# The inverse xgamma distribution: statistical properties and different
methods of estimation

Abhimanyu Singh Yadav a, Sudhansu S. Maiti b, Mahendra Saha a111Corresponding
author. e-mail<EMAIL_ADDRESS>
and Arvind Pandey a
aDepartment of Statistics, Central University of Rajasthan, Rajasthan, India
b Department of Statistics, Visva-Bharati University, Santiniketan, India

###### Abstract

This paper proposed a new probability distribution named as inverse xgamma
distribution (IXGD). Different mathematical and statistical properties,viz.,
reliability characteristics, moments, inverse moments, stochastic ordering and
order statistics of the proposed distribution have been derived and discussed.
The estimation of the parameter of IXGD has been approached by different
methods of estimation, namely, maximum likelihood method of estimation (MLE),
Least square method of estimation (LSE), Weighted least square method of
estimation (WLSE), Cramèr-von-Mises method of estimation (CME) and maximum
product spacing method of estimation (MPSE). Asymptotic confidence interval
(ACI) of the parameter is also obtained. A simulation study has been carried
out to compare the performance of the obtained estimators and corresponding
ACI in terms of average widths and corresponding coverage probabilities.
Finally, two real data sets have been used to demonstrate the applicability of
IXGD in real life situations.

Keywords: inverse xgamma distribution; survival properties; maximum likelihood
estimate; least square and weighted least square estimate; Cramer-Von Mises
estimate; maximum product spacing estimate; asymptotic confidence interval.

##  1 Introduction.

Sen et al. ($2016$) introduced a finite mixture of exponential ($\theta$) and
gamma ($3,\theta$) distributions with mixing proportion
$\pi_{1}=\theta/(1+\theta)$ and $\pi_{2}=1-\pi_{1}$, where $\pi_{1},\pi_{2}$
denote the mixing proportions that are non-negative and sum to one, to
obtained a probability distribution, named as xgamma distribution (XGD). The
probability density function (PDF) and cumulative distribution function(CDF)
of the XGD are, respectively, given by

$\displaystyle f(y;\theta)$ $\displaystyle=$
$\displaystyle\frac{\theta^{2}}{(1+\theta)}\left(1+\frac{\theta}{2}.y^{2}\right)e^{-\theta
y}~{}~{};y>0,~{}\theta>0$ $\displaystyle=$ $\displaystyle
0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{};\mbox{otherwise}.$
$\displaystyle F(y;\theta)$ $\displaystyle=$ $\displaystyle
1-\frac{\left(1+\theta+\theta
y+\frac{\theta^{2}y^{2}}{2}\right)}{(1+\theta)}e^{-\theta
y}~{}~{};y>0,~{}\theta>0$ $\displaystyle=$ $\displaystyle
0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{};\mbox{otherwise}.$

Sen et al. ($2016$) investigated mathematical, structural and survival
properties of the XGD and they have found that in many cases the XGD has more
flexibility than the exponential distribution. A new probability distribution,
namely, inverse xgamma distribution (IXGD) is introduced in this article. The
inverse of the XGD is considered in order to obtain the form of the inverse
xgamma distribution and hence the name proposed.

The objective of this article is two fold: First, we introduced a new
probability distribution and studied the several statistical properties of
IXGD, as the inverted version of XGD, introduced by Sen et al. ($2016$).
Second, different methods of estimation have been employed to estimate the
unknown parameter of IXGD. Further, $95\%$ asymptotic confidence interval
(ACI) of the parameter based on MPSE has been constructed. To the best of our
knowledge thus far, no attempt has been made to introduce the inverted version
of XGD. Our aim is to fill up this gap through this present study.

The rest of the article is organized as follows: In Section $2$, we have
developed a new probability distribution, called IXGD. In Section $3$, we have
studied different statistical properties and related measures of IXGD.
Different methods of estimation of the parameter of IXGD have been considered
in Section $4$. In Section $5$, ACI of the parameter of IXGD has been
obtained. Monte Carlo simulation has been carried out to see the performance
of the estimates of the parameters in mean squared error sense. Empirical
applications are presented and discussed in Section $6$. Finally, concluding
remarks are given in Section $7$.

##  2 The inverse xgamma distribution.

In recent years, the researcher has proposed number of methods to introduce a
new probability distribution. The inverse transfromation method of baseline
variables is one of them and the resulting distribution is parsimonious in
parameter, for example inverse exponential distribution (IED) [see, Keller and
Kamath ($1982$)], inverse Rayleigh distribution (IRD) [see, Voda ($1972$)],
inverse lindley distribution (ILD) [see, Sharma et al. ($2015$)] etc. The same
approach has been used to introduce the inverted form of XGD. If a random
variable $Y$ has XGD ($\theta$) with PDF given in (1), then the random
variable $X=(1/Y)$ is said to follow the inverse xgamma distribution (IXGD)
with PDF is of the following form:

$\displaystyle f(x)$ $\displaystyle=$
$\displaystyle\frac{\theta^{2}}{(1+\theta)}.\frac{1}{x^{2}}\left(1+\frac{\theta}{2}.\frac{1}{x^{2}}\right)e^{-\theta/x}~{}~{};x>0,~{}\theta>0$
$\displaystyle=$ $\displaystyle
0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{};\mbox{otherwise}.$

It is denoted by $X\sim\mbox{IXGD}(\theta)$. The CDF of IXGD is given by

$\displaystyle F(x)$ $\displaystyle=$
$\displaystyle\left(1+\frac{\theta}{(1+\theta)}.\frac{1}{x}+\frac{\theta^{2}}{2(1+\theta)}.\frac{1}{x^{2}}\right)e^{-\theta/x}~{}~{};x>0,~{}\theta>0$
(2.4)

The shape of the density and distribution functions for IXGD are presented in
Figure 1.

