)
0.7707(06) 0.7747(05) 0.9 3 0.7299(05) 0.7424(05) 0.7539(06) 0.7622(06)
0.7684(06) 0.7714(05) 0.7763(06) left-truncated log-logistic KS($\sqrt{N}D$)
significance level 99% p $\sqrt{\eta}$ 30 50 100 200 500 1000 10000 0 0
0.8780(11) 0.8918(14) 0.9015(15) 0.9093(13) 0.9160(14) 0.9180(13) 0.9227(14)
0.0323 0.18 0.8715(12) 0.8858(12) 0.8958(14) 0.9041(14) 0.9100(13) 0.9124(12)
0.9171(14) 0.1 0.33 0.8754(14) 0.8884(14) 0.9005(13) 0.9078(13) 0.9128(13)
0.9165(13) 0.9206(13) 0.2 0.5 0.8829(11) 0.8956(14) 0.9081(12) 0.9167(12)
0.9216(12) 0.9236(13) 0.9292(14) 0.3 0.65 0.8895(13) 0.9037(14) 0.9163(14)
0.9224(13) 0.9293(14) 0.9326(13) 0.9360(14) 0.4 0.82 0.8933(14) 0.9110(13)
0.9225(14) 0.9309(13) 0.9364(12) 0.9390(15) 0.9436(13) 0.5 1 0.8982(13)
0.9136(14) 0.9273(14) 0.9361(15) 0.9421(14) 0.9432(13) 0.9480(15) 0.6 1.22
0.9000(13) 0.9157(15) 0.9300(13) 0.9383(14) 0.9470(13) 0.9484(14) 0.9538(14)
0.7 1.53 0.9024(12) 0.9195(15) 0.9341(13) 0.9438(15) 0.9508(15) 0.9532(15)
0.9573(13) 0.8 2 0.9037(14) 0.9210(12) 0.9375(13) 0.9452(15) 0.9530(13)
0.9574(14) 0.9616(15) 0.8605 2.48 0.9042(14) 0.9205(14) 0.9381(16) 0.9477(15)
0.9560(13) 0.9591(16) 0.9641(11) 0.9 3 0.9040(11) 0.9238(12) 0.9375(13)
0.9480(15) 0.9562(15) 0.9598(12) 0.9666(15)

