# Statistical Inference of Odd Fréchet Inverse Lomax Distribution with Applications

Authors: Ramadan A. ZeinEldin; Muhammad Ahsan ul Haq; Sharqa Hashmi; Mahmoud Elsehety; M. Elgarhy
Journal: Complexity
 (2020)

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## Abstract

In this article, we propose and study a new three-parameter distribution, called the odd Fréchet inverse Lomax (OFIL) distribution, derived by combining the odd Fréchet-G family and the inverse Lomax distribution. Since Fréchet is a continuous distribution with wide applicability in extreme value theory, the new model contains these properties as well as the characteristics of the inverse Lomax distribution which make it more flexible and provide a good alternative for some well-known lifetime distributions. We initially present a linear representation of its functions and discussion on density and hazard rate function. Then, we study its various mathematical properties. Different estimation methods are used to estimate parameters of OFIL. The Monte Carlo simulation study is carried out to compare the efficiencies of different methods of estimation. The maximum likelihood estimation (MLE) method is used to estimate the OFIL parameters by considering three practical data applications. We show that the related model is the best in comparisons based on Akaike information criterion (AIC), Bayesian information criterion (BIC), and other goodness-of-fit measures.


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## Body

## 1. Introduction

Frèchet distribution is considered as an extreme value distribution and extensively applied in life study affected by earthquakes, floods, horse riding, vehicle racing, queues in supermarkets, wind speeds, and sea waves. A detailed study of Frèchet distribution and its applications was given by Nadarajah and Kotz [1]. Haq et al. [2] proposed the generalization of the Frèchet distribution named the transmuted Weibull Fréchet (TWFr) distribution and derived its main characteristics including probability-weighted moments, moments, incomplete moments, generating function, order statistics, and stress strength model. Hassan and Nassr [3] proposed the inverse Weibull-G family of distributions, and Hashmi and Gull [4] proposed and studied the Weibull–Lomax distribution. A comprehensive lifetime model is necessary to fit complex real data sets. The inverse Lomax (IL) distribution, being the member of the family of generalized beta distribution, is very useful in the fields of economics, geography, medical sciences, and actuarial sciences (see [5]). Kleiber [6] used the IL distribution in the model fitting of geophysical data especially on different sizes of land fires in forests of California (USA). He also presented Lorenz ordering among order statistics for some distributions including IL distribution. Singh et al. [7] considered the IL distribution and derived reliability estimates under Type-II censoring using the Markov chain Monte Carlo method. Bayesian estimation of parameters of the IL distribution based on the Type-I censoring scheme was discussed by Reyad and Othman [8]. The model said belongs to an inverted family of distributions with more flexibility to analyze nonmonotone behavior of the real hazard rate function. The IL distribution is obtained by considering a random variable X as X = 1/Y, where Y has a Lomax distribution Y∼Lx;α,β. Rahman et al. [9] presented Bayesian inference of model parameters of the IL distribution. A comparison study was conducted between Bayesian estimators and ML estimators. Yadav et al. [10] used this model for Type-II censoring. Later on, Yadav et al. [10] compared different estimation methods such as MLE and Bayes estimation under Jeffrey’s noninformative prior to both parameters using a hybrid censoring scheme (combination of Type-I and Type-II censoring). Jan and Ahmed [11] used different approximation techniques to obtain Bayesian estimates of model parameters of the IL distribution. Weibull IL distribution has been proposed by Hassan and Mohamed [12]. Maxwell et al. [13] proposed and studied Marshall–Olkin IL distribution and compared this new model with other competitive models using real data applications. ZeinEldin et al. [14] proposed and alpha-power transformed IL distribution. The application of both distributions: Frèchet and IL are wide-spreading in different fields, and by combining them, a new vital distribution is formed called the OFIL distribution. The motivations of proposing the OFIL distribution include more flexibility, closed expressions of its pdf and cdf, and applicability in more real-life situations. Bantan et al. [15] studied the estimation of entropy for the IL model using multiple censored samples. Thus, a random variable X is said to have IL distribution if the cumulative distribution function (cdf) and probability density function (pdf) of the IL distribution are given by Yadav et al. [10] as the following:(1)Gx;α,β=1+βx−α;x≥0,α,β>0,(2)gx;α,β=αβx21+βx−α−1;x≥0,α,β>0,with shape parameter α and scale parameter β of the distribution. In this article, the odd Frèchet-G (OF-G) family of distributions developed by Haq and Elgarhy [16] is having cdf and pdf given by(3)Fx;θ,ξ=e−1−Gx;ξ/Gx;ξθ,(4)fx;θ,ξ=θgx;ξ1−Gx;ξθ−1Gx;ξθ+1e−1−Gx;ξ/Gx;ξθ.OF-G family is used to boost the flexibility of the baseline distribution. Here,θ is the shape parameter, and gx;ξ considers a pdf of the baseline distribution. The random variable X with density (4) is denoted by X∼OF−Gx;θ,ξ.The hazard rate function (hrf) of the OF-G family is(5)hx:θ,ξ=θgx;ξ1−Gx;ξθ−1e−1−Gx;ξ/Gx;ξθGx;ξθ+11−e−1−Gx;ξ/Gx;ξθ.This article is planned as follows: in Section2, we define a new model called the odd Frèchet inverse Lomax (OFIL) distribution. In Section 3, some important mathematical properties are derived. Section 4 deals with characterization of the new model. Some different methods of estimation of model parameters are discussed in Section 5. Numerical study based on Monte Carlo simulations is conducted in Section 6. The analysis of real data sets is carried in Section 7, and conclusions are made in Section 8.


