# A New Family of Distributions to Encounter the Extrapolating Issues

Authors: Ahtasham Gul; Muhammad Mohsin; Zahra Almaspoor
Journal: Mathematical Problems in Engineering
 (2023)

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## Abstract

Traditionally, infinite models have been fitted on finite datasets which extrapolate the data, resulting in inadequate model fitting and predictions. To overcome this problem, we develop a new family of truncated distributions by introducing a new generator. In this article, a truncated random variableXtr “the transformer or input” is exerted to transform another random variable T “transformed or generator,” which yields a new T−Xtr family of distributions. Several characteristics of T−Xtr family of distributions are provided which are equally useful in engineering and biological sciences. For application purposes, a type-2 Gumbel-truncated exponential distribution is generated by using the proposed method along with its statistical properties. The efficacy of the new model is demonstrated by applying it to echophysiology and comparing the resulting outputs with those from the baseline models. Relevance of the Work. Indeed, the probability models have infinite domains but they are applied on the finite real datasets which may lead to exaggerated inferences and predictions. This problem can be solved by developing models that have finite domains. We propose finite models to analyze the finite data proficiently, provide reliable inferences, and save time.


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## Body

## 1. Introduction

ModelingT−Xtr of finite datasets using functions of infinite index has been a common practice, particularly in engineering and medical fields. It is observed that variables belonging to the fields of biology, ecology, engineering, and hydrology are used over finite ranges but fitted over infinite classical models. For instance, gamma-exponentiated exponential (GEE) distribution was applied by Ristić and Balakrishnan [1] to study the number of successive failures of the air conditioning system of a fleet of 13 jet airliners and the survival times of guinea pigs receiving a dose of tubercle bacilli. In engineering science, Sanhueza et al. [2] employed the generalized Birnbaum–Saunders distribution [3] and classical Birnbaum and Saunders [4] distributions to model the dataset corresponding to the cycles ×10−3 of aluminium articles. Similarly, Power Lindley distribution was fitted by Ghitany et al. [5] to model the tensile strength of 69 carbon fibers tested under tension at gauge lengths of 20 mm. All the above mentioned probability functions possess a domain from 0 to ∞ contrary to the dataset having a finite range.The idea of generating univariate continuous distributions has been arisen from the well-known differential equations developed by Pearson [6]. Afterwards, Johnson [7] developed continuous distributions by using the method of translation. Tukey [8] established the method of generating univariate continuous distributions by using a method based on quantile functions. Lee et al. [9] highlighted the fact that after 1980, the majority of the families of distributions generated were based on the method of compounding two existing distributions or embedding extra parameters into an existing distribution. For example, Amini et al. [10] proposed log-Gamma-generated families of distributions; Cordeiro et al. [11] constructed exponentiated half-logistic family of distributions. Eugene et al. [12] proposed a family of Beta generated distributions by utilizing the probability density function (PDF) of a beta random variable and the distribution function (CDF) of any distribution. The CDF of random variable Y is as follows:(1)Hy=∫0Fyrtdt,where rt is the PDF of Beta distribution, and F (y) is the CDF of any distribution having a PDF(2)hy=1Bθ,εfyFθ−1y1−FYε−1,0≤y≤1.Nadarajah and Kotz [13, 14] developed beta-Gumbel and betaexponential distributions, and Akinsete et al. [15] generated a beta-Pareto distribution by utilizing Eugene’s idea. Cordeiro and De Castro [16] and Jones [17] extended Eugene’s idea to generate the Kumaraswamy family of distributions by substituting the Beta distribution with the Kumaraswamy distribution [18]. The PDF of the generalized Kumaraswamy distribution is given by(3)hy=θβgyGθ−1y1−Gθyβ−1,y∈0,1.The method of generating a family of distributions by Eugene, Jones & Cordeiro, and Dae Castro was limited to the new distributions having domain (0, 1). Another common limitation in the abovementioned family of distributions was that the baseline distribution is fixed.Alzaatreh et al. [19] improved the idea of Eugene by replacing the PDF of Beta as a random variable with a PDF of any continuous random variable T as a generator. The CDF of the proposed family of distributions is expressed as follows:(4)Hy=∫0WFyrtdt,a<y<b,having PDF(5)hy=ddyWFyrWFy,a<y<b,and is known as T-X or Transformed-Transformer.According to Alzaatreh et al. [19], W. is the function of a cumulative distribution function (CDF) of any PDF and it should satisfy the following criteria:(1)
WFy∈a,b(2)
WFy should be nondecreasing function(3)
WFy⟶a as y⟶−∞ while WFy⟶b as y⟶∞.Tahir et al. [20] defined the Logistic-X family of probability densities using the methodology of Alzaatreh et al. [19].Generally, the probability distributions apply to an infinite domain, i.e., 0 to∞ but there are certain situations where we need the reduced or finite domains. The infinite domain of a probability distribution is reduced to a finite domain by truncation, and the resulting distribution is called the truncated distribution. The truncated distributions are constantly utilized in various fields like astronomy, medicine, epidemiology, biometry, engineering, hydrology, and economy. For example, in survival analysis, we desire to study how long people survive after taking cardiovascular attack treatment. The time of the cardiovascular attack is the beginning of our study. In engineering, the measurements are taken by using a detector which detects the signals above a specific limit, and the weak signals are not taken into account.The first attempt to estimate the parameters of a truncated Gamma distribution was accomplished by Chapman [21]. Broeder and Gerard [22] and Hegde and Dahiya [23] proposed different methods to estimate the parameters of truncated distributions. Jawitz [24] emphasized the significance of truncated distributions using hydrology data by analyzing the truncated moment expressions (TMEs) with complete moments using six standard distributions. The author pointed out that complete moments are a special case of general TMEs (for more [24]).The main motivation for writing this article is to model finite real data by using a truncated finite distribution to avoid extrapolating issues. The distributions developed by using different generators provide additional parameters which not only help to model the real-life data effectively but also increase the scope of compatibility. This research work is distinctive by the introduction of another new weight of tail parameter (kurtosis) obtained from the new truncated transformed distribution along with the usual induction of location and scale and/or shape parameters to control the shape (skewness). Moreover, the proposed family of distributions is also parsimonious in the context of estimating the model parameters and improving the goodness-of-fit of skewed data contrary to the baseline distributions. The distribution function ofT−Xtr family of distributions is in closed form which is more effective than those whose CDFs are not in closed form, for instance, Mc G and beta G distributions. The new proposed generator is easier to use than the traditional theories and techniques as it is more adaptable to the actual nature of information.The paper is arranged as follows: In Section2, a new method is proposed to generate a family of truncated distributions T−Xtr and some general distributional characteristics, for example, the hazard function, the cumulative hazard function, the mean, pth raw moment, the Shannon entropy, and the order statistics are derived. In Section 3, a type-2 Gumbel-truncated exponential distribution (G-TEXPD) is generated using the proposed technique, and its statistical properties are presented. In Section 4, model parameters of Type-2 Gumbel-truncated exponential distribution (G-TEXPD) are estimated using the MLE method. In Section 5, a Monte Carlo simulation study is carried out to examine the performance of a proposed model for different choices of the model parameters. In Section 6, the feasibility of the proposed model is studied by fitting it to the real dataset and comparing it with some other baseline models. Some concluding remarks are made in Section 7.