left-truncated log-logistic AD($A^{2}$) significance level 85% p $\sqrt{\eta}$
30 50 100 200 500 1000 10000 0 0 0.4998(07) 0.5010(07) 0.5011(07) 0.5013(07)
0.5021(07) 0.5021(07) 0.5021(07) 0.0323 0.18 0.4973(07) 0.4974(07) 0.4964(07)
0.4968(08) 0.4969(06) 0.4976(07) 0.4962(06) 0.1 0.33 0.5031(07) 0.5030(07)
0.5040(06) 0.5032(07) 0.5035(07) 0.5038(07) 0.5037(07) 0.2 0.5 0.5120(07)
0.5125(07) 0.5133(07) 0.5131(06) 0.5126(07) 0.5142(07) 0.5137(08) 0.3 0.65
0.5190(07) 0.5197(08) 0.5209(08) 0.5213(08) 0.5218(07) 0.5220(07) 0.5216(07)
0.4 0.82 0.5233(07) 0.5266(07) 0.5286(08) 0.5289(07) 0.5289(08) 0.5292(08)
0.5294(08) 0.5 1 0.5283(07) 0.5314(09) 0.5340(07) 0.5353(08) 0.5356(07)
0.5357(08) 0.5357(08) 0.6 1.22 0.5309(08) 0.5352(07) 0.5384(07) 0.5406(08)
0.5413(09) 0.5406(08) 0.5413(08) 0.7 1.53 0.5332(07) 0.5379(09) 0.5419(08)
0.5442(08) 0.5457(09) 0.5461(09) 0.5460(08) 0.8 2 0.5349(08) 0.5401(08)
0.5453(08) 0.5462(08) 0.5496(08) 0.5516(08) 0.5508(08) 0.8605 2.48 0.5361(08)
0.5404(09) 0.5461(08) 0.5483(08) 0.5516(09) 0.5536(08) 0.5537(09) 0.9 3
0.5360(09) 0.5422(08) 0.5462(09) 0.5497(09) 0.5526(09) 0.5541(10) 0.5549(09)
left-truncated log-logistic AD($A^{2}$) significance level 90% p $\sqrt{\eta}$
30 50 100 200 500 1000 10000 0 0 0.5589(08) 0.5601(09) 0.5608(08) 0.5616(09)
0.5620(08) 0.5629(08) 0.5628(09) 0.0323 0.18 0.5568(08) 0.5570(09) 0.5560(08)
0.5570(09) 0.5572(08) 0.5574(09) 0.5561(09) 0.1 0.33 0.5643(09) 0.5641(09)
0.5657(09) 0.5647(09) 0.5648(09) 0.5657(09) 0.5651(09) 0.2 0.5 0.5756(08)
0.5759(10) 0.5768(09) 0.5763(09) 0.5757(09) 0.5776(09) 0.5769(10) 0.3 0.65
0.5839(09) 0.5848(10) 0.5865(10) 0.5868(10) 0.5875(09) 0.5873(10) 0.5874(09)
0.4 0.82 0.5897(09) 0.5933(10) 0.5955(10) 0.5956(09) 0.5957(09) 0.5960(10)
0.5971(11) 0.5 1 0.5960(09) 0.5987(10) 0.6025(09) 0.6043(10) 0.6040(09)
0.6043(11) 0.6039(10) 0.6 1.22 0.5985(09) 0.6040(11) 0.6076(11) 0.6096(09)
0.6112(11) 0.6106(10) 0.6104(10) 0.7 1.53 0.6022(09) 0.6070(10) 0.6122(10)
0.6155(09) 0.6165(11) 0.6174(11) 0.6169(10) 0.8 2 0.6038(11) 0.6102(11)
0.6165(11) 0.6173(10) 0.6220(12) 0.6234(11) 0.6224(10) 0.8605 2.48 0.6056(09)
0.6106(11) 0.6171(10) 0.6201(11) 0.6235(11) 0.6255(11) 0.6261(11) 0.9 3
0.6053(11) 0.6123(10) 0.6170(11) 0.6218(11) 0.6247(10) 0.6268(12) 0.6278(11)
left-truncated log-logistic AD($A^{2}$) significance level 99% p $\sqrt{\eta}$
30 50 100 200 500 1000 10000 0 0 0.8917(29) 0,8956(30) 0,8981(28) 0,9035(30)
0,8998(31) 0,9050(28) 0,9027(32) 0,0323 0,18 0,8962(27) 0,8975(32) 0,8959(29)
0,8984(30) 0,8979(29) 0,8984(28) 0,8999(29) 0,1 0,33 0,9199(30) 0,9163(32)
0,9210(32) 0,9177(33) 0,9168(32) 0,9183(31) 0,9161(34) 0,2 0,5 0,9473(32)
0,9461(34) 0,9462(31) 0,9424(35) 0,9407(30) 0,9422(34) 0,9465(33) 0,3 0,65
0,9716(38) 0,9701(37) 0,9657(35) 0,9656(30) 0,9682(26) 0,9688(33) 0,9657(34)
0,4 0,82 0,9831(33) 0,9893(37) 0,9893(35) 0,9864(33) 0,9885(30) 0,9885(35)
0,9853(32) 0,5 1 0,9979(35) 1,0001(39) 1,0035(33) 1,0043(36) 1,0021(33)
1,0008(38) 1,0015(32) 0,6 1,22 1,0017(38) 1,0091(33) 1,0162(34) 1,0176(36)
1,0211(36) 1,0226(35) 1,0201(36) 0,7 1,53 1,0085(35) 1,0165(39) 1,0235(33)
1,0301(37) 1,0362(40) 1,0340(29) 1,0347(38) 0,8 2 1,0172(37) 1,0212(37)
1,0317(35) 1,0357(39) 1,0437(34) 1,0474(31) 1,0438(37) 0,8605 2,48 1,0154(37)
1,0241(38) 1,0343(40) 1,0452(40) 1,0485(39) 1,0544(35) 1,0524(37) 0,9 3
1,0180(36) 1,0319(34) 1,0350(34) 1,0436(40) 1,0523(36) 1,0540(32) 1,0605(32)