## 2. The Odd Fréchet Inverse Lomax Distribution

The cdf and pdf of the OFIL distribution are given by inserting (1) and (2) in (3) and (4) as follows:(6)Fx;α,θ,β=e−1+β/xα−1θ,(7)fx;α,θ,β=θαβx21+βxαθ−11−1+βx−αθ−1e−1+β/xα−1θ.The survival function, hrf, and cumulative hazard rate function are defined as follows:(8)Sx;α,θ,β=1−e−1+β/xα−1θ,hx;α,θ,β=αβθ1+β/xα−11+β/xα−1θ−1e1+β/xα−1θ−1x2,Hx;α,θ,β=−log1−e−1+β/xα−1θ.Now, a random variableX that follows density (7) is denoted by X∼OFILx;θ,α,β. Some graphic features which include pdf and hrf plots of X∼OFILx;θ,α,β are illustrated in Figures 1(a)–1(c).Figure 1
(a) The pdf plots of the OFIL distribution. (b) The hrf plots of the OFIL distribution. (c) The pdf and hrf plots of the OFIL distribution.
(a)(b)(c)The impact of values of parameters on density and hazard curve is significant. Figure1(a) shows density plots for specified values of parameters keeping one parameter fixed in each plot. The corresponding hazard rate functions are plotted in Figure 1(b). Figure 1(c) shows both of these plots for varying values of parameters. It is observed that the pdf and hazard curves start from zero or goes down at the origin.


### 2.1. Quantile Function and Median

The OFIL distribution can be easily simulated by inverting (6) as follows: if u∼U0,1, then the random variable X can be obtained from(9)Qu=β1+ln1/uθα−1,where Qu is the quantile function of the OFIL distribution. The quantile density function is obtained by taking the derivative of the quantile function, which is yet another way of prescribing a probability distribution. It is the reciprocal of the pdf composed with Qu. Therefore, the density function of quantile Qu is(10)qu=β1+ln1/u1/θ1/α−1ln1/u1/θ−1uαθ1+ln1/u1/θ1/α−12.Particularly, the median of the OFIL distribution is obtained from (9) considering u = 0.5. The median is defined as follows:(11)median=β1+ln2θα−1.


### 2.2. Skewness and Kurtosis

The skewness and kurtosis measures based on quantile measures are given, respectively, as follows:Bowley’s skewness measure is(12)S=Q0.25+Q0.75−2Q0.5Q0.75−Q0.25,and Moors’ kurtosis measure is(13)K=Q7/8−Q5/8+Q3/8−Q1/8Q6/8−Q2/8.S is the measure of the asymmetry of the OFIL distribution, and K is the measure of the heaviness of tails (in comparison to the normal distribution) of the OFIL distribution. These measures are less sensitive to outliers and defined by mathematical formulas. Moreover, they may exist for any distribution which does not have moments.Plots ofS and K for certain values of α as a function of θ and for certain choices of θ as a function of α at β = 0.5 are provided in Figures 2(a) and 2(b).Figure 2
(a) Skewness of the OFIL distribution: (A) as a function ofθ for certain values of α; (B) as a function of α for certain values of θ. (b) Kurtosis of the OFIL distribution: (A) as a function of θ for certain values of α; (B) as a function of α for certain values of θ.
(a)(b)