## 2. Method for Generating Family of TruncatedT−Xtr Distributions

LetXtr be a nonnegative random variable truncated on left having pdf fXtr and cdf FXtr on domain ξ,∞, and T be a random variable with pdf r (T) and cdf R (T) on interval (−∞≤a≤t≤b≤∞). Also if WFXtr is a function of the cdf FXtr of any nonnegative random variable then WFXtr satisfies the following conditions:(i)
WFXtr∈ξ,b∀ξ> 0(ii)
WFXtr is differentiable and monotonically increasing function(iii)
WFXtr⟶ξ as xtr⟶ 0: WFXtr⟶ b as xtr⟶∞.UsingWFXtr=−log1−FX|x>ξ, we demonstrate the cdf of the truncated distribution function of T−Xtr family as follows:(6)Gxtr=∫ξWFxtrrtdt,(7)Gxtr=∫ξ−log1−Fx|x>ςrtdt,(8)Gxtr=R−log1−Fxtr,(9)Gxtr=RHxtr,where Rt is the cdf of random variable T and Hxtr is the cumulative hazard function of random variable Xtr.The corresponding pdf ofT−Xtr family of distributions is as follows:(10)gxtr=rWFxtrddxWFxtr,(11)gxtr=r−log1−Fx|x>ξfx|x>ξ1−Fx|x>ξ,x>ξ,(12)gxtr=r−log1−Fxtrfxtr1−Fxtr,(13)gxtr=rHxtrhxtr,where hxtr is the hazard function of random variable Xtr


### 2.1. Remarks

(i)
The cdf and pdf are the hazard and cumulative hazard function of random variableXtr with cdf of truncated distribution Gxtr. Thus T−Xtr distribution can be considered as a family of distributions arising from weighted hazard function.(ii)
The pdf r (t) in (6) is transformed into new cdf Gxtr through the function WFxtr, which acts as a “transformer.” Hence, we shall refer to the distribution gxtr in (13) as “Transformed-Truncated Transformer” or T−Xtr distribution.(iii)
The random variableXtr may be discrete if Gxtr is a family of discrete distributions


