A New Generalized Two-Sided Class of the Distributions Via New Transmuted Two-Sided Bounded Distribution

O. Kharazmi; M. Zargar

doi:10.2991/jsta.d.190306.003

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Volume 18, Issue 2, June 2019, Pages 87 - 102

A New Generalized Two-Sided Class of the Distributions Via New Transmuted Two-Sided Bounded Distribution

Authors

O. Kharazmi, M. Zargar^*

Department of Statistics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran

^*Corresponding author. Email: zargar_m@vru.ac.ir

Corresponding Author

M. Zargar

Received 8 August 2017, Accepted 2 February 2018, Available Online 18 June 2019.

DOI: 10.2991/jsta.d.190306.003 How to use a DOI?
Keywords: Hazard rate function; order statistics; maximum likelihood estimator; transmutation map; transmuted two-sided distribution
Abstract: In the present paper, we first consider a generalization of the standard two-sided power distribution so-called the transmuted two-sided distribution, and then extend proposed idea to generalized two-sided class of distributions, introduced by Korkmaz and Genç [1]. Some statistical and reliability properties including explicit expressions for quantiles, hazard rate function, order statistics, and maximum likelihood estimation are obtained in general setting. Generalized transmuted two-sided exponential distribution is considered as a especial case and denoted with the name TTSG−G. A simulation study is presented to investigate the bias and mean square error of the maximum likelihood estimators. We use a real data set and obtain the maximum likelihood and parametric bootstrap estimator of the parameters of TTSG−G distribution. Finally, the superiority of the new model to some common statistical distributions is shown through the different criteria of selection model including log-likelihood values, Akaike information criterion, and Kolmogorov–Smirnov test statistic values.
Copyright: © 2019 The Authors. Published by Atlantis Press SARL.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

The statistical distribution theory has been widely explored by researchers in recent years. Given the fact that the data from our surrounding environment follow various statistical models, it is necessary to extract and develop appropriate high-quality models. One of the most important models in the statistical theory is the change point models. In the distribution theory, the change point distributions are used in the different branch of sciences such as economic, engineering, agriculture, and so on. Van Dorp and Kotz [2] introduced a family of the change point distributions so-called two-sided power (TSP) distribution with the probability density function (pdf),

(1)fx;α,β=αxβα−1,0<x≤β,α1−x1−βα−1,β≤x<1,

and with the cumulative distribution function cd f ,

(2)Fx;α,β=βxβα,0<x≤β,1−1−β1−x1−βα,β≤x<1,

where 0≤β≤1 and α>0. The parameter β is the location parameter called “turning point” and α is the shape parameter that control the shape of distribution on the left and right of β.

The TSP distribution for α>3 can be used for modeling unimodal phenomena on a bounded domain when a peak in data is observed. Van Dorp and Kotz [3] introduced an extension of the three-parameter triangular distribution utilized in risk analysis. Their model includes the TSP distribution as a special case. Van Dorp and Kotz [4] considered a family of continuous distributions on a bounded interval generated by convolutions of the TSP distributions. In recent years, a number of researchers have studied some generalization of the TSP distribution such as Nadarajah [5], Oruç and Bairamov [6], Vicari et al. [7], Herrerías-Velasco et al. [8], and Soltani and Homei [9]. Korkmaz and Genç [10] proposed a new generalization of Weibull distribution by making use of a transformation of the standard TSP distribution. Also, Korkmaz and Genç [11] considered the log transformation of the TSP distribution instead of uniform distribution and introduced a generalization of the exponential distribution. Korkmaz and Genç [1] extended the idea of two-sidedness to other ordinary distributions like normal and introduced the two-sided generalized normal distribution.

One of the interesting methods for constructing new distributions is the transmutation map approach. Recently, some new distributions have been generalized based on the transmutation method. The transmuted distribution based on the Gx is defined as

(3)Hx=1+λGx−λGx2, |λ|≤1,

where Gx is the cdf of the parent distribution.

Aryal and Tsokos [12] generated a flexible family of probability distributions taking extreme value distribution as the base value distribution by using the quadratic rank transmutation map (QRTM). Aryal and Tsokos [13] generalized the two-parameter Weibull distribution using the QRTM. Aryal [14] introduced a generalization of the log-logistic distribution so-called the transmuted log-logistic distribution. Abd El Hady [15] introduced a new generalization of the two-parameter Weibull distribution by using the QRTM. This new distribution is named exponentiated transmuted Weibull (ETW) distribution. Elgarhy and Shawki [16] introduced a new generalized version of the quasi Lindley distribution which is called the transmuted generalized quasi Lindley (TGQL) distribution.