##  Appendix C Proof of statement (2) in Lemma 3.

We follow the proof of [11] and adapt their notation and define the quantities
$a=-\ln{\lambda}$, $s_{i}=\ln{X_{i}}$ and
$S=\sum\limits_{i=1}^{N}\ln{X_{i}}=\sum\limits_{i=1}^{N}s_{i}$. We write now
for simplicity $\varphi(a,\beta)$ rather than $\varphi(\lambda,\beta)$ and use
the fact that all $s_{i}$ are positive because of the left-truncation equals
1. First, however, we start with two useful Propositions:

######  Proposition 1.

For any $m>0$ and $c\in\mathbb{R}$ let $A\in\mathbb{R}$ satisfy the following
inequalities

  1. (i)

$A\leq mc$

  2. (ii)

$A\leq-mc$

Then we have the following inequality

$\displaystyle A\leq-m|c|$

Proof:

We have $A\leq 0$ by adding up inequalities (i) and (ii). Consider first the
case when $c>0$ in which case inequality (ii) is tighter than inequality (i),
thus $A\leq-mc=-m|c|$. Likewise when $c<0$ the inequality (i) is tighter than
inequality (ii), thus $A\leq mc=m(-|c|)$. The case $c=0$ is trivial.
$\blacksquare$

######  Proposition 2.

For $z_{i}\in\mathbb{R}$ we have the following inequalities

$\displaystyle\frac{1}{N}\sum\limits_{i=1}^{N}z_{i}\leq\ln{\left(1+\exp{\left[\frac{1}{N}\sum\limits_{i=1}^{N}z_{i}\right]}\right)}\leq\frac{1}{N}\sum\limits_{i=1}^{N}\ln{\left(1+e^{z_{i}}\right)}\quad.$

Proof:

Note that the function $\psi(z)=\ln\left(1+\exp{[z]}\right)$ is convex because
$\psi^{\prime\prime}(z)=e^{z}/(1+e^{z})^{2}>0$ for all $z\in\mathbb{R}$. The
right-hand side inequality is Jensen’s inequality (e.g. [29], p. 74 Eq.
(3.8.1) and (3.8.2)) whereas the left-hand side follows from monotonicity of
the logarithm. $\blacksquare$

Proof of statement (2) in Lemma 3.

In our new notation the objective function Eq. (8) reads as

$\varphi(a,\beta)=N\ln{\left(1+e^{a}\right)}+N\ln{\beta}+Na+\beta
S-2\sum\limits_{i=1}^{N}\ln{\left[1+e^{a+\beta s_{i}}\right]}\quad.$ (8)

We shall derive from this equation various inequalities. Because the last term
is negative we have immediately the following inequality

$\displaystyle\varphi(a,\beta)$ $\displaystyle\leq$ $\displaystyle
N\ln{\left(1+e^{a}\right)}+N\ln{\beta}+Na+\beta S\quad.$ (51)

Using the entire inequality chain from Proposition 2. we obtain another
inequality from Eq. (8)

$\displaystyle\varphi(a,\beta)$ $\displaystyle\leq$ $\displaystyle
N\ln{\left(1+e^{a}\right)}+N\ln{\beta}-Na-\beta S\quad.$ (52)

Applying Proposition 1. to the inequalities Eq. (51) and (52) we get

$\displaystyle\varphi(a,\beta)$ $\displaystyle\leq$ $\displaystyle
N\ln{\left(1+e^{a}\right)}+N\ln{\beta}-\beta
N\left|\frac{a}{\beta}+\frac{S}{N}\right|\quad.$ (53)