### 2.3. Linear Representation

In this section, the representation of cdf (5) and pdf (7) of the OFIL distribution is obtained.Firstly, according to Haq and Elgarhy [16], equation (7) can be written as(14)fx:θ,ξ=∑k=0∞ηkgx;ξGx;ξk,where ηk=∑i,j=0∞θ−1i+k/i!θi+1+jjθi+1+j−1k.By inserting equations (1) and (2) in equation (14), the pdf of the OFIL distribution can be written as a linear combination of the IL distribution as(15)fx;θ,α,β=∑k=0∞ηkαβx21+βx−αk+1−1=∑k=0∞ηkgx;αk+1,β,where gx;αk+1,β denotes the pdf of the IL distribution given in equation (2) with parameters αk+1 and β.(16)fx;θ,α,β=∑k=0∞ηkαβx21+βx−αk+1−1.Secondly, according to Haq and Elgarhy [16], the expansion of cdf Fxh is given as follows:(17)Fxh=∑z=0∞SzGx;ξz,where Sz=∑q,u=0∞−1q+z/q!qθ+u−1uqθ+uz.(18)Fxh=∑z=0∞Sz1+βx−αz.


## 2.1. Quantile Function and Median

The OFIL distribution can be easily simulated by inverting (6) as follows: if u∼U0,1, then the random variable X can be obtained from(9)Qu=β1+ln1/uθα−1,where Qu is the quantile function of the OFIL distribution. The quantile density function is obtained by taking the derivative of the quantile function, which is yet another way of prescribing a probability distribution. It is the reciprocal of the pdf composed with Qu. Therefore, the density function of quantile Qu is(10)qu=β1+ln1/u1/θ1/α−1ln1/u1/θ−1uαθ1+ln1/u1/θ1/α−12.Particularly, the median of the OFIL distribution is obtained from (9) considering u = 0.5. The median is defined as follows:(11)median=β1+ln2θα−1.


## 2.2. Skewness and Kurtosis

The skewness and kurtosis measures based on quantile measures are given, respectively, as follows:Bowley’s skewness measure is(12)S=Q0.25+Q0.75−2Q0.5Q0.75−Q0.25,and Moors’ kurtosis measure is(13)K=Q7/8−Q5/8+Q3/8−Q1/8Q6/8−Q2/8.S is the measure of the asymmetry of the OFIL distribution, and K is the measure of the heaviness of tails (in comparison to the normal distribution) of the OFIL distribution. These measures are less sensitive to outliers and defined by mathematical formulas. Moreover, they may exist for any distribution which does not have moments.Plots ofS and K for certain values of α as a function of θ and for certain choices of θ as a function of α at β = 0.5 are provided in Figures 2(a) and 2(b).Figure 2
(a) Skewness of the OFIL distribution: (A) as a function ofθ for certain values of α; (B) as a function of α for certain values of θ. (b) Kurtosis of the OFIL distribution: (A) as a function of θ for certain values of α; (B) as a function of α for certain values of θ.
(a)(b)


## 2.3. Linear Representation

In this section, the representation of cdf (5) and pdf (7) of the OFIL distribution is obtained.Firstly, according to Haq and Elgarhy [16], equation (7) can be written as(14)fx:θ,ξ=∑k=0∞ηkgx;ξGx;ξk,where ηk=∑i,j=0∞θ−1i+k/i!θi+1+jjθi+1+j−1k.By inserting equations (1) and (2) in equation (14), the pdf of the OFIL distribution can be written as a linear combination of the IL distribution as(15)fx;θ,α,β=∑k=0∞ηkαβx21+βx−αk+1−1=∑k=0∞ηkgx;αk+1,β,where gx;αk+1,β denotes the pdf of the IL distribution given in equation (2) with parameters αk+1 and β.(16)fx;θ,α,β=∑k=0∞ηkαβx21+βx−αk+1−1.Secondly, according to Haq and Elgarhy [16], the expansion of cdf Fxh is given as follows:(17)Fxh=∑z=0∞SzGx;ξz,where Sz=∑q,u=0∞−1q+z/q!qθ+u−1uqθ+uz.(18)Fxh=∑z=0∞Sz1+βx−αz.