The transmutation method, as an important method for developing statistical distributions, hasn't been used for the change point distributions, yet. The main motivation of the present paper is to apply the transmutation technique for increasing the flexibility and usefulness of the TSP distribution and generalized TSP class of distributions.

This paper organized as follows. In Section 2, we introduce a new distribution so-called transmuted two-sided distribution. In Section 3, we propose a generalization of the transmuted two-sided distribution and consider the hazard function, quantiles, and order statistics of this distribution. We consider the exponential distribution as a parent distribution and introduce transmuted two-sided generalized exponential (TSGE) distribution, in Section 4. In this section, we plot the shape of density function and hazard function. In Section 5, the estimation of parameters of the generalized transmuted two-sided distribution are obtained by using two methods maximum likelihood estimation (MLE) and bootstrap estimation. Also, we study the performance of MLEs of parameters of the transmuted TSGE distribution via a simulation study. In Section 6, the superiority of new model to some competitor statistical models is shown through the different criteria of selection model. Finally, the paper is concluded in Section 7.

2. TRANSMUTED TWO-SIDED DISTRIBUTION

In this section, we introduce the transmuted two-sided distribution and then we consider its shape for different values of parameters. The main motivation for introducing this new family is to provide the more flexibility for the TSP distribution by compounding two-sided distributions family and transmutation map approach.

Let the random variable X1 have a beta distribution with pdf,

fX1x=axα−1, 0<x<1,α>0.

Assume that X be a random variable associated with the truncated X1 on the right at β, as

X=dX1|0<X1≤β,

where =d denotes identically distributed. The cdf of the random variable X is given by

(4)FXx=aβα, 0<x≤β,α>0.

Using Eqs. (3) and (4) the pdf of the transmuted truncated beta distribution is given by

(5)hx=1+λfXx−2λfXxFXx =1+λαβxβα−1−2λαβxβ2α−1, 0<x≤β.

Now, suppose that Z be a random variable with the density function hx. By defining

Y=1−1−ββZ,

the pd f of Y is given by

(6)hY(y)=α1−β1+λ1−y1−βα−1−2λ1−y1−β2α−1, β≤y<1.

According to relations Eqs. (5) and (6), a new distribution is defined by

(7)fx;α,β,λ=βhx,0<x≤β,1−βhYx,β≤x<1.

So, based on Eq. (7), we have the following definition.

Definition 2.1

A random variable X is said to be transmuted two-sided distribution if its pdf is given by

(8)fx;α,β,λ=α1+λxβα−1−2λxβ2α−1,0<x≤β,α1+λ1−x1−βα−1−2λ1−x1−β2α−1,β≤x<1,

and its cdf is given by

(9)Fx;α,β,λ=β1+λxβα−λxβ2α, 0<x≤β,131−1−β1+λ1−x1−βα−λ1−x1−β2α, β≤x<1,

where, α is a shape parameter, β is a scale parameter, and λ is a transmuted parameter. We denote the transmuted two-sided distribution by TTS(α,β,λ).

Remark 2.1.

If λ=0, we have the pdf and cdf of TSP distribution in Eqs. (1) and (2), respectively.

2.1. Density Shape of the TTS Distribution

Here, we consider a discussion about the shape of the proposed density function. In the end points of the support, the behaviour of the pdf Eq. (8) is given as follows:

limx→0,1fx;α,β,λ=∞,α<1,1+λ,α=1,0,α>1.

The derivative f′x;α,β,λ is

f′x;α,β,λ=αβx2xβα−4α+2 λxβα+1+λα−1 ,0<x≤β,αβ−1 x−121−x1−βα−4α+2λ1−x1−βα+1+λα−1 ,β≤x<1.

The right- and left-hand limits of f′ at x=β are given by

limx→β+f′x;α,β,λ=α21−3λ+αλ−1β,limx→β−f′x;α,β,λ=α21−3λ+αλ−1β−1.

These limits are different. So, f′β does not exist and f has a corner at x=β.

When α>1, λ>0, and 0<x≤β, if f′x=0 then x1=β1+λα−14α−2λ1/α and when α>1, λ>0, and β≤x<1, if f′x=0 then x2=1+β−11+λα−14α−2λ1/α. So, when α>1 and λ>0, the maximum of f occurs at points x1 and x2. It implies when α>1 and λ>0, f is a bimodal pdf.