A final upper bound is obtained from Eq. (8) using Proposition 2. for any
$i=1,2,...,N$

$\displaystyle\varphi(a,\beta)$ $\displaystyle\leq$ $\displaystyle
N\ln{\left(1+e^{a}\right)}+N\ln{\beta}-\sum\limits_{i=1}^{N}\ln{\left[1+e^{a+\beta
s_{i}}\right]}$ (54) $\displaystyle\leq$ $\displaystyle
N\ln{\left(1+e^{a}\right)}+N\ln{\beta}-\beta\left(\frac{a}{\beta}+s_{i}\right)\quad.$

Case I: Consider first $a\geq 0$. We can bound the positive quantity
$\ln{(1+e^{a})}$ above by $\ln{2}+a$ and thus inequality Eq. (52) yields
altogether

$\displaystyle\varphi(a,\beta)$ $\displaystyle\leq$ $\displaystyle
N\ln{2}+N\ln{\beta}-\beta S$ (55)

the desired upper bound independent of $a$.

Case II: Consider now $a<0$ We can bound the positive quantity
$\ln{(1+e^{a})}$ above by $\ln{2}$. In this part we shall use the inequality
Eq. (53) reading as

$\displaystyle\varphi(a,\beta)$ $\displaystyle\leq$ $\displaystyle
N\ln{2}+N\ln{\beta}-\beta N\left|\frac{a}{\beta}+\frac{S}{N}\right|$ (56)

and the inequality Eq. (54) reading as

$\displaystyle\varphi(a,\beta)$ $\displaystyle\leq$ $\displaystyle
N\ln{2}+N\ln{\beta}-\beta\left(\frac{a}{\beta}+s_{i}\right)\quad.$ (57)

Because not all $s_{i}$ are equal there exists an index $i_{0}$ with

$\displaystyle s_{i_{0}}-\frac{S}{N}=\varepsilon_{0}>0\quad.$

We have
$(-\infty,s_{i_{0}}-\varepsilon_{0})\cup(S/N+\varepsilon_{0},+\infty)=\mathbb{R}$.
Again we consider 2 sub-cases. If we have
$-a/\beta\in(S/N+\varepsilon_{0},+\infty)$ then

$\displaystyle-\frac{a}{\beta}>\frac{S}{N}+\varepsilon_{0}$ (58)

from which we obtain

$\displaystyle\left|\frac{a}{\beta}+\frac{S}{N}\right|=-\left(\frac{a}{\beta}+\frac{S}{N}\right)>\varepsilon_{0}\quad.$
(59)

Now we use this lower bound Eq. (59) in conjunction with inequality Eq. (56)
to obtain

$\displaystyle\varphi(a,\beta)$ $\displaystyle\leq$ $\displaystyle
N\ln{2}+N\ln{\beta}-\beta N\varepsilon_{0}\quad.$ (60)

Likewise if we have $-a/\beta\in(-\infty,s_{i_{0}}-\varepsilon_{0})$ then

$\displaystyle-\frac{a}{\beta}<s_{i_{0}}-\varepsilon_{0}$ (61)

from which we obtain

$\displaystyle\varepsilon_{0}<s_{i_{0}}+\frac{a}{\beta}\quad.$ (62)

Now we use this lower bound Eq. (62) in conjunction with inequality Eq. (57)

$\displaystyle\varphi(a,\beta)$ $\displaystyle\leq$ $\displaystyle
N\ln{2}+N\ln{\beta}-\beta\varepsilon_{0}\quad.$ (63)

Hence for any $a<0$ we obtain from Eq. (60) and (63) the desired bound
independent of $a$

$\displaystyle\varphi(a,\beta)$ $\displaystyle\leq$ $\displaystyle
N\ln{2}+N\ln{\beta}-\beta\varepsilon_{0}$ (64)

since $N\geq 2$.