Figure 1: Density and distribution function plot

##  3 Some statistical properties.

In the following subsections, different statistical properties, viz.,
reliability characteristics, moments and inverse-moments, stochastic ordering
and order statistics have been discussed.

###  3.1 Reliability Characteristics:

The basic tools for studying the ageing and associated characteristics of any
lifetime equipments are the reliability and hazard functions. The reliability
and the hazard function of IXGD($\theta$) are given below:

  * •

The reliability function $R(t)$, which is the probability that an item not
failing prior to some time $t(\geq 0)$, is defined by

$\displaystyle R(t)$ $\displaystyle=$ $\displaystyle P[X\geq t]$ (3.5)
$\displaystyle=$ $\displaystyle 1-F(t)$ $\displaystyle=$ $\displaystyle
1-\left\\{\left(1+\frac{\theta}{(1+\theta)}.\frac{1}{x}+\frac{\theta^{2}}{2(1+\theta)}.\frac{1}{x^{2}}\right)e^{-\theta/x}\right\\}.$

  * •

The hazard rate function (or failure rate function) for a continuous
distribution with PDF $f(t)$, CDF $F(t)$, and survival function (SF) $S(t)$ is
the conditional probability of failure, given it has survived up to time
$t(\geq 0)$ and is defined as

$\displaystyle H(t)$ $\displaystyle=$ $\displaystyle\frac{f(t)}{1-F(t)}$
$\displaystyle=$ $\displaystyle\frac{f(t)}{S(t)}.$

For the IXGD, the hazard rate function is given by

$\displaystyle H(t)$ $\displaystyle=$
$\displaystyle\frac{\left\\{\frac{\theta^{2}}{(1+\theta)}.\frac{1}{t^{2}}\left(1+\frac{\theta}{2}.\frac{1}{t^{2}}\right)e^{-\theta/t}\right\\}}{\left[1-\left\\{\left(1+\frac{\theta}{(1+\theta)}.\frac{1}{t}+\frac{\theta^{2}}{2(1+\theta)}.\frac{1}{t^{2}}\right)e^{-\theta/t}\right\\}\right]}.$
(3.6)

  * •

The reverse hazard rate function (or reverse failure rate function) for a
continuous distribution with PDF $f(t)$, CDF $F(t)$ is defined as

$\displaystyle h(t)$ $\displaystyle=$ $\displaystyle\frac{f(t)}{F(t)}$ (3.7)
$\displaystyle=$
$\displaystyle\frac{\theta^{2}(2t^{2}+\theta)}{(2t^{2}+2\theta t^{2}+2\theta
t+\theta^{2})}.\frac{1}{t^{2}}.$

The shape of the reliability and hazard functions for IXGD are presented in
Figure $2$. From Figure $2$, it is clear that the proposed distribution
accommodate the shape of non-monotone failure rate pattern. Such pattern of
failure rate is very obvious in clinical trial studies and in reliability
studies, thus, IXGD can be an alternative choice to analyse such data set.

Figure 2: Reliability and hazard function plot

###  3.2 Moments and related measures:

The $r$-th order moment about origin of IXGD is given by

$\displaystyle\mu^{{}^{\prime}}_{r}$ $\displaystyle=$ $\displaystyle
E\left(X^{r}\right)$ $\displaystyle=$
$\displaystyle\int\limits_{0}^{\infty}x^{r}.\frac{\theta^{2}}{(1+\theta)}.\frac{1}{x^{2}}\left(1+\frac{\theta}{2}.\frac{1}{x^{2}}\right)e^{-\theta/x}.dx$
$\displaystyle=$
$\displaystyle\left\\{\frac{\theta^{r+1}}{(1+\theta)}\Gamma(1-r)+\frac{\theta^{r}}{2(1+\theta)}\Gamma(3-r)\right\\}~{};~{}~{}r<1.$

The above expression indicate that moment of IXGD will exist only when $r<1$.
Therefore, the evaluation of inverse moments may be of interest. The $r$-th
order inverse moment about origin of IXGD is given by

$\displaystyle\mu^{{}^{\prime}}_{r^{-1}}$ $\displaystyle=$ $\displaystyle
E\left(\frac{1}{X^{r}}\right)$ (3.8) $\displaystyle=$
$\displaystyle\int\limits_{0}^{\infty}\frac{1}{x^{r}}.\frac{\theta^{2}}{(1+\theta)}.\frac{1}{x^{2}}\left(1+\frac{\theta}{2}.\frac{1}{x^{2}}\right)e^{-\theta/x}.dx$
$\displaystyle=$
$\displaystyle\frac{\theta^{2}}{2(1+\theta)}\left[\frac{2\Gamma(r+1)}{\theta^{r+1}}+\frac{\Gamma(r+3)}{\theta^{r+2}}\right]~{}~{}~{}~{};r=1,2,3~{}\&~{}4.$

###  3.3 Harmonic mean and other moments:

The harmonic mean for the density function as expressed in (2) is obtained by

$E\left(\frac{1}{x}\right)=\int_{x}\frac{1}{x}f(x,\theta)dx.$ (3.9)