When α≥1, λ≤0, and 0<x≤β, we have f′x>0 and when α≥1, λ≤0, and β≤x<1, we have f′x<0. So, for α≥1, λ≤0, f is a unimodal pdf. The density shapes of the TTS distribution for different choices of the parameters are plotted in Fig. 1.

3. TRANSMUTED TWO-SIDED GENERALIZED-G FAMILY OF THE DISTRIBUTIONS

Consider a continuous random variable with cdf Gx;ξ and pdf gx;ξ. Using Definition 2.1, we define a generalization of the transmuted two-sided distribution as follows:

Definition 3.1

A random variable X is said to be transmuted two-sided generalized-G (TSG-G) distribution if its pdf is given by

fx;α,β,λ,ξ=αgx;ξ1+λGx;ξβα−1−2λGx;ξβ2α−1,−∞<x≤Gx;ξ−1β,αgx;ξ1+λ1−Gx;ξ1−βα−1−2λ1−Gx;ξ1−β2α−1,Gx;ξ−1β≤x<∞,

and its cdf is given by

Fx;α,β,λ,ξ=β1+λGx;ξβα−λGx;ξβ2α,−∞<x≤Gx;ξ−1β,1−1−β1+λ1−Gx;ξ1−βα−λ1−Gx;ξ1−β2α,Gx;ξ−1β≤x<∞,

where, −∞<x<∞, ξ is a parameter vector in the cdf Gx;ξ and Gx;ξ−1. is its inverse.

We denote transmuted TSG-G family of distributions by TTSG−Gα,β,λ,ξ.

Remark 3.1.

If λ=0, we have the pdf and cdf of the TSG-G family introduced by Korkmaz and Genç [1]. If α=1, β=1, and λ=0, we have the pdf of the base distribution.

In the next subsections, we study hazard function, random variate generation, order statistics and relative entropy of the TTSG−G distribution.

3.1. Hazard Function

The hazard rate is a fundamental tools in reliability modelling for evaluation the ageing process. Knowing the shape of the hazard rate is important in reliability theory, risk analysis, and other disciplines. The concepts of increasing, decreasing, bathtub-shaped (first decreasing and then increasing) and upside-down bathtub-shaped (first increasing and then decreasing) hazard rate functions are very useful in reliability analysis. The lifetime distributions with these ageing properties are designated as the increasing failure rate (IFR), decreasing failure rate (DFR), bathtub-shape (BUT), and upside-down bathtub-shaped (UBT) distributions, respectively. The hazard function of the TTSG−G distribution is given by

rx=fx1−Fx =αgx;ξ1+λGx;ξβα−1−2λGx;ξβ2α−11−β1+λGx;ξβα−λGx;ξβ2α,−∞<x≤Gx;ξ−1β,αgx;ξ1+λ1−Gx;ξ1−βα−1−2λ1−Gx;ξ1−β2α−11−β1+λ1−Gx;ξ1−βα−λ1−Gx;ξ1−β2α,Gx;ξ−1β≤x<∞.

3.2. Random Variate Generation

For generating random variables from the TTSG−G distribution, we use the inverse transformation method. The quantile of order q of the TTSG−G distribution is

xq=F−1q;α,β,λ,ξ=Gx;ξ−1β1+λ−1+λ2−4λqβ2λ1α,0<q≤β, Gx;ξ−11−1−β1+λ−1+λ2−4λ1−q1−β2λ1α,β≤q<1.

Let U be a random variable generated from a uniform distribution on 0,1, then

X=Gx;ξ−1β1+λ−1+λ2−4λUβ2λ1α,0<U≤β, Gx;ξ−11−1−β1+λ−1+λ2−4λ1−U1−β2λ1α,β≤U<1,

is a random variable generated from the TTSG−G distribution by the probability integral transform.

3.3. Order Statistics

Order statistics play a vital role in the theory of probability and statistics. Let X1,X2,…,Xn be a random sample from the TTSG−G distribution. Let Xi:n denote the ith order statistics. Then the pdf of Xi:n is given by

fi:nx=∑t1=0n−i∑t2=0i−1+t1n−it1i−1+t1t2−1t1−λi−1+t1−t21+λt2βi−1+t1 × gx;ξ1+λGx;ξβα2i+2t1−t2−1−1−2λGx;ξβα2i+2t1−t2−1,x≤Gx,;ξ−1β,∑t1=0i−1∑t2=0n−t1−1i−1t1n−t1−1t2−1i−1−t1−λn−t1−t2−11+λt21−βn−t1−1 × gx;ξ1+λ1−Gx;ξ1−βα2n−2t1−t2−1−1−2λ1−Gx;ξ1−βα2n−2t1−t2−1,Gx;ξ−1β≤x,

where A=n!αi−1!n−i!.