From Case I Eq. (55) and Case II Eq. (65) we obtain altogether

$\displaystyle\varphi(a,\beta)$ $\displaystyle\leq$ $\displaystyle
N\ln{2}+N\ln{\beta}-\beta\min{\\{S,\varepsilon_{0}\\}}$ (65)

and thus the existence of a sufficiently large $\beta_{2}>\beta_{1}>0$,
independent of $a$ such that $\varphi(a,\beta)<-M$ for $\beta>\beta_{2}$ and
all $a\in\mathbb{R}$. This concludes the proof. $\blacksquare$

## References

  * [1] Fisk, P.R. (1961) ”The Graduation of Income Distributions”, _Econometrica_ 29(2), pp. 171-185.
  * [2] Shoukri, M.M., Mian, I.U.H. and Tracy, D.S. (1988) ”Sampling properties of estimators of the log-logistic distribution with application to Canadian precipitation data”, _The Canadian Journal of Statistics_ 16 (3), pp. 223-236.
  * [3] Ahmad, M.I., Sinclair, C.D. and Werritty, A. (1988) ”Log-logistic flood frequency analysis”, _Journal of Hydrology_ 98, pp. 205-224
  * [4] Reath, J. , Dong, J. and Wang, M. (2018) ”Improved parameter estimation of the log-logistic distribution with applications”, _Computational Statistics_ , 33 (1), pp 339-356
  * [5] He, X. , Chen, W. , Qian,W. (2020) ”Maximum likelihood estimators of the parameters of the log-logistic distribution”, _Statistical Papers_ 61(5), pp 1875-1892
  * [6] Kendall, M. and Stuart, A. (1979) ”The advanced theory of statistics II - Inference and relationship”, 4th revised edition, Griffin London.
  * [7] Cohen, C. (1991) ”Truncated and Censored Samples -Theory and Applications”, 1st edition, CRC Press Boca Raton
  * [8] Guscott, J.C. (2018) ”Reliable Statistical Methods and their Applications for Testing Incomplete Multidisciplinary Data”, MPhil Thesis, Physics Department, University of Adelaide, South Australia, https://hdl.handle.net/2440/127169
  * [9] Castillo, J. del (1994) ”The singly truncated normal distribution: A non-steep exponential family”, _Annals of the Institute of Statistical Mathematics_ 45 (1), pp. 57-66
  * [10] Kreer, M., Kizilersu A., Thomas A.W. and dos Reis A.E. (2015). ”Goodness-of-fit tests and applications for left-truncated Weibull distributions to non-life insurance.” _European Actuarial Journal_ , 5 (1), pp 139-163.
  * [11] Gupta, R.C., Akman, O. and Levin, S. (1999) ”A study of log-logistic model in survival analysis”, _Biometrical Journal_ 41, pp 431-443.
  * [12] Antle, C.E., Klimko, L. and Harkness, W. (1970) ”Confidence intervals for the parameters of the logistic distribution”, _Biometrika_ 57 (2), 397-402.
  * [13] Kolmogorov, A. (1933) ”Sulla detenninazione empirica di una legge di distribuzione” , _Giornale dell’Istituto Italiano degli Attuari_ , 4, pp. 83–91.
  * [14] Smirnov, N. (1948) ”Table for estimating the goodness of fit of emperical distributions”, _The Annals of Mathematical Statistics_ , 19(2), pp. 279–281.