The above equation can also be calculated from the expression of inverse of
moment by putting $r=1$. Hence, after simplification we get

$\displaystyle E\left(\frac{1}{x}\right)=\dfrac{\theta+3}{\theta(1+\theta)}.$
(3.10)

###  3.4 Stochastic Ordering:

Stochastic ordering of a positive random variable is a very important property
to study the comparative behaviour of a random variable. Recall some basic
definitions: A random variable X is said to be smaller than a random variable
Y in the

  * •

stochastic order $(X\leq_{st}Y)$ if $F_{X}(x)\geq F_{Y}(x)$ for all $x$

  * •

hazard rate order $(X\leq_{hr}Y)$ if $h_{X}(x)\geq h_{Y}(x)$ for all $x$

  * •

mean residual life order $(X\leq_{mrl}Y)$ if $m_{X}(x)\leq m_{Y}(x)$ for all
$x$

  * •

likelihood ratio order $(X\leq_{lr}Y)$ if $f_{X}(x)/f_{Y}(x)$ decreases in $x$

The following implications based on this property are illustrated by Shaked &
Shanthikumar (1994).

$X\leq_{lr}Y\Rightarrow X\leq_{hr}Y\Rightarrow X\leq_{mrl}Y$

and hence

$X\leq_{hr}Y\Rightarrow X\leq_{st}Y$

The following theorem shows that IXGD is ordered with respect to the strongest
likelihood ratio ordering.
Theorem: Let $X\sim IXGD(\theta_{1})$ and $Y\sim IXGD(\theta_{2})$. If
$\theta_{1}>\theta_{2}$, then $X\leq_{lr}Y$ and hence it implies oredering in
others also.

Proof:

$\displaystyle\phi(x)$ $\displaystyle=$
$\displaystyle\frac{f_{\theta_{1}}(x)}{f_{\theta_{2}}(x)}$ $\displaystyle=$
$\displaystyle\frac{\theta^{2}_{1}(1+\theta_{2})}{\theta^{2}_{2}(1+\theta_{1})}\left\\{\left(\frac{2x^{2}+\theta_{1}}{2x^{2}+\theta_{2}}\right)e^{-\frac{1}{x}(\theta_{1}-\theta_{2})}\right\\}$

Taking logarithm both sides, we can write

$\displaystyle\ln\phi(x)$ $\displaystyle=$ $\displaystyle\ln
K+\ln(2x^{2}+\theta_{1})-\ln(2x^{2}+\theta_{2})-\frac{1}{x}(\theta_{1}-\theta_{2})$

Taking partial derivative both sides, we have

$\displaystyle\frac{\partial\ln\phi(x)}{\partial x}$ $\displaystyle=$
$\displaystyle\frac{1}{\phi(x)}.\phi^{{}^{\prime}}(x)$ $\displaystyle=$
$\displaystyle\left\\{\frac{4x}{(2x^{2}+\theta_{1})(2x^{2}+\theta_{2})}(\theta_{2}-\theta_{1})-\frac{(\theta_{2}-\theta_{1})}{x^{2}}\right\\},$

that implies

$\phi^{{}^{\prime}}(x)=\phi(x)(\theta_{2}-\theta_{1})\left\\{\frac{4x}{(2x^{2}+\theta_{1})(2x^{2}+\theta_{2})}-\frac{1}{x^{2}}\right\\}$

Now, if $\theta_{1}>\theta_{2}$, then $\phi^{{}^{\prime}}(x)<0$. Hence,
$\phi(x)$ decreases in $x$ and this implies $X\leq_{st}Y$.

###  3.5 Order Statistics:

Let $X_{(1)},~{}X_{(2)},~{}...,~{}X_{(n)}$ are the $n$ ordered random sample
observed from density function (2). Then, the distribution of $r$-th order
statistic is obtained by using the following expressions as follows:

$\displaystyle f_{r}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{\beta(r,n-r+1)}.\sum\limits_{k=0}^{n-r}(-1)^{k}{{n-r}\choose
k}F^{r+k-1}(x).f(x)$ (3.11)

and the $r$-th order CDF $F_{r}(x)$ is

$\displaystyle F_{r}(x)$ $\displaystyle=$
$\displaystyle\sum\limits_{j=r}^{n}\sum\limits_{k=0}^{n-j}{{n}\choose
j}{{n-j}\choose k}(-1)^{k}F^{j+k}(x)$ (3.12)

Hence, using Equations (2), (2.4) in (3.11), the PDF and the CDF of $r$th
order statistics are, respectively, given by

$\displaystyle f_{r}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{\beta(r,n-r+1)}.\frac{\theta^{2}}{(1+\theta)}\sum\limits_{k=0}^{n-r}(-1)^{k}{{n-r}\choose
k}\left(1+\frac{\theta}{(1+\theta)}.\frac{1}{x}+\frac{\theta^{2}}{2(1+\theta)}.\frac{1}{x^{2}}\right)^{r+k-1}\times$
(3.13)
$\displaystyle\left(\frac{1}{x^{2}}\left(1+\frac{\theta}{2}.\frac{1}{x^{2}}\right)\right)e^{-\theta(r+k)/x}$
$\displaystyle F_{r}(x)$ $\displaystyle=$
$\displaystyle\sum\limits_{j=r}^{n}\sum\limits_{k=0}^{n-j}{{n}\choose
j}{{n-j}\choose
k}(-1)^{k}\left(1+\frac{\theta}{(1+\theta)}.\frac{1}{x}+\frac{\theta^{2}}{2(1+\theta)}\frac{1}{x^{2}}\right)^{j+k}\times$
(3.14) $\displaystyle e^{-\theta(j+k)/x}.$

The distributions (PDF & CDF) of the smallest and the largest order statistics
in case of IXGD are obtained by putting $r=1$ and $r=n$ in Equations (3.13)
and (3.14) respectively.