3.4. Kullback–-Leibler Divergence

The Kullback–Leibler divergence (or relative entropy) is an informational measure for comparing the similarity between two pdfs. The Kullback–Leibler divergence between the proposed distribution TTSG−G and the TSG−G distribution, introduced by Korkmaz and Genç [1], is obtained as

(10)KlfTTSG−G|| fTSG−G=∫−∞∞fTTSG−G(x)logfTTSG−GxfTSG−Gxdx =∫011+λ−2λulog 1+λ−2λudu =−1−λ2log1−λ+1+λ2log1+λ−2λ4λ.

On the other hand, if U be a uniform random variable on (0,1) then the corresponding transmuted distribution is given as

fUu=1+λ−2λu, 0<u<1.

The Shannon entropy of transmuted uniform distribution is computed by

(11)HU=−∫01fUulogfUudu =−∫011+λ−2λulog1+λ−2λudu =−−1−λ2log 1−λ+1+λ2log 1+λ−2λ4λ.

From the relations, Eqs. (10) and (11) we see that the Kullback–Leibler divergence between the TTSG−G and the TSG−G distributions is equal to −HU. So, this informational measure is free of the parent distribution G.

4. TRANSMUTED TSGE DISTRIBUTION

The TTSG−G distribution is specialized by taking G as the well-known distribution. We suppose that the base distribution G has an exponential distribution with cdf and inverse cdf function Gx;θ=1−e−xθ,x>0,θ>0 and G−1x;θ=−θlog1−x, respectively. The pdf of the parent distribution is gx;θ=1θe−xθ. By considering this distribution, the pdf of the TTSG−G distribution can be given as

fx;α,β,λ,θ=α1θe−xθ1+λ1−e−xθβα−1−2λ1−e−xθβ2α−1,0<x≤−θlog1−β,α1θe−xθ1+λe−xθ1−βα−1−2λe−xθ1−β2α−1,−θlog1−β≤x<∞,

and its cdf is given by

Fx;α,β,λ,θ=β1+λ1−e−xθβα−λ1−e−xθβ2α,0<x≤−θlog1−β,1−1−β1+λe−xθ1−βα−λe−xθ1−β2α,−θlog1−β≤x<∞.

We call this distribution the transmuted TSGE distribution and denote by TTSG−E(α,β,λ,θ).

Remark 3.1.

If λ=0, we have the pdf and cdf of the TSGE distribution introduced by Korkmaz and Genç [11]. If α=1, β=1 and λ=0, we have the pdf of the base distribution.

4.1. Density Shape of the TTSG — E Distribution

In the end points of the support, the behaviour of the pdf of the TTSG−E distribution is given as follows:

limx→0fx;α,β,λ,θ=∞,α<1,1+λθ,α=1, limx→∞fx;α,β,λ,θ=0, ∀α>0.0,α>1,

The derivative f′x;α,β,λ,θ is

f′(x;α,β,λ,θ)=α1−e−xθα−2e−xθθ2βα−1[(2λ−4λαe−xθ)(1−e−xθβ) +[(1+λ)(αe−xθ−1)],0<x≤−θlog(1−β),α2e−xθαθ2(1−β)α−1[4λe−xθ1−βα−λ−1],−θlog(1−β)≤x<∞.

The right- and left-hand limits of f′ at x=−θlog1−β are given by

limx→−θlog 1−β+f′x;α,β,λ,θ=α1−βθ2βαβ−13λ−1+λ−1,

limx→−θlog 1−β−f′x;α,β,λ,θ=α21−β3λ−1θ2.

These limits are not equal. So, f′−θlog1−β does not exit and the TTSG−E distribution has a corner at x=−θlog1−β. The shape of the TTSG−E distribution for different values of the parameters are plotted in Figs. 2 and 3. Figures 2 and 3 indicate that for α>1 and λ>0, the TTSG−E distribution is a bimodal distribution and for α≥1 and λ≤0, the TTSG−E distribution is a unimodal distribution.

In the next section, we consider the hazard shape of the TTSG−E distribution.