  * [15] Anderson, T.W. and Darling, D.A. (1952) ”Asymptotic theory of certain ’goodness of fit’ criteria based on stochastic processes”, _The Annals of Mathematical Statistics_ , 23(2), pp. 193–212.
  * [16] Mosteller, F. (1946). ”On some useful inefficient statistics”. _The Annals of Mathematical Statistics_ 17 (4), pp 377-–408.
  * [17] Cuddington, J.T. and Navidi, W. (2011) ”A critical assessment of simulated critical values” _Communications in Statistics-Simulation and Computation_ , 40, pp. 719–727.
  * [18] Stephens, M.A. (1979) ”Test of fit for the logistic distribution based on the empirical distribution function”, _Biometrika_ 66 (3) pp. 591-595
  * [19] Lee, E.T. and Wang, J. W. (2003) ”Statistical Methods for Survival Data Analysis”, 3rd edition, John Wiley & Sonds Inc. Hoboken, New Jersey.
  * [20] Al-Shomrani, A. A, Shawky, A.I., Arif, O.H. and Aslam, M. (2016) ”Log-logistic distribution for survival data analysis using MCMC”, _SpringerPlus_ 5(1), pp 1774–1790 .
  * [21] Ashkar, F. and Mahdi, S. (2006) ”Fitting the log-logistic distribution by generalized moments”, _Journal of Hydrology_ , 328, pp 694–703.
  * [22] Climate data base of the DWD Deutscher Wetterdienst (German Meteorology Office), Offenbach Germany, http://opendata.dwd.de/climate_environment/CDC/regional_averages_DE/annual/precipitation/regional_averages_rr_year.txt
  * [23] Toronto Weather Statistics (2022), ”Total Precipitation - Annual data Jul 1-Jun 30 for Toronto”, toronto.weatherstats.ca/charts/precipitation-wyearly.html
  * [24] http://www.bom.gov.au/jsp/ncc/cdio/weatherData/av?p_nccObsCode=136&p_display_type=dailyDataFile&p_startYear=1884&p_c=-106001191&p_stn_num=023011 http://www.bom.gov.au/climate/data/index.shtml
  * [25] Wingo, D. R. (1983) “Maximum likelihood methods for fitting the Burr type XII distribution to life test data”, _Biometrical Journal_ Vol. 25 (1), pp. 77-84.
  * [26] Shao, Q. (2004) “Notes on maximum likelihood estimation for the three-parameter Burr XII distribution”, _Computational Statistics $\&$ Data Analysis_, 45 (3) pp 675 – 687.
  * [27] Wang , F.-K. and Cheng Y.-F. (2010) “Robust regression for estimating the Burr XII parameters with outliers”, _Journal of Applied Statistics_ 37 (5), pp. 807-819.
  * [28] Ostrowski, A.M. (1960), ”Solution of Equations and Systems of Equations”, Academic Press New York and London.
  * [29] Hardy, G., Littlewood, J.E. and Polya, G. ”Inequalities”, 2nd edition reprinted 1994, Cambridge University Press, Cambridge.
  * [30] Harris, G. and Martin, C. (1987). ”Shorter notes: The roots of a polynomial vary continuously as a function of the coefficients.” _Proceedings of the American Mathematical Society_ 100 (2), pp 390–392.
  * [31] Antle, C.E. and Bain, L.J. (1969) ”A property of maximum likelihood estimators of location and scale parameters”, _SIAM Review_ 11 (2), pp. 251-253 .