## 3. Some Statistical Properties of the OFIL Distribution

### 3.1. Probability-Weighted Moments (PWMs)

Probability-weighted moments are often considered to be superior to ordinary moments. They are less sensitive to outliers and uniquely defined. They are sometimes used when maximum likelihood estimates are unavailable or difficult to compute. They may also be considered as initial values for maximum likelihood estimates. PWMs are denoted byτr,s, which can be defined as(19)τr,s=EXrFxs=∫−∞∞xrFxsfxdx.The PWMs of OFIL are obtained by substituting (14) and (17) into (19) as follows:(20)τr,s=∑z=0∞∑k=0∞ηkSz∫0∞xrgx;ξGx;ξz+kdx,where gx;ξ and Gx;ξ have pdf (1) and cdf (2), so we obtain τr,s as follows:(21)τr,s=∑z=0∞∑k=0∞ηkSzαβ∫0∞xr−21+βx−α−11+βx−αz+kdx,τr,s=∑z,k=0∞ηkSzτr,z+k,where τr,z+k=∫0∞xrgx;ξGx;ξz+kdx=αβ∫0∞xr−21+β/x−αz+k+1−1dx.According to Hassan and Mohamed [12], the moments about the origin of the IL distribution are given as(22)τr,z+k=αβrΓr+αz+k+1Γ1−rΓαz+k+1+1,where Γ0=−γ,Γ−r=−1r/r!φr−−1r/r!γforr=1,2,…,γ denotes Euler’s constant and φr=∑k=1r1/k (see [17]).


### 3.2. Moments

IfX has pdf (16), then the rth moment can be obtained as(23)μr′=∫0∞xrfxdx=∫0∞∑k=0∞ηkgx;ξGx;ξkdx.Substituting (16) in (23) gives(24)μr′=∑k=0∞ηkαβ∫0∞xr−21+βx−αk+1−1dx=∑k=0∞ηkτr,k,where τr,k is the PWM of the IL distribution. The mean of X can be obtained using equation (22) by putting r = 1 and z = 0:(25)μ=μ1′=EX=∑k=0∞ηkβαγ,where γ denotes Euler’s constant. The rth central moment μr of X is derived as(26)μr=EX−μ1′r=∑s=0rrs−1r−sμr−s′μ1′s.The variance (Var) of the OFIL distribution can be easily obtained as(27)σ2=μ2′−μ1′2=∑k=0∞ηkαβ2Γ2+αk+1Γ−1Γαk+1+1−μ1′2.The moment-generating function (mgf) is derived as(28)MXt=EetX=∑r=0∞trr!μr′=∑r=0∞trr!∑k=0∞ηkτr,k=∑r,k=0∞trr!ηkαβrΓr+αk+1Γ1−rΓαk+1+1.The basic measures are obtained, as shown in Table1, for particular values of parameters.Table 1
Moments for some selected valued parameters.
θαβμ1′μ2′μ3′μ4′VarSkewnessKurtosis0.50.50.50.13650.06990.04630.03460.05121.96826.10630.80.510.19950.10740.07180.05360.06761.33473.767410.51.50.22990.13210.09030.06810.07921.05472.94811.20.52.50.22550.14400.10350.08010.09321.01972.63810.510.50.16430.08880.05960.04460.06181.60774.62040.8110.19700.12650.09100.07050.08771.21583.0774111.50.15000.10970.08510.06900.08711.65254.11171.212.50.04050.03400.02910.02530.03234.317320.0600.51.50.50.16970.09810.06730.05080.06931.48454.07900.81.510.13360.09780.07590.06150.07991.83674.774411.51.50.04260.03570.03060.02670.03394.189618.9491.21.52.50.00040.00040.00030.00030.000450.04852514.5We can observe from Table1 that(1)
For lower values ofθ and β < 1, skewness and kurtosis decrease as α increases(2)
For higher values ofθ > 1 and β > 1, skewness and kurtosis increase as α increases


### 3.3. Rènyi Entropy

The Rènyi entropy is an important measure to find the amount of uncertainty in the data. The larger the uncertainty, the larger the value of Rènyi entropy.By definition, Rènyi entropy is(29)IRδ=11−δlogIδ,where Iδ=∫0∞fδxdx,δ>0andδ≠1. From Haq and Elgarhy [16], the Rènyi entropy of OF-G is(30)IRδ=1/1−δlog∑k=0∞wk∫−∞∞gx,ζδGx,ζkdx,where wk=θδδk−1i+k/i!θi+δ+δ+j−1jθi+δ+j−δk.Therefore, the Rényi entropy of OFIL is given by(31)Iδ=αβδ∑k=0∞wk∫0∞x−2δ1+βx−αk+δ−δdx=αδβ−δ−1∑k=0∞wkB2δ−1,αk+δ−δ+1.Therefore, Rènyi entropy of the OFIL distribution is given as(32)IRδ=11−δlogαδβ−δ−1∑k=0∞wkB2δ−1,αk+δ−δ+1,where B(…) is the beta function.