4.2. Hazard Function of the TTSG — E Distribution

The hazard function of the TTSG−E distribution is

rx=fx;α,β,λ,θ1−Fx;α,β,λ,θ =α1θe−xθ1+λ1−e−xθβα−1−2λ1−e−xθβ2α−11−β1+λ1−e−xθβα−λ1−e−xθβ2α,0<x≤−θlog1−β,α1θe−xθ1+λe−xθ1−βα−1−2λe−xθ1−β2α−11−β1+λe−xθ1−βα−λe−xθ1−β2α,−θlog1−β≤x<∞.

Because of complicated form of the hazard function, we couldn't explore this function analytically. We only consider the end points of the support. The behaviour of the hazard function in the end points is given as follows:

limx→0rx=∞,α<1,1+λθ,α=1, limx→∞rx=αθ, ∀α>0.0,α>1,

Some shapes of the hazard function for the selected values of parameters is given in Figs. 4 and 5. Figures 4 and 5 show that the hazard rate function of the TTSG−E distribution can be IFR, DFR, BUT, and UBT.

5. ESTIMATION OF THE PARAMETERS OF THE TTSG — G DISTRIBUTION

In this section, we obtain the estimation of parameters the TTSG−G distribution by using two methods: MLE and bootstrap estimation. Also, a simulation study is conducted for MLEs of parameters of the TTSG−E distribution.

5.1. Maximum Likelihood Estimation

Let X1,X2,…,Xn be a random sample of size n from the TTSG−G distribution and X1:n≤X2:n≤…≤Xn:n denote the corresponding order statistics. The log-likelihood function is given by

ℓα,β,λ,ξ=nlog α+∑i=1nloggxi;ξ + log ∏i=1r1+λGxi:n;ξβα−1−2λGxi:n;ξβ2α−1 × ∏i=r+1n1+λ1−Gxi:n;ξ1−βα−1−2λ1−Gxi:n;ξ1−β2α−1,

where xr:n≤Gx;ξ−1β<xr+1:n for r=1,2,…,n and X0:n=−∞, Xn+1:n=∞.

For estimating the parameters, we obtain the partial derivatives of the log-likelihood function with respect to the parameters. At the corner point β, the log-likelihood function for the TTSG−G distribution is not differentiable and we can not find the estimate of β in a regular way. According to Van Dorp and Kotz [2], we can find the MLE of parameters. We first consider the MLE's of α and β when the parameters λ and ξ are known. Without loss of generality, we assume that λ=0. So, the log-likelihood function will be

ℓα,β,λ,ξ=nlogα+∑i=1nloggxi;ξ+log ∏i=1rGxi:n;ξβα−1∏i=r+1n1−Gxi:n;ξ1−βα−1 =nlogα+∑i=1nloggxi;ξ+α−1log∏i=1rGxi:n;ξ∏i=r+1n1−Gxi:n;ξβr1−βn−r.

According to Van Dorp and Kotz [2] and Korkmaz and Genç [1], the MLEs of α and β are as follows:

α^=−nlogMr^,ξ, β^=Gxr^:n;ξ,

where r^=argmaxMr,ξ, r∈1,2,…,n with Mr,ξ=∏i=1r−1Gxi:n;ξGxr:n;ξ∏r+1n1−Gxi:n;ξ1−Gxr:n;ξ.

By taking the derivative of the log-likelihood function with respect to parameter vector ξ and parameter λ, the MLEs of parameters ξ and λ are obtained by equating it to zero. These derivatives are given as

∂ℓα,β,λ,ξ∂λ=∑i=1r^Gxi:n;ξβα−1−2Gxi:n;ξβ2α−11+λGxi:n;ξβα−1−2λGxi:n;ξβ2α−1 +∑i=r^+1n1−Gxi:n;ξ1−βα−1−21−Gxi:n;ξ1−β2α−11+λ1−Gxi:n;ξ1−βα−1−2λ1−Gxi:n;ξ1−β2α−1,

∂ℓα,β,λ,ξ∂ξk=∑i=1ng′xi:n;ξgxi:n;ξ +∑i=1r^G′xi:n;ξβ.1+λα−1Gxi:n;ξβα−2−2λ2α−1Gxi:n;ξβ2α−21+λGxi:n;ξβα−1−2λGxi:n;ξβ2α−1 +∑i=r^+1n−G′xi:n;ξ1−β.1+λα−11−Gxi:n;ξ1−βα−2−2λ2α−11−Gxi:n;ξ1−β2α−21+λ1−Gxi:n;ξ1−βα−1−2λ1−Gxi:n;ξ1−β2α−1,

where g′t;ξ=∂gt;ξ∂ξk and G′t;ξ=∂Gt;ξ∂ξk.