##  4 Different methods of estimation of parameter $\theta$.

In this section, we have used five methods of estimation to estimate the
unknown parameter $\theta$, namely, maximum likelihood method of estimation
(MLE), least squares method of estimation (LSE), weighted least squares method
of estimation (WLSE), Cramèr-von-Mises estimator method estimation (CME) and
maximum product of spacings method of estimation (MPSE) respectively.

###  4.1 Maximum likelihood method of estimation:

Let $(x_{1},~{}x_{2},~{}...,~{}x_{n})$ be the random sample of size $n$ drawn
from the IXGD, given in Equation (2). The maximum likelihood estimator (MLE)
of $\theta$ for given $x$ is obtained as follows:
The likelihood function of $\theta$ is given by

$\displaystyle L(\theta\mid x)$ $\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{n}\left\\{\frac{\theta^{2}}{(1+\theta)}.\frac{1}{x_{i}^{2}}\left(1+\frac{\theta}{2}.\frac{1}{x_{i}^{2}}\right)e^{-\theta/x_{i}}\right\\}.$

Taking logarithm both sides, the log-likelihood function is given by

$\displaystyle l(\theta\mid x)$ $\displaystyle=$ $\displaystyle\ln
L(\theta\mid x)$ $\displaystyle=$ $\displaystyle
2n\ln\theta-n\ln(1+\theta)-2\sum\limits_{i=1}^{n}\ln
x_{i}+\sum\limits_{i=1}^{n}\ln\left(1+\frac{\theta}{2x^{2}_{i}}\right)-\theta\sum\limits_{i=1}^{n}\frac{1}{x_{i}}$

The resulting partial derivative of the log-likelihood function

$\frac{\partial l(\theta\mid
x)}{\partial\theta}=\sum\limits_{i=1}^{n}\left\\{\frac{1}{2x^{2}_{i}+\theta}\right\\}+\frac{2n}{\theta}-\frac{n}{(1+\theta)}-\sum\limits_{i=1}^{n}\frac{1}{x_{i}}=0$
(4.15)

yield the MLE of $\theta$. Equating this partial derivative to zero does not
yield closed-form solution for MLE of $\theta$ and thus a numerical method is
used for solving this equation.

###  4.2 Ordinary and weighted least square methods of estimation:

The least square estimator (LSE) and the weighted least square estimator
(WLSE) were proposed by Swain et al. ($1988$) to estimate the parameters of
Beta distributions. Suppose $F(x_{(j)})$ denotes the distribution function of
the ordered random variables $x_{(1)}<x_{(2)}<\cdots<x_{(n)}$, where,
$\\{x_{1},x_{2},\cdots,x_{n}\\}$ is a random sample of size $n$ from a
distribution function $F(\cdot)$. Therefore, in this case, the LSE of
$\theta$, say, $\hat{\theta}_{LSE}$ can be obtained by minimizing

$S(\theta)=\sum_{i=1}^{n}\left[F\left(x_{i:n}|\theta\right)-\frac{i}{n+1}\right]^{2}$

with respect to $\theta$, where, $F(\cdot)$ is the CDF, given in Equation
(2.4). Equivalently, it can be obtained by solving:

$\displaystyle\displaystyle\sum_{i=1}^{n}\left[F\left(x_{i:n}\mid\theta\right)-\frac{i}{n+1}\right]\eta_{1}\left(x_{i:n}\mid\theta\right)=0,$
(4.16)

where,

$\eta_{1}\left(x_{i:n}\mid\theta\right)=-\frac{\theta
e^{-\theta/x_{i}}}{2x_{i}^{3}(1+\theta)^{2}}\left(4x_{i}^{2}+2\theta
x_{i}^{2}+\theta x_{i}+\theta+\theta^{2}\right)$ (4.17)

The WLSE of $\theta$, say, $\hat{\theta}_{WLSE}$, can be obtained by
minimizing

$\displaystyle\displaystyle
W\left(\kappa,\lambda\right)=\sum_{i=1}^{n}\frac{\left(n+1\right)^{2}\left(n+2\right)}{i\left(n-i+1\right)}\left[F\left(x_{i:n}\mid\theta\right)-\frac{i}{n+1}\right]^{2}.$
(4.18)

This estimator can also be obtained by solving:

$\displaystyle\displaystyle\sum_{i=1}^{n}\frac{\left(n+1\right)^{2}\left(n+2\right)}{i\left(n-i+1\right)}\left[F\left(x_{i:n}\mid\theta\right)-\frac{i}{n+1}\right]\eta_{1}\left(x_{i:n}\mid\theta\right)=0,$
(4.19)

where, $\eta_{1}\left(x_{i:n}\mid\theta\right)$ is defined in (4.17).