## 3.1. Probability-Weighted Moments (PWMs)

Probability-weighted moments are often considered to be superior to ordinary moments. They are less sensitive to outliers and uniquely defined. They are sometimes used when maximum likelihood estimates are unavailable or difficult to compute. They may also be considered as initial values for maximum likelihood estimates. PWMs are denoted byτr,s, which can be defined as(19)τr,s=EXrFxs=∫−∞∞xrFxsfxdx.The PWMs of OFIL are obtained by substituting (14) and (17) into (19) as follows:(20)τr,s=∑z=0∞∑k=0∞ηkSz∫0∞xrgx;ξGx;ξz+kdx,where gx;ξ and Gx;ξ have pdf (1) and cdf (2), so we obtain τr,s as follows:(21)τr,s=∑z=0∞∑k=0∞ηkSzαβ∫0∞xr−21+βx−α−11+βx−αz+kdx,τr,s=∑z,k=0∞ηkSzτr,z+k,where τr,z+k=∫0∞xrgx;ξGx;ξz+kdx=αβ∫0∞xr−21+β/x−αz+k+1−1dx.According to Hassan and Mohamed [12], the moments about the origin of the IL distribution are given as(22)τr,z+k=αβrΓr+αz+k+1Γ1−rΓαz+k+1+1,where Γ0=−γ,Γ−r=−1r/r!φr−−1r/r!γforr=1,2,…,γ denotes Euler’s constant and φr=∑k=1r1/k (see [17]).


## 3.2. Moments

IfX has pdf (16), then the rth moment can be obtained as(23)μr′=∫0∞xrfxdx=∫0∞∑k=0∞ηkgx;ξGx;ξkdx.Substituting (16) in (23) gives(24)μr′=∑k=0∞ηkαβ∫0∞xr−21+βx−αk+1−1dx=∑k=0∞ηkτr,k,where τr,k is the PWM of the IL distribution. The mean of X can be obtained using equation (22) by putting r = 1 and z = 0:(25)μ=μ1′=EX=∑k=0∞ηkβαγ,where γ denotes Euler’s constant. The rth central moment μr of X is derived as(26)μr=EX−μ1′r=∑s=0rrs−1r−sμr−s′μ1′s.The variance (Var) of the OFIL distribution can be easily obtained as(27)σ2=μ2′−μ1′2=∑k=0∞ηkαβ2Γ2+αk+1Γ−1Γαk+1+1−μ1′2.The moment-generating function (mgf) is derived as(28)MXt=EetX=∑r=0∞trr!μr′=∑r=0∞trr!∑k=0∞ηkτr,k=∑r,k=0∞trr!ηkαβrΓr+αk+1Γ1−rΓαk+1+1.The basic measures are obtained, as shown in Table1, for particular values of parameters.Table 1
Moments for some selected valued parameters.
θαβμ1′μ2′μ3′μ4′VarSkewnessKurtosis0.50.50.50.13650.06990.04630.03460.05121.96826.10630.80.510.19950.10740.07180.05360.06761.33473.767410.51.50.22990.13210.09030.06810.07921.05472.94811.20.52.50.22550.14400.10350.08010.09321.01972.63810.510.50.16430.08880.05960.04460.06181.60774.62040.8110.19700.12650.09100.07050.08771.21583.0774111.50.15000.10970.08510.06900.08711.65254.11171.212.50.04050.03400.02910.02530.03234.317320.0600.51.50.50.16970.09810.06730.05080.06931.48454.07900.81.510.13360.09780.07590.06150.07991.83674.774411.51.50.04260.03570.03060.02670.03394.189618.9491.21.52.50.00040.00040.00030.00030.000450.04852514.5We can observe from Table1 that(1)
For lower values ofθ and β < 1, skewness and kurtosis decrease as α increases(2)
For higher values ofθ > 1 and β > 1, skewness and kurtosis increase as α increases


## 3.3. Rènyi Entropy

The Rènyi entropy is an important measure to find the amount of uncertainty in the data. The larger the uncertainty, the larger the value of Rènyi entropy.By definition, Rènyi entropy is(29)IRδ=11−δlogIδ,where Iδ=∫0∞fδxdx,δ>0andδ≠1. From Haq and Elgarhy [16], the Rènyi entropy of OF-G is(30)IRδ=1/1−δlog∑k=0∞wk∫−∞∞gx,ζδGx,ζkdx,where wk=θδδk−1i+k/i!θi+δ+δ+j−1jθi+δ+j−δk.Therefore, the Rényi entropy of OFIL is given by(31)Iδ=αβδ∑k=0∞wk∫0∞x−2δ1+βx−αk+δ−δdx=αδβ−δ−1∑k=0∞wkB2δ−1,αk+δ−δ+1.Therefore, Rènyi entropy of the OFIL distribution is given as(32)IRδ=11−δlogαδβ−δ−1∑k=0∞wkB2δ−1,αk+δ−δ+1,where B(…) is the beta function.