However, these equations are nonlinear and there are no explicit solutions. Thus, they have to be solved numerically. So, the optim package is used for estimating the parameters in R software.

5.2. Bootstrap Estimation

The parameters of the fitted distribution can be estimated by parametric (resampling from the fitted distribution) bootstrap resampling (see Efron and Tibshirani [17]). The parametric bootstrap procedure is described as follows:

Parametric bootstrap procedure:

1
Estimate θ (vector of unknown parameters), say θ^, by using the MLE procedure based on a random sample.
2
Generate a bootstrap sample {X1∗,…,Xm∗} using θ^ and obtain the bootstrap estimate of θ, say θ*^, from the bootstrap sample based on the MLE procedure.
3
Repeat Step 2 NBOOT times.
4
Order θ*^1,…,θ*^NBOOT as θ*^1,…,θ*^NBOOT. Then obtain γ-quantiles and 1001−α% confidence intervals (CIs) of parameters.

In case of the TTSG−G distribution, the parametric bootstrap estimators (PBs) of α,β,λ, and ξ, say α^PB,β^PB,λ^PB, and ξ^PB, respectively.

5.3. Simulation

Here, we assess the performance of the MLEs of the parameters with respect to sample size n for the TTSG−E distribution. The assessment of performance is based on a simulation study by using the Monte Carlo method. Let α^,β^,λ^, and θ^ be the MLEs of the parameters α,β,λ, and θ, respectively. We calculate the mean square error (MSE) and bias of the MLEs of the parameters, α,β,λ and θ based on the simulation results of 3000 independence replications. results are summarised in Table 1 for different values of, n,α,β,λ and θ. From Table 1 the results verify that MSE of the MLEs of the parameters decrease with respect to sample size n for all parameters. So, we can see the MLEs of α,β,λ, and θ are consistent estimators.

		α=0.5	β=0.3	λ=0.5	θ=0.5
n	30	0.0274 (0.0316)	0.0165 (0.0099)	0.0855 (0.0481)	0.0272 (0.0229)
	50	0.0153 (0.0269)	0.0092 (0.0034)	0.0721 (0.0502)	0.0141 (0.0147)
	100	0.0081 (0.0174)	0.0045 (0.0030)	0.0514 (0.0451)	0.0063 (0.0071)
	200	0.0035 (0.0098)	0.0020 (0.0018)	0.0243 (0.0245)	0.0033 (0.0046)
		α=0.5	β=0.3	λ=0.5	θ=1.5
n	30	0.0286 (0.0357)	0.0181 (0.0093)	0.0870 (0.0552)	0.2462 (0.0742)
	50	0.0156 (0.0294)	0.0097 (0.0057)	0.0720 (0.0552)	0.1245 (0.0412)
	100	0.0079 (0.0202)	0.0042 (0.0011)	0.0484 (0.0470)	0.0632 (0.0352)
	200	0.0035 (0.0097)	0.0019 (−0.0007)	0.0264 (0.0299)	0.0285 (0.0138)
		α=2	β=0.3	λ=0.75	θ=0.5
n	30	32.5268 (0.6036)	0.0291 (0.0479)	0.0914 (−0.0422)	2.9212 (0.1673)
	50	13.0018 (0.3243)	0.0167 (0.0174)	0.0449 (0.0192)	1.1566 (0.0768)
	100	0.1474 (0.0976)	0.0063 (0.0019)	0.0236 (0.0454)	0.0132 (0.0142)
	200	0.0412 (0.0524)	0.0022 (−0.0016)	0.0138 (0.0327)	0.0025 (0.0072)
		α=2	β=0.3	λ=0.75	θ=1.5
n	30	13.8834 (0.5349)	0.0291 (0.0320)	0.0946 (−0.0477)	16.7689 (0.4470)
	50	26.9195 (0.4725)	0.0174 (0.0109)	0.0460 (0.0255)	12.5857 (0.3298)
	100	2.0210 (0.1251)	0.0067 (−0.0021)	0.0249 (0.0396)	1.5966 (0.0752)
	200	0.0467 (0.0624)	0.0023 (−0.0025)	0.0150 (0.0388)	0.0256 (0.0241)
		α=0.5	β=0.3	λ=−0.5	θ=0.5
n	30	16.0098 (0.6477)	0.0589 (0.0013)	0.8003 (0.8346)	3.5771 (0.2223)
	50	0.5018 (0.2976)	0.0619 (0.0477)	0.7082 (0.7889)	0.2463 (0.0554)
	100	0.2836 (0.2171)	0.0600 (0.0730)	0.4518 (0.6418)	0.1114 (0.0304)
	200	0.0460 (0.1789)	0.0595 (0.0878)	0.3221 (0.5549)	0.0172 (0.0254)
		α=0.5	β=0.3	λ=−0.5	θ=1.5
n	30	5.7339 (0.5136)	0.0664 (0.0481)	0.8871 (0.8801)	13.4951 (0.4215)
	50	2.1495 (0.3433)	0.0707 (0.0714)	0.6986 (0.7841)	8.0247 (0.2459)
	100	0.0741 (0.2105)	0.0666 (0.0826)	0.4685 (0.6529)	0.2054 (0.0803)
	200	0.0449 (0.1787)	0.0615 (0.0871)	0.3225 (0.5558)	0.1457 (0.0802)
		α=2	β=0.3	λ=−0.75	θ=0.5
n	30	83.3071 (3.0259)	0.0518 (0.0820)	0.8762 (0.8905)	1.5715 (0.1644)
	50	24.7184 (1.9937)	0.0413 (0.0725)	0.9042 (0.9040)	0.4906 (0.0413)
	100	4.6772 (1.5489)	0.0344 (0.0633)	0.9048 (0.9032)	0.0876 (0.0049)
	200	2.0712 (1.3450)	0.0255 (0.0435)	0.7545 (0.8364)	0.0120 (−0.0076)