### 2.2. Important Characteristics ofT−Xtr Family of Distributions

(i)
The hazard function ofT−Xtr family of distributions is expressed as follows:(14)hXtr=fXtrr−log1−FXtr1−FXtr1−R−log1−Fxtr.(ii)
The cumulative hazard function ofT−Xtr family of distributions is expressed as follows:(15)Hxtr=−log1−RHxtr.(iii)
Thepth percentile is used to compute median, multiple statistics and to generate random numbers as well. The pth percentile of T−Xtr family of distributions is computed as follows:(16)R−log1−Fxtr=P,Xtr=F−11−e−R−1P.(iv)
The first raw moment ofT−Xtr family of distributions is computed as follows:SinceXtr=F−11−e−T,(17)EXtr=EF−11−e−T.(v)
The moment generating function ofT−Xtr distribution is obtained as follows:(18)Mxtrt=Eetxtr,Mxtrt=Eetxtrr−log1−FXtr1−FXtr.(vi)
We can generate a family of discrete distributions by introducing the random variableX as a discrete distribution. It is defined as follows:(19)gxtr=Gxtr−Gxtr−1,gxtr=R−log1−Fxtr−R−log1−Fxtr−1.Theorem 1.
If a random variableXtr follows the truncated family of distributions gxtr=r−log1−Fxtrfxtr/1−Fxtr, then the Shannon entropy of a random variable Xtr is given by ηxtr=−EF−11−e−T+μT+ηT.Proof.
We haveT=−log1−Fxtr.(20)gxtr=r−log1−FXtrfxtr1−FXtr,ξ>x,ηxtr=−EF−11−e−T+μT+ηT,where μT and ηT are the mean and Shannon entropy of random variable T respectively having pdf r(t).Theorem 2.
Thessth order statistic gs;k (Xtr) in a random sample of size k from the T−Xtr distribution is given by(21)gs;kXtr=Cs;k−ss+11−FXtr∑j=0k−s−1jk−sjr−log1−FXtrfs+1;s+jXtr.Proof.
By definition,(22)gs;kXtr=Cs;kgXtrGXtrs−11−GXtrk−s.
Since(23)gXtr=fXtrr−log1−FXtr1−FXtr,Gxtr=R−log1−FXtr,gs;kXtr=Cs;kr−log1−FXtrfXtr1−FXtrGXtrs−11−GXtrk−s,gs;kXtr=Cs;k−ss+11−FXtr∑j=0k−s−1jk−sjr−log1−FXtrfs+1;s+jXtr.
Table1 provides the new families of distributions by using different T distributions as generator.Table 1
Families of generalized truncated distributions for differentXtr distributions.
NameFXtr|Xtr>ξpdf of family ofgXtrSupport ofXtrTruncatedexponential1−e−θx−ξθrθx−ξIξ,∞Truncatedpareto1−b/xccxrclnx/bIb,∞TruncatedRayleigh1−e−x2−a2/2b2xb2rx2−a2/2b2Ib,∞TruncatedWeibull1−e−x/αββxβ−1/βrx/αβIb,∞TruncatedKumaraswammyβlog1−aα/1−xααβxα−1/1−xαrβlog1−aα/1−xαIa,1Truncatedburrβlog1+xα/1+aααβxα−1/1+xα2rβlog1+xα/1+aαIa,∞


## 2.1. Remarks

(i)
The cdf and pdf are the hazard and cumulative hazard function of random variableXtr with cdf of truncated distribution Gxtr. Thus T−Xtr distribution can be considered as a family of distributions arising from weighted hazard function.(ii)
The pdf r (t) in (6) is transformed into new cdf Gxtr through the function WFxtr, which acts as a “transformer.” Hence, we shall refer to the distribution gxtr in (13) as “Transformed-Truncated Transformer” or T−Xtr distribution.(iii)
The random variableXtr may be discrete if Gxtr is a family of discrete distributions


## 2.2. Important Characteristics ofT−Xtr Family of Distributions

(i)
The hazard function ofT−Xtr family of distributions is expressed as follows:(14)hXtr=fXtrr−log1−FXtr1−FXtr1−R−log1−Fxtr.(ii)
The cumulative hazard function ofT−Xtr family of distributions is expressed as follows:(15)Hxtr=−log1−RHxtr.(iii)
Thepth percentile is used to compute median, multiple statistics and to generate random numbers as well. The pth percentile of T−Xtr family of distributions is computed as follows:(16)R−log1−Fxtr=P,Xtr=F−11−e−R−1P.(iv)
The first raw moment ofT−Xtr family of distributions is computed as follows:SinceXtr=F−11−e−T,(17)EXtr=EF−11−e−T.(v)
The moment generating function ofT−Xtr distribution is obtained as follows:(18)Mxtrt=Eetxtr,Mxtrt=Eetxtrr−log1−FXtr1−FXtr.(vi)
We can generate a family of discrete distributions by introducing the random variableX as a discrete distribution. It is defined as follows:(19)gxtr=Gxtr−Gxtr−1,gxtr=R−log1−Fxtr−R−log1−Fxtr−1.Theorem 1.
If a random variableXtr follows the truncated family of distributions gxtr=r−log1−Fxtrfxtr/1−Fxtr, then the Shannon entropy of a random variable Xtr is given by ηxtr=−EF−11−e−T+μT+ηT.Proof.
We haveT=−log1−Fxtr.(20)gxtr=r−log1−FXtrfxtr1−FXtr,ξ>x,ηxtr=−EF−11−e−T+μT+ηT,where μT and ηT are the mean and Shannon entropy of random variable T respectively having pdf r(t).Theorem 2.
Thessth order statistic gs;k (Xtr) in a random sample of size k from the T−Xtr distribution is given by(21)gs;kXtr=Cs;k−ss+11−FXtr∑j=0k−s−1jk−sjr−log1−FXtrfs+1;s+jXtr.Proof.
By definition,(22)gs;kXtr=Cs;kgXtrGXtrs−11−GXtrk−s.
Since(23)gXtr=fXtrr−log1−FXtr1−FXtr,Gxtr=R−log1−FXtr,gs;kXtr=Cs;kr−log1−FXtrfXtr1−FXtrGXtrs−11−GXtrk−s,gs;kXtr=Cs;k−ss+11−FXtr∑j=0k−s−1jk−sjr−log1−FXtrfs+1;s+jXtr.
Table1 provides the new families of distributions by using different T distributions as generator.Table 1
Families of generalized truncated distributions for differentXtr distributions.
NameFXtr|Xtr>ξpdf of family ofgXtrSupport ofXtrTruncatedexponential1−e−θx−ξθrθx−ξIξ,∞Truncatedpareto1−b/xccxrclnx/bIb,∞TruncatedRayleigh1−e−x2−a2/2b2xb2rx2−a2/2b2Ib,∞TruncatedWeibull1−e−x/αββxβ−1/βrx/αβIb,∞TruncatedKumaraswammyβlog1−aα/1−xααβxα−1/1−xαrβlog1−aα/1−xαIa,1Truncatedburrβlog1+xα/1+aααβxα−1/1+xα2rβlog1+xα/1+aαIa,∞


## 3.T-Truncated Exponential Distribution (T-TEXPD) with Different T Distributions

In this section, we present truncated exponential distribution with differentT random variables. Some properties of each case are also discussed. Let Xtr be the random variable of the truncated exponential model having pdf given as follows:(24)gXtr=θrθx−ξ,Iξ,∞,ξ>0,and cdf as follows:(25)GXtr=Rθx−ξ,where ξ is the truncation point. The generalized form of Shannon entropy for T-TEXPD is given by(26)ηXtr=−logθ+θηT,where ηT is Shannon entropy of random variable T.