## 4. Characterizations

Characterization of distribution is a key aspect that has fascinated the researchers, and it is also helpful in finding an appropriate model. In distribution theory, characterizations of probability models are very useful. A characterization of probability models plays a vital role in statistical studies in various fields of natural sciences, physical sciences, and applied sciences. Characterizing a probability distribution is a characteristic or feature which helps in identifying it, or it is the only distribution that satisfies a specified characteristic. Many researchers have investigated the characterizations of absolutely continuous probability distributions over the years. For instance, the characterization of distributions by truncated moments was studied by Glänzel [18, 19]. Hamedani and Ahsanullah [20–23] discussed various techniques of characterizations of probability distributions.


### 4.1. Characterization Based on Two Truncated Moments

Glänzel [18, 19] presented a theorem based on the ratio of two truncated moments. By utilizing this theorem, we obtain characterization of the OFIL distribution.Proposition 1.
LetX: Ω⟶0,∞ be distributed as equation (7), and(33)q1x=1+βx−αθ+1exp1−1+β/x−α1+β/x−αθ,(34)q2x=q1x1−1+βx−αθ,forx>0.
The random variable (r. v.)X follows OFIL distribution if and only if the function η is of the form(35)ηx=121−1+βx−αθ.Proof.
LetX be a random variable with density equation (7); then,(36)1−FxEq1XX≥x=1−1+βx−αθ,x>0,1−FxEq2XX≥x=1−1+β/x−α2θ2,x>0,and finally,(37)ηx=121−1+βx−αθ.
Asηxq1x−q2x=−1/2e1+β/x−α/1−1+β/x−α−θ1+β/x−α1+θ1−1+β/x−αθ<0 for all x, the proof follows.
Conversely, givenq1x,q2x,andηx, we show that the random variable X has OFIL distribution.
Here,(38)s′x=η′xq1xηxq1x−q2x=θαβ/x21+β/x−α−11−1+β/x−α,x>0,and hence, sx=−ln1−1+β/x−αθ,x>0.
Now, in view of TheoremA.1 in Appendix, X has density (7):(39)Fx=d∫0xη′uηuq1u−q2uexp−sudu,Fx=∫0xθαβx21+βx−α−11+βu−α−θ−11−1+βu−αθ−1e−1+β/u−α/1−1+β/u−α−θdu,which can be simplified to ∫0xfOFILudu=FOFILx.Corollary 1.
LetX be distributed as equation (7), and let q1x be as Proposition 1. The pdf of X be (6) if and only if there exist functions q2xandηx defined in Proposition 1 (equations (34) and (35)) satisfying the differential equation(40)η′xq1xηxq1x−q2x=θαβ/x21+β/x−α−11−1+β/x−α,x>0.


### 4.2. Characterization Based on the Hazard Function

It is known that the hrf,hF, of a twice differentiable distribution function, F, satisfies the first-order differential equation(41)f′xfx=hF′xhFx−hFx.This may be the only characterization available in terms of hrf for many univariate continuous distributions.Proposition 2.
The pdf of the OFIL distribution is (7) if and only if its hrf satisfies the following differential equation:(42)h′x;ξ−2x+1−αβxx+βhx;ξ=αβx+β/x2α−1x+β/xα−1θ−2e−1+1+β/xαθ−12x2x+β2×θ−2+x+βxα1+θ+e−1+x+β/xαθ2+x+βxα−1+−1+x+βxαθθ−1.Proof.
IfX has pdf (7), then clearly (42) holds.
Now, if (42) holds, then(43)ddxgx;ξ−1hx=ddx1+β/x2α1+β/xα−1θ−1θe1+β/xα−1θ−1,or equivalently(44)hx=αβθ1+β/xα−11+β/xα−1θ−1e1+β/xα−1θ−1x2.