MLE, maximum likelihood estimation; MSE, mean square error.

Table 1

MSE and bias (values in parentheses) of the MLEs of the parameters α, β, λ, and θ.

6. APPLICATION OF THE TTSG — E DISTRIBUTION

To investigate the advantage of the proposed distribution, we consider a real data set provided by Bjerkedal [18]. This real data set consists of survival times of 72 guinea pigs injected with different amount of tubercle. This species of guinea pigs are known to have high susceptibility of human tuberculosis, which is one of the reasons for choosing. We consider only the study in which animals in a single cage are under the same regimen. The data represents the survival times of guinea pigs in days. The data are given below:

12 15 22 24 24 32 32 33 34 38 38 43 44 48 52 53 54 54 55 56 57 58 58 59 60 60 60 60 61 62 63 65 65 67 68 70 70 72 73 75 76 76 81 83 84 85 87 91 95 96 98 99 109 110 121 127 129 131 143 146 146 175 175 211 233 258 258 263 297 341 341 376.

6.1. Bootstrap Inference for Parameters of the TTSG — E Distribution

In this section, we obtain point and %95 CI estimation of parameters of the TTSG−E distribution by parametric bootstrap method for the real data set. We provide results of bootstrap estimation based on 10,000 bootstrap replicates in Table 2. It is interesting to look at the joint distribution of the bootstrapped values in a scatter plot in order to understand the potential structural correlation between parameters (see Fig. 6).

	Point estimation	CI
α	2.223	(1.354, 4.117)
β	0.298	(0.157, 0.505)
λ	−0.307	(−0.846, 0.734)
θ	161.075	(111.722, 268.019)

CI, confidence interval.

Table 2

Parametric bootstrap point and interval estimation of the parameters α, β, λ, and θ.

6.2. MLE Inference and Comparing with Other Models

We fit the proposed distribution to the real data set by MLE method and compare the results with the gamma, Weibull, TSGE, generalized exponential (GE), and weighted exponential (WE) distributions with respective densities

fgammax=1Γαλαxα−1e−λx, x>0

fWeibullx=βλβxβ−1e−xλβ, x>0

fTSGEx=α1θe−xθ1−e−xθβα−1,0<x≤−θlog1−β,α1θe−xθe−xθ1−βα−1,−θlog1−β≤x<∞,

fGEx=αλe−λx1−e−λxα−1, x>0

fWEx=α+1αλe−λx1−e−αλx, x>0.