###  4.3 Cramèr-von-Mises method of estimation:

To motivate our choice of Cramer-von Mises type minimum distance estimators,
MacDonald ($1971$) provided empirical evidence that the bias of the estimator
is smaller than the other minimum distance estimators. Thus, The Cramer-von
Mises estimators of $\theta$, say, $\hat{\theta}_{CME}$ can be obtained by
minimizing the function

$\displaystyle
C(\theta)=\frac{1}{12n}+\sum_{i=1}^{n}\left[F\left(x_{i:n}\mid\theta\right)-\frac{2i-1}{2n}\right]^{2}$
(4.20)

with respect to $\theta$. The estimator can also be obtained by solving the
non linear equation

$\displaystyle\displaystyle\sum_{i=1}^{n}\left[F\left(x_{i:n}\mid\theta\right)-\frac{2i-1}{2n}\right]\eta_{1}\left(x_{i:n}\mid\theta\right)=0,$
(4.21)

where, $\eta_{1}\left(x_{i:n}\mid\theta\right)$ is defined in (4.17).

###  4.4 Maximum product of spacings method of estimation:

The maximum product spacing method has been introduced by Cheng and Amin
($1979,\\\ ~{}1983$) as an alternative to MLE for the estimation of the
unknown parameters of continuous univariate distributions. This method was
also derived independently by Ranneby ($1984$) as an approximation to the
Kullback-Leibler measure of information. To motivate our choice, Cheng and
Amin ($1983$) proved that this method is as efficient as the MLE and
consistent under more general conditions. Let us define

$D_{i}(\theta)=F\left(x_{i:n}\mid\theta\right)-F\left(x_{i-1:n}\mid\theta\right),\qquad
i=1,2,\ldots,n,$ (4.22)

where, $F(x_{0:n}\mid\theta)=0$ and $F(x_{n+1:n}\mid\theta)=1.$ Clearly,
$\sum_{i=1}^{n+1}D_{i}(\theta)=1.$. The MPSE $\hat{\theta}_{MPS}$, of the
parameter $\theta$ are obtained by maximizing the geometric mean of the
spacings with respect to $\theta$, given as

$G\left(\theta\right)=\left[\prod\limits_{i=1}^{n+1}D_{i}(\theta)\right]^{\frac{1}{n+1}},$
(4.23)

or, equivalently, by maximizing the function

$H\left(\theta\right)=\frac{1}{n+1}\sum_{i=1}^{n+1}\log D_{i}(\theta).$ (4.24)

The estimator $\hat{\theta}_{MPS}$ of the parameter $\theta$ can be obtained
by solving the non-linear equation

$\displaystyle\frac{\partial}{\partial\theta}H\left(\theta\right)=\frac{1}{n+1}\sum\limits_{i=1}^{n+1}\frac{1}{D_{i}(\theta)}\left[\eta_{1}(x_{i:n}|\theta)-\eta_{1}(x_{i-1:n}|\theta)\right]$
$\displaystyle=$ $\displaystyle 0$ (4.25)

where, $\eta_{1}\left(\cdot\mid\theta\right)$ is given by (4.17).

##  5 Asymptotic confidence interval of $\theta$.

Here, we consider the asymptotic confidence interval (ACI) based on MPSE, as
MPSE of $\theta$ performed better in mean squared error (MSE) sense among the
other estimates (MLE, LSE, WLSE and CME). Cheng and Amin ($1979$), Ghosh and
Jammalamadaka ($2001$) already mentioned and shows that the maximum product
spacing method also shows asymptotic properties as MLE. Keeping this in mind,
we have considered the Fisher information and is obtained as;

$I(\hat{\theta})=-E\left[\dfrac{\partial^{2}\ln
L(\theta)}{\partial\theta^{2}}\right]$ (5.26)

where,

$\dfrac{\partial^{2}\ln
L(\theta)}{\partial\theta^{2}}=-\sum\limits_{i=1}^{n}\left\\{\frac{1}{(2x^{2}_{i}+\theta)^{2}}\right\\}-\frac{2n}{\theta^{2}}+\frac{n}{(1+\theta)^{2}}.$

Therefore, the asymptotic variance $\sigma^{2}_{\theta\theta}$ of $\theta$ is
obtained as

$\sigma^{2}_{\theta\theta}=\left[\dfrac{1}{I(\hat{\theta})}\right]_{MPSE}$

The $100(1-\alpha)$% ACI based on MPSE of $\theta$ is given by

$\left\\{\hat{\theta}_{MPSE}\mp
Z_{(\alpha/2)}\sqrt{\left(\sigma^{2}_{\theta\theta}\right)}\right\\},$

where, $Z_{(\alpha/2)}$ is the upper $(\alpha/2)$-th point of the standard
normal distribution.

Also, to study the confidence interval (CI), we have considered the estimated
average widths and coverage probabilities of ACI using MSE of $\theta$. In
addition, the average widths of ACIs are calculated based on the $B=1,000$
different trials. The average widths and corresponding coverage probabilities
are given by

$\mbox{Average
width}=\frac{\sum\limits_{i=1}^{B}\left(U_{i}-L_{i}\right)}{B},$

and

$\mbox{Coverage probability}=\frac{\mbox{Number}\left(L_{W}\leq\theta\leq
U_{p}\right)}{B},$

where, $L_{W}$ and $U_{P}$ denote the $100(1-\alpha)\%$ CIs based on $B$
replicates.