### 3.1. Type-2 Gumbel-Truncated Exponential Distribution (G-TEXPD)

LetT be the Gumbel pdf having two parameters α and β is as follows:(27)gt=αβt−α−1e−βt−α,t>0,and cdf is(28)Gt=e−βt−α.Using (24), the resulting Gumbel-truncated exponential distribution (G-TEXPD) is as follows:(29)gXtr=αβθ−αe−βθ−αxtr−a−αxtr−a−α−1,forxtr>a.The cdf of G-TEXPD is as follows:(30)GXtr=e−βθ−αxtr−a−α.Figure1 displays different shapes of Gumbel-TEXPD for different values of parameters.Figure 1
The pdf of the Gumbel-TEXPD for different values ofα, β, and θ. (a) Different values of α. (b) Different values of β. (c) Different values of θ. (d) Different values of parameters.
(a)(b)(c)(d)The abovementioned Figure1(a) is sketched at different values of α which connotes that by increasing the value of α, the curve becomes platykurtic (thinner tail). Similarly, by increasing the value of β, the shape of Figure 1(b) tends towards leptokurtic i.e. a heavy/fatter tail resulting in a capturing of extreme values. The additional parameter θ in Figure 1(c) clearly gives evidence that the shape of G-TEXPD tends to be mesokurtic by increasing the value of θ. Furthermore, Figure 1(d) is sketched at different values of the parameters.Figure 2
hXtr of G-TEXPD for different values of α,β,andθ.


#### 3.1.1. Distributional Properties of the G-TEXPD

(i)
The hazard function of G-TEXPD is as follows:(31)hXtr=αβθ−αt−a−α−1e−βθ−αt−a−α1−e−βθ−αt−a−α.The hazard is the probability of the event occurring during any given time point. Figure2 depicts that the G-TEXPD is useful to model lifetime data having an upside down bath-tub, concave shape, or decreasing failure rate function.(ii)
The cumulative hazard function of G-TEXPD is as follows:(32)HXtr=−log1−e−βθ−αt−a−α.(iii)
Thepth percentile of G-TEXPD distribution is as follows:(33)xtr=β1/α−logp−1/αθ+a.Theorem 3.
LetXtr be a random variable that follows G-TEXPD, then the Shannon entropy is given by(34)ηXtr=1+1α+logβα−logθ−logα.Proof.
We know that(35)ηXtr=−EloggXtr,using (29) in the above equation, we get(36)ηXtr=−Elogα−Elogβ+αElogθ+α+1Elogxtr−a+βθαExtr−a−α,ηXtr=1+1α+logβα−logθ−logα.Theorem 4.
ThePth raw moment of G-TEXPD is as follows:(37)EXtrp=βp/αθp∑k=0ppkθaβ1/αp−k1−kα¯.Proof.
By definition(38)EXtrp=αβθ−α∫a∞xtrpxtr−a−α−1e−βθ−αxtr−a−αdxtr,EXtrp=βp/αθp∫0∞θaβ1/α+u−1/αpe−udu,EXtrp=βp/αθp∑k=0ppkθaβ1/αp−k1−kα¯.Theorem 5.
Thesth order statistic fs;k (xtr) in a random sample of size k from the G-TEXPD distribution, is given by(39)fs;kxtr=Cs;kαβθ−αxtr−a−α−1e−βθ−αxtr−a−αs1−e−βθ−αxtr−a−αk−s.Proof.
By definition,(40)fs;kxtr=Cs;kgxtrGxtrs−11−Gxtrk−s.
Using (29) and (30) in (40), we get(41)fs;kxtr=Cs;kαβθ−αxtr−a−α−1e−βθ−αxtr−a−αs1−e−βθ−αxtr−a−αk−s.


## 3.1. Type-2 Gumbel-Truncated Exponential Distribution (G-TEXPD)

LetT be the Gumbel pdf having two parameters α and β is as follows:(27)gt=αβt−α−1e−βt−α,t>0,and cdf is(28)Gt=e−βt−α.Using (24), the resulting Gumbel-truncated exponential distribution (G-TEXPD) is as follows:(29)gXtr=αβθ−αe−βθ−αxtr−a−αxtr−a−α−1,forxtr>a.The cdf of G-TEXPD is as follows:(30)GXtr=e−βθ−αxtr−a−α.Figure1 displays different shapes of Gumbel-TEXPD for different values of parameters.Figure 1
The pdf of the Gumbel-TEXPD for different values ofα, β, and θ. (a) Different values of α. (b) Different values of β. (c) Different values of θ. (d) Different values of parameters.
(a)(b)(c)(d)The abovementioned Figure1(a) is sketched at different values of α which connotes that by increasing the value of α, the curve becomes platykurtic (thinner tail). Similarly, by increasing the value of β, the shape of Figure 1(b) tends towards leptokurtic i.e. a heavy/fatter tail resulting in a capturing of extreme values. The additional parameter θ in Figure 1(c) clearly gives evidence that the shape of G-TEXPD tends to be mesokurtic by increasing the value of θ. Furthermore, Figure 1(d) is sketched at different values of the parameters.Figure 2
hXtr of G-TEXPD for different values of α,β,andθ.