For each model, Table 3 includes the MLE's of parameters, Kolmogorov–Smirnov (K−S) distance between the empirical distribution and the fitted model, its corresponding p-value, log-likelihood, and Akaike information criterion (AIC) for the real data set. We fit the TTSG−E distribution to the real data set and compare it with the mentioned distributions. The selection criterion is that the lowest AIC and K−S statistic corresponding to the best fitted model. The TTSG−E distribution provides the best fit for the data set as it has lower AIC and K−S statistic than the other competitor models. The histogram of data set, fitted pdf of the TTSG−E distribution and fitted pdfs of other competitor distributions for the real data set are plotted in Fig. 7. Also, the plots of empirical and fitted cdfs functions, P−P plots and Q−Q plots for the TTSG−E and other fitted distributions are displayed in Fig. 7. These plots also support the results in Table 3. The asymptotic covariance matrix of MLEs for TTSG−E model parameters which is the inverse of the Fisher information matrix, is given by

0.158260.000060.082470.000720.000060.000110.00005−0.000130.082470.000050.071360.000070.00072−0.000130.000070.08243

and the 95% two-sided asymptotic CIs for α, β, λ, and θ are given by 1.950±0.7797, 0.303±0.0205, −0.423±0.5236, and 160.67±0.5627, respectively.

Model	Estimation	Log-likelihood	AIC	K−S statistic	p-value
TTSG−E	α^,β^,λ^,θ^=(1.950,0.303,−0.423,160.67)	−388.063	784.127	0.097	0.508
gamma	α^,λ^=(2.812,0.020)	−394.247	792.495	0.138	0.127
Weibull	β^,λ^=(1.392,110.529)	−397.147	798.295	0.146	0.091
TSGE	α^,β^,θ^=(2.561,0.270,177.911)	−389.549	785.099	0.130	0.171
WE	α^,λ^=(1.626,0.0138)	−393.568	791.138	0.117	0.274
GE	α^,λ^=(2.476,0.017)	−393.110	790.220	0.133	0.159

AIC, Akaike information criterion; GE, generalized exponential; K -- S, Kolmogorov -- Smirnov; MLE, maximum likelihood estimation; TSGE, two-sided generalized exponential, WE, weighted exponential.

Table 3

The MLEs of parameters for real data set.

6.3. Likelihood Ratio Test

We use the likelihood ratio test (LRT) for testing the null hypothesis that the TSGE distribution, proposed by Korkmaz and Genç [11], is equally close to the pig data against the alternative hypothesis that the TTSG−E distribution is closer. That is, we wish to test

H0:X∼TTSG−Eα,β,0,θ≡TSGEα,β,θH1:X∼TTSG−Eα,β,λ,θ,

and equivalently, by considering the estimated value of the parameter λ in Table 3 we should test a one-tailed test as

H0:λ=0H1:λ<0.

According to the LRT, the test statistic is given by

Λx=supH0ℓα,β,λ,θsupH1ℓα,β,λ,θ=ℓα^0,β^0,0,θ^0ℓα^,β^,λ^,θ^,

where ℓ(α,β,λ,θ) is the log-likelihood function of TTSG−E distribution. Based on Table 3, −2log Λx=2.972. Since −2log Λx is distributed asymptotically chi-squared distribution with 1 degrees of freedom, we can conclude that the null hypothesis is rejected in significance level α=0.1. Also, the p-value is 0.085.

7. CONCLUDING REMARKS

In this paper, we propose a new family of distributions that is a compounding of two-sided distributions family and transmuted technique. The proposed model generalizes TSP distribution and generalized two-sided family of distributions and contains these distributions as its submodels. Some reliability and statistical properties of the proposed family of distribution are discussed through the paper. Estimation and inference procedure for distribution parameters are investigated by two well-known maximum likelihood and bootstrap methods in general setting. The TTSG−E distribution considered as a special case of this family. One of the advantage of this new distribution is that it can be fitted to the data sets with one or two modes. Data analysis shows that the TTSG−E distribution provides the best fit and the best performance. The proposed distribution may be a better alternative than the other well-known distributions commonly used in literature for fitting statistical data.

ACKNOWLEDGMENTS

The authors would like to thank the associate editor and referees for their constructive comments that improved presentation of the paper.

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 18 - 2
Pages: 87 - 102
Publication Date: 2019/06/18
ISSN (Online): 2214-1766
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TY  - JOUR
AU  - O. Kharazmi
AU  - M. Zargar
PY  - 2019
DA  - 2019/06/18
TI  - A New Generalized Two-Sided Class of the Distributions Via New Transmuted Two-Sided Bounded Distribution
JO  - Journal of Statistical Theory and Applications
SP  - 87
EP  - 102
VL  - 18
IS  - 2
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.190306.003
DO  - 10.2991/jsta.d.190306.003
ID  - Kharazmi2019
ER  -

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