##  6 Simulation and Discussion.

In this section, we have carried out a Monte Carlo simulation study to assess
the performance of the proposed estimators (MLE, LSE, WLSE, CME and MPSE) of
the parameter $\theta$ for IXGD, discussed in Section $3$. In particular, we
have considered the different variations of sample sizes
$n=10,~{}20,~{}50,~{}100$ and the parameter values
$\theta=0.1,~{}0.5,~{}1.0,~{}1.5$ respectively. For each design, sample with
each of size $n$ are drawn from the original sample and replicated $5,000$
times. First, we have calculated the average estimates (AV) of the parameter
$\theta$ using MLE, LSE, WLSE, MPSE and CME along with the corresponding MSEs.
The results are reported in Table $1$.

$\displaystyle AV=\frac{1}{5000}\sum_{j=1}^{5000}\theta_{j}$ $\displaystyle
MSE=\frac{1}{5000}\sum_{j=1}^{5000}(\theta_{j}-\theta)^{2}.$

From Table $1$, it has been observed that as the sample sizes increases, the
MSEs of all estimators are decreases. It verifies the consistency of all the
estimators that we have considered. It has been also observed that the MSEs of
MPSE of $\theta$ are less for all the considered choices of $n$ and $\theta$.
Further, the interval estimation of the parameter is also considered and
corresponding results are reported in Table $2$. Table $2$ showed the
estimated average widths and coverage probabilities of $95\%$ ACI of the
parameter $\theta$ for IXGD using MPSE. Here also, it has been observed that
as the sample sizes increases, the average widths decreases. All simulations
were performed using programs written in the open source statistical package
$R$ [see, Ihaka and Gentleman, ($1996$)].

Table 1: True value of $\theta$ along with average estimates and corresponding
MSEs for IXGD.

|  | Estimates of $\theta$ and corresponding MSEs
---|---|---
$n$ | $\theta$ | MLE | LSE | WLSE | CME | MPSE
$10$ | $0.10$ | $0.102854$ | $0.101512$ | $0.101270$ | $0.102144$ | $0.097421$
|  | $0.000385$ | $0.000457$ | $0.000439$ | $0.000457$ | $0.000344$
$20$ | $0.10$ | $0.101592$ | $0.101217$ | $0.101025$ | $0.101560$ | $0.098079$
|  | $0.000197$ | $0.000228$ | $0.000218$ | $0.000229$ | $0.000183$
$50$ | $0.10$ | $0.100781$ | $0.100311$ | $0.100357$ | $0.100455$ | $0.098992$
|  | $0.000072$ | $0.000083$ | $0.000077$ | $0.000083$ | $0.000069$
$100$ | $0.10$ | $0.100284$ | $0.100091$ | $0.100151$ | $0.100164$ | $0.099238$
|  | $0.000035$ | $0.000040$ | $0.000038$ | $0.000040$ | $0.000035$
$10$ | $0.50$ | $0.521981$ | $0.751896$ | $2.324485$ | $0.789168$ | $0.487656$
|  | $0.0130953$ | $1.782553$ | $191.100200$ | $2.254722$ | $0.011006$
$20$ | $0.50$ | $0.513173$ | $0.718121$ | $2.284905$ | $0.774732$ | $0.491954$
|  | $0.005711$ | $1.626593$ | $190.9996$ | $1.518383$ | $0.005195$
$50$ | $0.50$ | $0.506318$ | $0.527461$ | $1.978845$ | $0.528769$ | $0.495569$
|  | $0.002262$ | $0.179746$ | $78.79531$ | $0.182699$ | $0.002158$
$100$ | $0.50$ | $0.500999$ | $0.504581$ | $1.37044$ | $0.505124$ | $0.494758$
|  | $0.001028$ | $0.027757$ | $76.54193$ | $0.027899$ | $0.001002$
$10$ | $1.00$ | $1.067974$ | $1.072587$ | $5.07754$ | $1.185889$ | $0.986976$
|  | $0.064716$ | $2.146959$ | $241.826$ | $2.146005$ | $0.053416$
$20$ | $1.00$ | $1.028478$ | $1.185968$ | $4.71047$ | $1.095143$ | $0.979043$
|  | $0.026007$ | $1.264714$ | $166.107$ | $1.312496$ | $0.022954$
$50$ | $1.00$ | $1.010089$ | $1.237631$ | $3.19992$ | $1.084109$ | $0.985321$
|  | $0.0091631$ | $1.07522$ | $121.167$ | $1.17262$ | $0.008626$
$100$ | $1.00$ | $1.004157$ | $1.426787$ | $2.11971$ | $1.08019$ | $0.989713$
|  | $0.004993$ | $0.832140$ | $111.24400$ | $1.106002$ | $0.004813$
$10$ | $1.50$ | $1.61502$ | $1.570825$ | $5.144353$ | $1.185889$ | $0.986976$
|  | $0.1655082$ | $0.251852$ | $241.826234$ | $2.146005$ | $0.053416$
$20$ | $1.50$ | $1.564556$ | $1.554483$ | $1.75837$ | $1.564219$ | $1.482625$
|  | $0.073945$ | $0.1023667$ | $153.753451$ | $1.103020$ | $0.044771$
$50$ | $1.50$ | $1.514692$ | $1.520526$ | $1.637223$ | $1.524409$ | $1.473679$
|  | $0.025980$ | $0.101587$ | $76.340366$ | $0.811026$ | $0.024301$
$100$ | $1.50$ | $1.507803$ | $1.413624$ | $1.468612$ | $1.453052$ | $1.483952$
|  | $0.011712$ | $0.080194$ | $14.224941$ | $0.278361$ | $0.011236$