## 4.1. Characterization Based on Two Truncated Moments

Glänzel [18, 19] presented a theorem based on the ratio of two truncated moments. By utilizing this theorem, we obtain characterization of the OFIL distribution.Proposition 1.
LetX: Ω⟶0,∞ be distributed as equation (7), and(33)q1x=1+βx−αθ+1exp1−1+β/x−α1+β/x−αθ,(34)q2x=q1x1−1+βx−αθ,forx>0.
The random variable (r. v.)X follows OFIL distribution if and only if the function η is of the form(35)ηx=121−1+βx−αθ.Proof.
LetX be a random variable with density equation (7); then,(36)1−FxEq1XX≥x=1−1+βx−αθ,x>0,1−FxEq2XX≥x=1−1+β/x−α2θ2,x>0,and finally,(37)ηx=121−1+βx−αθ.
Asηxq1x−q2x=−1/2e1+β/x−α/1−1+β/x−α−θ1+β/x−α1+θ1−1+β/x−αθ<0 for all x, the proof follows.
Conversely, givenq1x,q2x,andηx, we show that the random variable X has OFIL distribution.
Here,(38)s′x=η′xq1xηxq1x−q2x=θαβ/x21+β/x−α−11−1+β/x−α,x>0,and hence, sx=−ln1−1+β/x−αθ,x>0.
Now, in view of TheoremA.1 in Appendix, X has density (7):(39)Fx=d∫0xη′uηuq1u−q2uexp−sudu,Fx=∫0xθαβx21+βx−α−11+βu−α−θ−11−1+βu−αθ−1e−1+β/u−α/1−1+β/u−α−θdu,which can be simplified to ∫0xfOFILudu=FOFILx.Corollary 1.
LetX be distributed as equation (7), and let q1x be as Proposition 1. The pdf of X be (6) if and only if there exist functions q2xandηx defined in Proposition 1 (equations (34) and (35)) satisfying the differential equation(40)η′xq1xηxq1x−q2x=θαβ/x21+β/x−α−11−1+β/x−α,x>0.


## 4.2. Characterization Based on the Hazard Function

It is known that the hrf,hF, of a twice differentiable distribution function, F, satisfies the first-order differential equation(41)f′xfx=hF′xhFx−hFx.This may be the only characterization available in terms of hrf for many univariate continuous distributions.Proposition 2.
The pdf of the OFIL distribution is (7) if and only if its hrf satisfies the following differential equation:(42)h′x;ξ−2x+1−αβxx+βhx;ξ=αβx+β/x2α−1x+β/xα−1θ−2e−1+1+β/xαθ−12x2x+β2×θ−2+x+βxα1+θ+e−1+x+β/xαθ2+x+βxα−1+−1+x+βxαθθ−1.Proof.
IfX has pdf (7), then clearly (42) holds.
Now, if (42) holds, then(43)ddxgx;ξ−1hx=ddx1+β/x2α1+β/xα−1θ−1θe1+β/xα−1θ−1,or equivalently(44)hx=αβθ1+β/xα−11+β/xα−1θ−1e1+β/xα−1θ−1x2.


## 5. Statistical Inference

The parameters of the OFIL distribution can be estimated using maximum likelihood (ML), least square (LS), percentile (PC), and Anderson–Darling (AD) methods of estimation.


### 5.1. ML Estimation

Using ML estimation technique, letX1, …, Xn indicate observed values from the OFIL distribution, and then log-likelihood function, say ℓ, can be written as(45)ℓ=∏i=1nfxi=nlnα+nlnθ+nlnβ−2∑i=1nlogxi+αθ−1∑i=1nln1+βxi+θ−1∑i=1nln1−1+βxi−α−∑i=1n1+βxiα−1θ.The ML equations of the OFIL distribution are obtained by taking the partial derivative with respect to parametersα, θ, and β. We have(46)∂ℓ∂α=nα+θ∑i=1nln1+βxi+∑i=1n1+β/xi−αln1+β/xi1−1+β/xi−α−θ∑i=1n1+βxiα−1θ−11+βxiαln1+βxi,∂ℓ∂θ=nθ+α∑i=1nln1+βxi+∑i=1nln1−1+βxi−α−∑i=1n1+βxiα−1θln1+βxiα−1,∂ℓ∂β=nβ+αθ−1∑i=1n1xi+β+αθ−1∑i=1n1/xi1+β/xi−α−11−1+β/xi−α−θα∑i=1n1xi1+βxiα−11+βxiα−1θ−1.Equating∂ℓ/∂α, ∂ℓ/∂θ, and ∂ℓ/∂β with zeros and solving simultaneously, we obtain the ML estimators of α, θ, and β.


### 5.2. Ordinary LS Estimators (LSEs)

SupposeX1,X2,…,Xn be a random sample (r.s.) of size n from OFIL distribution with a corresponding ordered sample which is X1,X2,…,Xn. The mean and variance of OFIL are independent of the unknown parameters, given by(47)EFXi=in+1,VarFXi=in−i+1n+12n+2,where FXi is the cdf of ith order statistic Xi. The LS estimators can be obtained by minimizing the following sum of square of errors:(48)∑i=1nFxi−in+12,with respect to α, θ, and β. So, the LS estimators of α, θ, and β of the LS model are obtained by minimizing the following:(49)∑i=1ne−1+β/xiα−1θ−in+12,with respect to α, θ, and β (see [24]).