* (132) Zhengyi Ma et al.  One chatbot per person: Creating personalized chatbots based on implicit user profiles.  In Proceedings of the 44th international ACM SIGIR conference on research and development in information retrieval, pages 555–564, 2021.
  * (133) Hanxun Zhong et al.  Less is more: Learning to refine dialogue history for personalized dialogue generation.  In Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 5808–5820, Seattle, United States, July 2022. Association for Computational Linguistics.
  * (134) Haoyu Song et al.  Generate, delete and rewrite: A three-stage framework for improving persona consistency of dialogue generation.  In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pages 5821–5831, Online, July 2020. Association for Computational Linguistics.
  * (135) Yihong Tang et al.  Enhancing personalized dialogue generation with contrastive latent variables: Combining sparse and dense persona.  In Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 5456–5468, Toronto, Canada, July 2023. Association for Computational Linguistics.
  * (136) Sebastian Borgeaud et al.  Improving language models by retrieving from trillions of tokens.  In International conference on machine learning, pages 2206–2240. PMLR, 2022.
  * (137) Gautier Izacard and Edouard Grave.  Leveraging passage retrieval with generative models for open domain question answering.  In Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume, pages 874–880, Online, April 2021. Association for Computational Linguistics.
  * (138) Leonard Adolphs et al.  Reason first, then respond: Modular generation for knowledge-infused dialogue.  In Findings of the Association for Computational Linguistics: EMNLP 2022, pages 7112–7132, Abu Dhabi, United Arab Emirates, December 2022. Association for Computational Linguistics.
  * (139) Devendra Singh et al.  End-to-end training of multi-document reader and retriever for open-domain question answering.  Advances in Neural Information Processing Systems, 34:25968–25981, 2021.
  * (140) Kenton Lee, Ming-Wei Chang, and Kristina Toutanova.  Latent retrieval for weakly supervised open domain question answering.  In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pages 6086–6096, Florence, Italy, July 2019. Association for Computational Linguistics.
  * (141) Mojtaba Komeili, Kurt Shuster, and Jason Weston.  Internet-augmented dialogue generation.  In Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 8460–8478, Dublin, Ireland, May 2022. Association for Computational Linguistics.
  * (142) Antoine Bordes, Y-Lan Boureau, and Jason Weston.  Learning end-to-end goal-oriented dialog.  arXiv preprint arXiv:1605.07683, 2016.
  * (143) Matthew Henderson et al.  Efficient natural language response suggestion for smart reply.  arXiv preprint arXiv:1705.00652, 2017.
  * (144) Oriol Vinyals and Quoc Le.  A neural conversational model.  arXiv preprint arXiv:1506.05869, 2015.
  * (145) Ilya Sutskever, Oriol Vinyals, and Quoc V Le.  Sequence to sequence learning with neural networks.  Advances in neural information processing systems, 27, 2014.
  * (146) Lifeng Shang, Zhengdong Lu, and Hang Li.  Neural responding machine for short-text conversation.  In Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International Joint Conference on Natural Language Processing (Volume 1: Long Papers), pages 1577–1586, Beijing, China, July 2015. Association for Computational Linguistics.
  * (147) Xueliang Zhao et al.  Knowledge-grounded dialogue generation with pre-trained language models.  In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 3377–3390, Online, November 2020. Association for Computational Linguistics.
  * (148) Alessandro Sordoni et al.  A neural network approach to context-sensitive generation of conversational responses.  In Proceedings of the 2015 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 196–205, Denver, Colorado, May–June 2015. Association for Computational Linguistics.
  * (149) Zhao Yan et al.  Docchat: An information retrieval approach for chatbot engines using unstructured documents.  In Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 516–525, 2016.
  * (150) Jing Xu, Arthur Szlam, and Jason Weston.  Beyond goldfish memory: Long-term open-domain conversation.  In Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 5180–5197, Dublin, Ireland, May 2022. Association for Computational Linguistics.
  * (151) Matthew Henderson, Blaise Thomson, and Steve Young.  Word-based dialog state tracking with recurrent neural networks.  In Proceedings of the 15th annual meeting of the special interest group on discourse and dialogue (SIGDIAL), pages 292–299, 2014.
  * (152) Chien-Sheng Wu et al.  Transferable multi-domain state generator for task-oriented dialogue systems.  In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pages 808–819, Florence, Italy, July 2019. Association for Computational Linguistics.