### 3.1.1. Distributional Properties of the G-TEXPD

(i)
The hazard function of G-TEXPD is as follows:(31)hXtr=αβθ−αt−a−α−1e−βθ−αt−a−α1−e−βθ−αt−a−α.The hazard is the probability of the event occurring during any given time point. Figure2 depicts that the G-TEXPD is useful to model lifetime data having an upside down bath-tub, concave shape, or decreasing failure rate function.(ii)
The cumulative hazard function of G-TEXPD is as follows:(32)HXtr=−log1−e−βθ−αt−a−α.(iii)
Thepth percentile of G-TEXPD distribution is as follows:(33)xtr=β1/α−logp−1/αθ+a.Theorem 3.
LetXtr be a random variable that follows G-TEXPD, then the Shannon entropy is given by(34)ηXtr=1+1α+logβα−logθ−logα.Proof.
We know that(35)ηXtr=−EloggXtr,using (29) in the above equation, we get(36)ηXtr=−Elogα−Elogβ+αElogθ+α+1Elogxtr−a+βθαExtr−a−α,ηXtr=1+1α+logβα−logθ−logα.Theorem 4.
ThePth raw moment of G-TEXPD is as follows:(37)EXtrp=βp/αθp∑k=0ppkθaβ1/αp−k1−kα¯.Proof.
By definition(38)EXtrp=αβθ−α∫a∞xtrpxtr−a−α−1e−βθ−αxtr−a−αdxtr,EXtrp=βp/αθp∫0∞θaβ1/α+u−1/αpe−udu,EXtrp=βp/αθp∑k=0ppkθaβ1/αp−k1−kα¯.Theorem 5.
Thesth order statistic fs;k (xtr) in a random sample of size k from the G-TEXPD distribution, is given by(39)fs;kxtr=Cs;kαβθ−αxtr−a−α−1e−βθ−αxtr−a−αs1−e−βθ−αxtr−a−αk−s.Proof.
By definition,(40)fs;kxtr=Cs;kgxtrGxtrs−11−Gxtrk−s.
Using (29) and (30) in (40), we get(41)fs;kxtr=Cs;kαβθ−αxtr−a−α−1e−βθ−αxtr−a−αs1−e−βθ−αxtr−a−αk−s.


## 3.1.1. Distributional Properties of the G-TEXPD

(i)
The hazard function of G-TEXPD is as follows:(31)hXtr=αβθ−αt−a−α−1e−βθ−αt−a−α1−e−βθ−αt−a−α.The hazard is the probability of the event occurring during any given time point. Figure2 depicts that the G-TEXPD is useful to model lifetime data having an upside down bath-tub, concave shape, or decreasing failure rate function.(ii)
The cumulative hazard function of G-TEXPD is as follows:(32)HXtr=−log1−e−βθ−αt−a−α.(iii)
Thepth percentile of G-TEXPD distribution is as follows:(33)xtr=β1/α−logp−1/αθ+a.Theorem 3.
LetXtr be a random variable that follows G-TEXPD, then the Shannon entropy is given by(34)ηXtr=1+1α+logβα−logθ−logα.Proof.
We know that(35)ηXtr=−EloggXtr,using (29) in the above equation, we get(36)ηXtr=−Elogα−Elogβ+αElogθ+α+1Elogxtr−a+βθαExtr−a−α,ηXtr=1+1α+logβα−logθ−logα.Theorem 4.
ThePth raw moment of G-TEXPD is as follows:(37)EXtrp=βp/αθp∑k=0ppkθaβ1/αp−k1−kα¯.Proof.
By definition(38)EXtrp=αβθ−α∫a∞xtrpxtr−a−α−1e−βθ−αxtr−a−αdxtr,EXtrp=βp/αθp∫0∞θaβ1/α+u−1/αpe−udu,EXtrp=βp/αθp∑k=0ppkθaβ1/αp−k1−kα¯.Theorem 5.
Thesth order statistic fs;k (xtr) in a random sample of size k from the G-TEXPD distribution, is given by(39)fs;kxtr=Cs;kαβθ−αxtr−a−α−1e−βθ−αxtr−a−αs1−e−βθ−αxtr−a−αk−s.Proof.
By definition,(40)fs;kxtr=Cs;kgxtrGxtrs−11−Gxtrk−s.
Using (29) and (30) in (40), we get(41)fs;kxtr=Cs;kαβθ−αxtr−a−α−1e−βθ−αxtr−a−αs1−e−βθ−αxtr−a−αk−s.