Table 2: $\theta$ and its estimated average widths and coverage probabilities
of ACI for IXGD.

|  | Confidence Limits |  |
---|---|---|---|---
$n$ | $\theta$ | $L$ | $U$ | Average | Cov.
|  |  |  | Width | Prob.
$10$ | $0.10$ | $0.0000$ | $0.2170$ | $0.2170$ | $0.939$
$20$ | $0.10$ | $0.0000$ | $0.2166$ | $0.2166$ | $0.941$
$50$ | $0.10$ | $0.0000$ | $0.2164$ | $0.2164$ | $0.941$
$100$ | $0.10$ | $0.0000$ | $0.2143$ | $0.2143$ | $0.943$
$10$ | $0.50$ | $0.0000$ | $1.1510$ | $1.1510$ | $0.938$
$20$ | $0.50$ | $0.0000$ | $1.1418$ | $1.1418$ | $0.941$
$50$ | $0.50$ | $0.0000$ | $1.1334$ | $1.1334$ | $0.941$
$100$ | $0.50$ | $0.0000$ | $1.1166$ | $1.1166$ | $0.944$
$10$ | $1.00$ | $0.0000$ | $2.4052$ | $2.4052$ | $0.939$
$20$ | $1.00$ | $0.0000$ | $2.3219$ | $2.3219$ | $0.940$
$50$ | $1.00$ | $0.0000$ | $2.3094$ | $2.3094$ | $0.941$
$100$ | $1.00$ | $0.0000$ | $2.2732$ | $2.2732$ | $0.942$
$10$ | $1.50$ | $0.0000$ | $3.5353$ | $3.5353$ | $0.938$
$20$ | $1.50$ | $0.0000$ | $3.4840$ | $3.4840$ | $0.940$
$50$ | $1.50$ | $0.0000$ | $3.4460$ | $3.4460$ | $0.941$
$100$ | $1.50$ | $0.0000$ | $3.3722$ | $3.3722$ | $0.942$

##  7 Real Life Examples.

Here, we consider two real data sets to show the practical applicability of
the proposed model. We check whether the considered data sets actually come
from the IXGD or not by goodness of fit test. For this purpose, we compared
the newly introduced model IXGD with well known one parameter inverted family
of distributions, namely, inverse exponential distribution (IED), inverse
Rayleigh distribution (IRD), inverse Lindley distribution (ILD). This
procedure is based on the Kolmogorov-Smirnov (K-S) statistic and it compares
an empirical and a theoretical model by computing the maximum absolute
difference between the empirical and theoretical distribution functions and is
defined as $D_{n}=Sup_{x}|F_{n}(x)-F(x;\alpha)|$, where $Sup_{x}$ is the
supremum of the set of distances, $F_{n}(x)$ is the empirical distribution
function and $F(x;\alpha)$ is the CDF. Note that, K-S statistic to be used
only to verify the goodness-of-fit and not as a discrimination criteria.
Therefore, we consider four discrimination criteria based on the log-
likelihood function evaluated at the maximum likelihood estimates. The
criteria are: AIC (Akaike Information Criterion), BIC (Bayesian Information
Criterion), CAIC (Consistent Akaike Information Criterion) and HQIC (Hannan-
Quinn Information Criterion). These statistics are given by
$AIC=-2l(\hat{\alpha})+2p$, $BIC=-2l(\hat{\alpha})+2\ln(n)$,
$CAIC=-2l(\hat{\alpha})+p(\ln(n)+1)$ and
$HQIC=-2l(\hat{\alpha})+2p\ln(\ln(n))$, where $l(\hat{\alpha})$ denotes the
log-likelihood function evaluated at the MLEs, $p$ is the number of model
parameters and $n$ is the sample size. The model with lowest values for these
statistics could be chosen as the best model to fit the data. The values of
MLE of the parameter, $l(\hat{\alpha})$, AIC, BIC, HQIC, CAIC & K-S Statistic
are displayed in Table $3$. Among all other competitive models, it is to be
noted that the IXGD($\alpha$) has the lowest values of $l(\hat{\alpha})$, AIC,
BIC, HQIC, CAIC & K-S and so it could be chosen as the best model to fit the
given data sets.

  * •

Data Set I: Postate cancer data taken from Collett ($2003$) of size $38$. For
this data, we have compared the proposed model IXGD with well known one
parameter inverted family of distributions, viz., inverted Exponential
distribution (IED), inverse Lindley distribution (ILD), inverse Rayleigh
distribution (IRD) and found that IXGD is the better choice for the considered
data set [see, Table $3$].

  * •

Data Set II: Item failure data represents the $46$ repair times (in hours)
have taken from Chhikara and Folks ($1997$), initially considered by Chhikara
and Folks ($1977$), for an airborne communication transceiver. Here also, we
have compared the proposed model IXGD with well known one parameter inverted
family of distributions, viz., inverted Exponential distribution (IED),
inverse Lindley distribution (ILD), inverse Rayleigh distribution (IRD) and
found that IXGD is the better choice for the considered data set [see, Table
$3$].

Table 3: The model fitting summary for the considered data sets based on MLE.