  * (153) Abhinav Rastogi et al.  Towards scalable multi-domain conversational agents: The schema-guided dialogue dataset.  In Proceedings of the AAAI conference on artificial intelligence, volume 34, pages 8689–8696, 2020.
  * (154) Kishore Papineni et al.  Bleu: a method for automatic evaluation of machine translation.  In Proceedings of the 40th annual meeting of the Association for Computational Linguistics, pages 311–318, 2002.
  * (155) Jiwei et al. Li.  A diversity-promoting objective function for neural conversation models.  In Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 110–119, San Diego, California, June 2016. Association for Computational Linguistics.
  * (156) Paweł Budzianowski et al.  MultiWOZ - a large-scale multi-domain Wizard-of-Oz dataset for task-oriented dialogue modelling.  In Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, pages 5016–5026, Brussels, Belgium, October-November 2018. Association for Computational Linguistics.
  * (157) Mihail Eric et al.  MultiWOZ 2.1: A consolidated multi-domain dialogue dataset with state corrections and state tracking baselines.  In Proceedings of the Twelfth Language Resources and Evaluation Conference, pages 422–428, Marseille, France, May 2020. European Language Resources Association.
  * (158) Xiaoxue Zang et al.  MultiWOZ 2.2 : A dialogue dataset with additional annotation corrections and state tracking baselines.  In Proceedings of the 2nd Workshop on Natural Language Processing for Conversational AI, pages 109–117, Online, July 2020. Association for Computational Linguistics.
  * (159) Ting Han et al.  Multiwoz 2.3: A multi-domain task-oriented dialogue dataset enhanced with annotation corrections and co-reference annotation.  In Natural Language Processing and Chinese Computing: 10th CCF International Conference, NLPCC 2021, Qingdao, China, October 13–17, 2021, Proceedings, Part II 10, pages 206–218. Springer, 2021.
  * (160) Fanghua Ye, Jarana Manotumruksa, and Emine Yilmaz.  MultiWOZ 2.4: A multi-domain task-oriented dialogue dataset with essential annotation corrections to improve state tracking evaluation.  In Proceedings of the 23rd Annual Meeting of the Special Interest Group on Discourse and Dialogue, pages 351–360, Edinburgh, UK, September 2022. Association for Computational Linguistics.
  * (161) Jun Quan et al.  RiSAWOZ: A large-scale multi-domain Wizard-of-Oz dataset with rich semantic annotations for task-oriented dialogue modeling.  In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 930–940, Online, November 2020. Association for Computational Linguistics.
  * (162) Qi Zhu et al.  Crosswoz: A large-scale chinese cross-domain task-oriented dialogue dataset.  Transactions of the Association for Computational Linguistics, 8:281–295, 2020.
  * (163) Xuewei Wang et al.  Persuasion for good: Towards a personalized persuasive dialogue system for social good.  In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pages 5635–5649, Florence, Italy, July 2019. Association for Computational Linguistics.
  * (164) Nikola Mrkšić et al.  Neural belief tracker: Data-driven dialogue state tracking.  In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 1777–1788, Vancouver, Canada, July 2017. Association for Computational Linguistics.
  * (165) Tsung-Hsien Wen et al.  A network-based end-to-end trainable task-oriented dialogue system.  In Proceedings of the 15th Conference of the European Chapter of the Association for Computational Linguistics: Volume 1, Long Papers, pages 438–449, Valencia, Spain, April 2017. Association for Computational Linguistics.
  * (166) Mihail et al. Eric.  Key-value retrieval networks for task-oriented dialogue.  In Proceedings of the 18th Annual SIGdial Meeting on Discourse and Dialogue, pages 37–49, Saarbrücken, Germany, August 2017. Association for Computational Linguistics.
  * (167) Saizheng Zhang et al.  Personalizing dialogue agents: I have a dog, do you have pets too?  In Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 2204–2213, Melbourne, Australia, July 2018. Association for Computational Linguistics.
  * (168) Jiazhan Feng et al.  MMDialog: A large-scale multi-turn dialogue dataset towards multi-modal open-domain conversation.  In Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 7348–7363, July 2023.
  * (169) Yanran Li et al.  DailyDialog: A manually labelled multi-turn dialogue dataset.  In Proceedings of the Eighth International Joint Conference on Natural Language Processing (Volume 1: Long Papers), pages 986–995, Taipei, Taiwan, November 2017. Asian Federation of Natural Language Processing.
  * (170) Hongjin Qian et al.  Pchatbot: a large-scale dataset for personalized chatbot.  In Proceedings of the 44th international ACM SIGIR conference on research and development in information retrieval, pages 2470–2477, 2021.
  * (171) Yinhe Zheng et al.  Personalized dialogue generation with diversified traits.  arXiv preprint arXiv:1901.09672, 2019.