## 4. Estimation of Model Parameters by Using the Maximum Likelihood (ML) Method

In this section, we estimated the unknown parameters of G-TEXPD by using maximum likelihood estimation method as proposed by Johnson et al. [25]. The pdf of G-TEXPD is as follows:(42)gXtr=αβθ−αxtr−a−α−1e−βθ−αxtr−a−α,forxtr>a.The log-likelihood function of the G-TEXPD is given by(43)logLa,α,β,θ;xtr=log∏j=1nαβθ−αxtrj−a−α−1e−βθ−αxtrj−a−α,logLa,α,β,θ;xtr=nlogα+nlogβ−nαlogθ−α∑j=1nlogxtrj−a−∑j=1nlogxtrj−a−βθα∑j=1nxtrj−a−α.Now, computing the first partial derivatives of (43) with respect to a,α,β,andθ and equating the results to zero, we have(44)∂logLa,α,β,θ;xtr∂a=Minxtrj,j=0,1,2………,n,or(45)a∧=Minxtrj,(46)∂logLa,α,β,θ;xtr∂α=nα−nlogθ−∑j=1nlogxtrj−a−β∑j=1nlogxtrj−a−αθ−αlogθ−βθ−α∑j=1nxtrj−a−α∑j=1nlogxtrj−a=0,(47)∂logLa,α,β,θ;xtr∂β=nβ−∑j=1nxtrj−a−αθα=0,(48)∂logLa,α,β,θ;xtr∂θ=αβθα+1∑j=1nxtrj−a−α−nαθ=0.Since the equations (46)–(48) are not in closed form, we use a well-known iterative method, i.e., the Newton Raphson to obtain the approximate ML estimates for the parameters θ,α, and β.


### 4.1. Asymptotic Confidence Bounds

It is observed that ML estimates of the unknown parametersα,β,andθ of G-TEXPD are not in closed forms. In this situation, we compute the asymptotic confidence bounds of G-TEXPD based on the asymptotic distribution of the MLE.The Fisher information matrix can be used for interval estimation and hypothesis testing. For G-TEXPD, the information matrix is obtained by computing the second partial derivatives of the equations (46)–(48) as follows:(49)In=IααIαβIαθIβαIββIβθIθαIθβIθθ.The entries of Fisher information matrix of G-TEXPD are as follows:(50)Iαα=∂2logLa,α,β,θ;xtr∂α2=−nα2−βθα∑j=1nxtrj−alog2xtrj−a−βθαlogθ∑j=1nxtrj−alogxtrj−a,Iββ=∂2logLa,α,β,θ;xtr∂β2=−nβ2,Iθθ=∂2logLa,α,β,θ;xtr∂θ2=nαθ2−αβα+1θα+2∑j=1nxtrj−a−α,Iαβ=∂logLa,α,β,θ;xtr∂α∂β=−θ−α∑j=1nlogxtrj−axtrj−a−α−θ−αlogθ∑j=1nxtrj−a−α,Iαθ=∂logLa,α,β,θ;xtr∂α∂θ=−nθ+βθα+1∑j=1nxtrj−a−α+αβθα+1logθ∑j=1nxtrj−a−α+αβθα+1∑j=1nxtrj−a−αlogxtrj−a,Iθβ=∂logLa,α,β,θ;xtr∂θ∂β=αθα+1∑j=1nxtrj−a−α.The asymptotic confidence intervals are obtained by using either the approximate normal distribution or the approximate log-normal distribution of the ML estimatesg∧=α∧,β∧,θ∧. The estimated standard errors of α∧,β∧, and θ∧ are expressed as follows:(51)σα∧,β∧,θ∧=∑jj,where∑jj=In−1.For instance, the expressions for confidence interval ofα is calculated by using the approximate normal distribution and log-normal distribution is as follows:(52)α^±δξ/2σα^,α^exp±δξ/2σα^α^,respectively, where δ1−ξ/2 is the 1−ξ/2 percentile of the standard normal distribution. The log-normal approximation works well if the standard error of the parameters is greater than half of their point estimate.


## 4.1. Asymptotic Confidence Bounds

It is observed that ML estimates of the unknown parametersα,β,andθ of G-TEXPD are not in closed forms. In this situation, we compute the asymptotic confidence bounds of G-TEXPD based on the asymptotic distribution of the MLE.The Fisher information matrix can be used for interval estimation and hypothesis testing. For G-TEXPD, the information matrix is obtained by computing the second partial derivatives of the equations (46)–(48) as follows:(49)In=IααIαβIαθIβαIββIβθIθαIθβIθθ.The entries of Fisher information matrix of G-TEXPD are as follows:(50)Iαα=∂2logLa,α,β,θ;xtr∂α2=−nα2−βθα∑j=1nxtrj−alog2xtrj−a−βθαlogθ∑j=1nxtrj−alogxtrj−a,Iββ=∂2logLa,α,β,θ;xtr∂β2=−nβ2,Iθθ=∂2logLa,α,β,θ;xtr∂θ2=nαθ2−αβα+1θα+2∑j=1nxtrj−a−α,Iαβ=∂logLa,α,β,θ;xtr∂α∂β=−θ−α∑j=1nlogxtrj−axtrj−a−α−θ−αlogθ∑j=1nxtrj−a−α,Iαθ=∂logLa,α,β,θ;xtr∂α∂θ=−nθ+βθα+1∑j=1nxtrj−a−α+αβθα+1logθ∑j=1nxtrj−a−α+αβθα+1∑j=1nxtrj−a−αlogxtrj−a,Iθβ=∂logLa,α,β,θ;xtr∂θ∂β=αθα+1∑j=1nxtrj−a−α.The asymptotic confidence intervals are obtained by using either the approximate normal distribution or the approximate log-normal distribution of the ML estimatesg∧=α∧,β∧,θ∧. The estimated standard errors of α∧,β∧, and θ∧ are expressed as follows:(51)σα∧,β∧,θ∧=∑jj,where∑jj=In−1.For instance, the expressions for confidence interval ofα is calculated by using the approximate normal distribution and log-normal distribution is as follows:(52)α^±δξ/2σα^,α^exp±δξ/2σα^α^,respectively, where δ1−ξ/2 is the 1−ξ/2 percentile of the standard normal distribution. The log-normal approximation works well if the standard error of the parameters is greater than half of their point estimate.