Data | Model | Estimate | Negative | AIC | BIC | CAIC | HQIC | K-S
---|---|---|---|---|---|---|---|---
Set |  |  of $\theta$ | Log-likelihood |  |  |  |  | Statistic
I | IED | $24.97312$ | $200.1996$ | $402.3996$ | $404.0369$ | $405.0369$ | $402.9819$ | $0.31005$
| ILD | $25.90154$ | $200.2675$ | $402.5350$ | $404.1726$ | $405.1726$ | $403.1176$ | $0.29978$
| IRD | $120.6323$ | $256.2503$ | $514.5006$ | $516.1382$ | $517.1382$ | $515.0832$ | $0.67427$
| IXGD | $26.82069$ | $199.4590$ | $400.9181$ | $402.5570$ | $403.5570$ | $401.5007$ | $0.30710$
|  |  |  |  |  |  |  |
II | IED | $1.13620$ | $100.6971$ | $203.3941$ | $205.2228$ | $206.2281$ | $204.0792$ | $0.06790$
| ILD | $1.57712$ | $100.1692$ | $204.3385$ | $206.1671$ | $207.1671$ | $205.0235$ | $0.08128$
| IRD | $0.60590$ | $128.0239$ | $258.0478$ | $259.8763$ | $260.8765$ | $258.7329$ | $0.33770$
| IXGD | $1.90130$ | $101.1312$ | $204.2623$ | $206.0909$ | $207.0909$ | $204.9473$ | $0.06720$

Also, the estimates of the parameter $\theta$ and reliability characteristics
of IXGD for both the data sets are computed using different methods of
estimation [see, Table $4$].

Table 4: Widths of ACI of the parameter $\theta$ for IXGD

Data Set | Model | $\hat{\theta}_{MLE}$ | $\hat{\theta}_{LSE}$ | $\hat{\theta}_{WLSE}$ | $\hat{\theta}_{CME}$ | $\hat{\theta}_{MPSE}$ | ACI
---|---|---|---|---|---|---|---
I | IXGD | $26.82069$ | $33.22330$ | $26.63840$ | $33.19222$ | $26.75462$ | $0.5346$
II | IXGD | $01.90130$ | $01.98668$ | $01.95813$ | $01.99161$ | $01.88956$ | $0.8752$

##  8 Concluding Remarks.

In this article, we have proposed a new probability model, namely, IXGD by
considering the inverse of XGD. Different statistical characteristics and
properties have been discussed. Different methods of estimation have been
discussed for estimating the unknown parameter of the proposed model. The
comparison among the considered estimators of $\theta$ have been carried out
using Monte Carlo simulation study and it has been noticed that MPSE of
$\theta$ performed better in MSE sense. Further, $95\%$ ACI of $\theta$ has
been calculated using MPSE of $\theta$ as it performed better among the other
estimators. Finally, two real data sets have been analyzed for illustration
purposes of the study.

## References

  1. 1.

Cheng R. C. H., Amin N. A. K. (1979). Maximum product-of-spacings estimation
with applications to the log-normal distribution, University of Wales IST,
Math Report, 79-1.

  2. 2.

Cheng, R. C. H. and Amin, N.A.K. (1983). Estimating parameters in continuous
univariate distributions with a shifted origin. Journal of the Royal
Statistical Society: Series B Statistical Methodology, 3, 394-403.

  3. 3.

Chhikara, R. S. and Folks, J. L. (1977). The inverse gaussian distribution as
a lifetime model. Technometrics, 19, 461-468.

  4. 4.

Collett (2003). Modelling Survival Data in Medical Research. 2, Chapman and
Hall/CRC.

  5. 5.

Ghosh, S. R. Jammalamadaka, (2001). A general estimation method using
spacings. Journal of Statistical Planning and Inference, 93.

  6. 6.

Ihaka, R. and Gentleman, R. (1996). R: A language for data analysis and
graphics. Journal of Computational and Graphical Statistics, 5, 299-314.

  7. 7.

Keller, A. Z and Kamath, A. R (1982). Reliability analysis of CNC Machine
tools. Reliability Engineering, 3, 449-473.
http://dx.doi.org/10.1016/0143-8174(82)90036-1.

  8. 8.

MacDonald, P. D. M. (1971). Comment on an estimation procedure for mixtures of
distributions by Choi and Bulgren. Journal of Royal Statistical Society:
Series B, 33(2), 326-329.

  9. 9.

Ranneby, B. (1984). The maximum spacing method. an estimation method related
to the maximum likelihood Method. Scandinavian Journal of Statistics, 11(2),
93-112.

  10. 10.

Sen, S., Maiti, S. S. and Chandra, N. (2016). The xgamma Distribution:
statistical properties and Application Journal of Modern Applied Statistical
Methods, 15(1), 774-788.

  11. 11.

Shaked, M., and Shanthikumar, J. G. (1994). Stochastic orders and their
applications. New York, NY: Academic Press.

  12. 12.

Sharma, V. K., Singh, S. K., Singh, U. and Agarwal, V. (2015). The inverse
Lindley distribution: a stress-strength reliability model with application to
head and neck cancer data. Journal of Industrial and Production Engineering,
32(3), 162-173.

  13. 13.

Swain, J., Venkatraman, S. and Wilson, J. (1988). Least squares estimation of
distribution function in Johnsons translation system. Journal of Statistical
Computation and Simulation, 29, 271-297.

  14. 14.

Voda, V. G. (1972). On the inverse Rayleigh random variable. Rep. Stat. Appl.
Res. Jues, 19(4), 13-21.