## 5. Simulation Study

In this section, a Monte Carlo simulation is carried out to study the stability of the model parameters. The Monte Carlo simulation is run 5000 times for four different combinations of the parameters to draw the random samples of size n each from the G-TEXPDα,β,andθ. The model parameters are estimated by the maximum likelihood (ML) method. Table 2 presents the average maximum likelihood estimates (MLEs) of the three parameters with standard errors (SEs), biases, mean square errors (MSEs), and the corresponding 99% coverage probability for approximate confidence intervals for samples of sizes 20, 50, 100, and 200. A fixed seed is used to generate such random numbers, implying that all results of these studies can always be exactly replicated. The Monte Carlo simulation is performed in the following two steps:(1)
Generate 5000 samples of sizen = 20, 50, 100 and 200 each using (33)(2)
Compute the average MLEs, the SEs, MSEs and CIs for each sample, i.e.,α∧,β∧,θ∧The bias and the MSE are computed by using(53)bias=15000∑i=15000α∧i−α,MSE=15000∑i=15000σα2∧+biasα∧i2,respectively.(3)
Similarly, the two-sided asymptotic (1−ε)% CI for the parameter α is computed by using(54)α∧±Zε/2σ∧α25000,whereZε represents the (1−ε)% percentile of the standard normal distribution.(4)
The simulated coverage probabilities for two-sided approximate at 99% confidence intervals Prα∈I∧ (the parameter of interest α is estimated by α∧ and the confidence interval I∧) based on the normal-approximate distribution are computed. Same algorithm is repeated for the other two population parameters β and θ, respectively.Table 2
Average estimated values, corresponding SEs (given in parentheses) bias, MSE, and 99% coverage probability for approximate confidence intervals of the model parameters.
Actual valuesnAverage estimate (S.E)BiasMSE99% coverage probabilityα^β^θ^α^β^θ^α^β^θ^α^β^θ^α = 0.1β = 0.5θ = 2.0200.11486 (0.00002)0.47287 (0.00010)2.06053 (0.00443)0.01486−0.027130.060530.001850.008170.3169777%78%78%500.11635 (0.00001)0.46799 (0.00006)1.97171 (0.00079)0.01635−0.03201−0.028290.001170.005360.0569583%68%79%1000.12051 (0.00009)0.46067 (0.00004)1.95342 (0.00082)0.02051−0.03933−0.046580.001120.004920.0603186%65%95%2000.14845 (0.00007)0.44590 (0.00003)3.49963 (0.00109)0.02675−0.04863−0.062470.001260.005040.0810894%60%99%α = 0.1β = 0.5θ = 3.5200.11060 (0.00001)0.48123 (0.00008)3.61586 (0.01562)0.01060−0.018770.115860.001500.006201.1180184%85%88%500.11156 (0.00001)0.47670 (0.00005)3.46154 (0.00217)0.01156−0.02330−0.038460.000880.004240.1552290%85%97%1000.11446 (0.00008)0.47130 (0.00004)3.42917 (0.00231)0.01446−0.02870−0.070830.000810.003790.1682587%79%96%2000.14845 (0.00002)0.44590 (0.00008)3.49963 (0.00004)0.04845−0.05410−0.000370.003620.008470.0030688%73%99%α = 0.25β = 1.0θ = 3.5200.24995 (0.00000)1.00020 (0.00000)3.50088 (0.00001)−0.000050.000200.000880.000020.000050.0007799%99%99%500.25012 (0.00000)1.00006 (0.00000)3.50100 (0.00001)0.000120.000060.001000.000010.000030.0006598%99%99%1000.25011 (0.00000)1.00003 (0.00000)3.50102 (0.00001)0.000110.000030.001020.000000.000010.0006498%99%99%2000.25011 (0.00002)1.00008 (0.00008)3.50176 (0.00004)0.000110.000080.001760.000000.000020.0015198%99%99%α = 0.5β = 1.5θ = 3.5200.50000 (0.00000)1.50000 (0.00000)3.50000 (0.00000)0.000010.000030.000000.000000.000000.0000099%99%100%500.50000 (0.00000)1.50000 (0.00000)3.50000 (0.00000)0.000000.000000.000000.000000.000000.00000100%100%100%1000.50000 (0.00000)1.50000 (0.00000)3.50000 (0.00000)0.000000.000000.000000.000000.000000.00000100%100%100%2000.50000 (0.00000)1.50000 (0.00000)3.50000 (0.00000)0.000000.000000.000000.000000.000000.00000100%100%100%Table2 contains the results of the simulation study for different values of the G-TEXPD parameters. We can depict from the Table 2 that standard error (S.E) of additional parameter θ systematically become lower than those of α and β. Moreover, the simulation results connote that 99% coverage probability for approximate confidence intervals of the true parameter based on MLEs gives satisfactory results. It is interesting to mention here that the estimate of β tends to the true parameter value provided β is large. We observed based on coverage probability results that the estimate of α,β and θ can be statistically meaningful even in rather small samples.


## 6. Application: Ecophysiological Data

The competence of the Gumbel-truncated exponential distribution is established by modeling ecophysiological data used by Potvin et al. [26]. The data consist of nonlinear growth curves regarding cold tolerance for 3 plants each from 2 locations (Quebec and Mississippi) and 2 treatments (controlled and chilled) at 7 levels of CO2 concentration. The variable CO2 uptake rate is selected for model fitting. The dataset of 18 observations is recorded as: 10.79, 10.80, 10.86, 10.93, 10.99, 10.96, 10.98, 11.03, 11.08, 11.10, 11.19, 11.25, 11.40, 11.61, 11.69, 11.91, 12.07, and 12.32.The descriptive statistic ofCO2 uptake rate (μL/L) given in Table 3 shows the initial analysis which suggests that the data is skewed and right tailed.Table 3
Descriptive statistics ofCO2 uptake rate (μL/L).
MinQ1Q2MeanQ3MaxVarSkewnessKurtosis10.7910.9611.0911.2711.5612.320.211.063.11Table4 provides the estimated model parameters of CO2 uptake rate (μL/L) by using the ML method, along with the comparison of G-TEXPD with Gumbel-exponential (G-EXPD) using the methodology of Alzaatreh et al. [19], truncated exponential (TEXPD), Weibull, Gamma, and exponential distributions. The negative log-likelihood l∧, Akaike information criterion (AIC), and Bayesian information criterion (BIC) are computed to compare the models. The model with the highest negative log-likelihood value and the smallest AIC and BIC values indicates the better model. The competence of the proposed model is evidenced in Table 4 where the proposed model has the highest value of negative log-likelihood and the lowest values of AIC and BIC for the given data.Table 4
Negative log-likelihood function, MLEs of model parameters, the corresponding SEs (in parentheses), and the statistics AIC and BIC forCO2 uptake rate (μL/L).
ModelsEstimatesStatisticl^a^α^β^θ^AICBICG-TEXPD−7.31510.794.665 (0.901)0.028 (0.133)0.377 (0.381)20.63123.302G-EXPD−8.159—36.80338.2320.09922.31924.990TEXPD−24.891—0.681 (0.160)——51.78352.673Weibull−13.767—11.505 (0.124)23.167 (3.895)—31.53433.315Gamma−10.786—58.023 (19.344)654.253 (218.033)—25.57327.354Exponential−61.607—0.088 (0.020)——125.214126.105The bold values are representing the proposed distribution (i.e. G-TEXPD) values and rest of the distributions values are un-bold in the Tables.Table5 shows the values of different statistics for the distributions in comparison to check the goodness-of-fit. The distribution having the smallest values of test statistics fits the best. It is obvious from Table 5 that G-TEXPD provides statistically the best fit to the data.Table 5
Goodness-of-fit statistic ofCO2 uptake rate (μL/L).
ModelsG-TEXPDG-EXPDTEXPDWeibullGammaExponentialK-Smirnov0.1190.1550.4880.2180.2060.615C-Von0.0470.0840.9170.2200.1631.677A-Darling0.3180.5344.3891.220.9177.667The bold values are representing the proposed distribution (i.e. G-TEXPD) values and rest of the distributions values are un-bold in the Tables.Figure3 shows that the pdf of G-TEXPD fits the histogram of CO2 uptake rate (μL/L) more adequately than the other models. Also, the observed probabilities are plotted versus the predicted probabilities to check the goodness of fit. It is observed that the G-TEXPD follows the diagonal line more closely than the others.Figure 3
The fitted pdfs of the Gumbel-TEXPD, Gumbel-exponential, truncated-exponential, Weibull, gamma, and exponential models on the histogram ofCO2 uptake rate (μL/L) along with their probability plots. (a) Histogram and theoretical densities. (b) P−P plot.
(a)(b)


## 7. Concluding Remarks

Conventionally infinite distributions have been used to model the finite datasets which results in extrapolating the data. Extrapolating may cause a greater element of uncertainty in the estimation, prediction and interpretation. We made an attempt to encounter this issue by proposing a newT−Xtr family, of truncated models. Different characteristics of the proposed T−Xtr family such as hazard function, cumulative hazard function, quantile function, Shannon entropy, and order statistics, are derived to study the behavior of the distribution. We generated Type-2 Gumbel-truncated exponential distribution (G-TEXPD) along with different properties by applying the proposed methodology. The Monte Carlo simulation of the G-TEXPD is conducted for different combinations of the parameters for different sample sizes to study the average ML estimates, standard errors, biases, mean square errors, and corresponding coverage probabilities at 99% confidence intervals. The simulation results connote that 99% coverage probability for approximate confidence intervals of the true parameter based on MLEs gives satisfactory results. It is also observed based on coverage probability results that the estimate of α,β,and θ can be statistically meaningful even in rather small samples. A real-life example from ecophysiology is presented to compare the performance of our model with the truncated and untruncated contemporary models. Several statistical criteria, i.e., negative log-likelihood, AIC, BIC, K-Smirnov, C-Von, A-Darling, and pp-plot are used to collect enough evidence for the better performance of G-TEXPD. Indeed, the use of truncated models for finite datasets certainly helps to have precise estimation, prediction, and interpretation.The authors have extracted the new directions to the readers and researchers for further research. This methodology can be utilized to generate a new family ofT−Xtr distributions by limiting the on upper or both domains of Xtr random variable. New families of exponentiated, transmuted, and weighted distributions can be originated using the proposed function. This study can be extended to compare the estimators for censored data. In addition to, the existing estimation algorithms, bootstrapping, and the Bayesian techniques can be implemented for distributions generating from T−Xtr family. Furthermore, a bivariate and multivariate version of truncated distributions can be developed following the proposed technique.



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