### 5.3. PC Estimators (PCEs)

LetX1, …, Xn be a r.s. from the OFIL distribution with the corresponding order statistics X1⋯X2<⋯<Xn. Using PC technique of estimation, the estimators of α, θ, and β are derived by minimizing the following:(50)∑i=1nlnin+1−lne−1+β/xiα−1θ2,with respect to α, θ, and β.


### 5.4. Anderson–Darling Estimators (ADEs)

Anderson and Darling [25] introduced the technique of Anderson–Darling estimation. By utilizing it in the OFIL model, the Anderson–Darling estimators (ADEs) of α, θ, and β can be obtained by minimizing the function given by(51)AD=−n−1n∑i=1n2i−1logFxi;α,θ,β+log1−Fxn+1−i;α,θ,β,with respect to α, θ, and β.


## 5.1. ML Estimation

Using ML estimation technique, letX1, …, Xn indicate observed values from the OFIL distribution, and then log-likelihood function, say ℓ, can be written as(45)ℓ=∏i=1nfxi=nlnα+nlnθ+nlnβ−2∑i=1nlogxi+αθ−1∑i=1nln1+βxi+θ−1∑i=1nln1−1+βxi−α−∑i=1n1+βxiα−1θ.The ML equations of the OFIL distribution are obtained by taking the partial derivative with respect to parametersα, θ, and β. We have(46)∂ℓ∂α=nα+θ∑i=1nln1+βxi+∑i=1n1+β/xi−αln1+β/xi1−1+β/xi−α−θ∑i=1n1+βxiα−1θ−11+βxiαln1+βxi,∂ℓ∂θ=nθ+α∑i=1nln1+βxi+∑i=1nln1−1+βxi−α−∑i=1n1+βxiα−1θln1+βxiα−1,∂ℓ∂β=nβ+αθ−1∑i=1n1xi+β+αθ−1∑i=1n1/xi1+β/xi−α−11−1+β/xi−α−θα∑i=1n1xi1+βxiα−11+βxiα−1θ−1.Equating∂ℓ/∂α, ∂ℓ/∂θ, and ∂ℓ/∂β with zeros and solving simultaneously, we obtain the ML estimators of α, θ, and β.


## 5.2. Ordinary LS Estimators (LSEs)

SupposeX1,X2,…,Xn be a random sample (r.s.) of size n from OFIL distribution with a corresponding ordered sample which is X1,X2,…,Xn. The mean and variance of OFIL are independent of the unknown parameters, given by(47)EFXi=in+1,VarFXi=in−i+1n+12n+2,where FXi is the cdf of ith order statistic Xi. The LS estimators can be obtained by minimizing the following sum of square of errors:(48)∑i=1nFxi−in+12,with respect to α, θ, and β. So, the LS estimators of α, θ, and β of the LS model are obtained by minimizing the following:(49)∑i=1ne−1+β/xiα−1θ−in+12,with respect to α, θ, and β (see [24]).


## 5.3. PC Estimators (PCEs)

LetX1, …, Xn be a r.s. from the OFIL distribution with the corresponding order statistics X1⋯X2<⋯<Xn. Using PC technique of estimation, the estimators of α, θ, and β are derived by minimizing the following:(50)∑i=1nlnin+1−lne−1+β/xiα−1θ2,with respect to α, θ, and β.


## 5.4. Anderson–Darling Estimators (ADEs)

Anderson and Darling [25] introduced the technique of Anderson–Darling estimation. By utilizing it in the OFIL model, the Anderson–Darling estimators (ADEs) of α, θ, and β can be obtained by minimizing the function given by(51)AD=−n−1n∑i=1n2i−1logFxi;α,θ,β+log1−Fxn+1−i;α,θ,β,with respect to α, θ, and β.


## 6. Simulation Study

This section consists of the comparison of the proposed estimators MLE, LSE, PCE, and ADE for OFILD in terms of simulated risks. Comparisons among the estimators have been made through simulation using 5000 simulated samples. We took samples of sizen = 10, 20, 50, 100, 200, and 500 from the OFIL distribution with different variations of parameters: set 1 = (α=2,θ=2,β=0.5), set 2 = (α=2,θ=3,β=0.5), set 3 = (α=1,θ=3,β=0.5), set 4 = (α=0.5,θ=2,β=0.5), set 5 = (α=3,θ=2,β=1.5), and set 6 = (α=0.5,θ=2,β=1.5). The MLEs, LSEs, PCEs, and ADEs of α, θ, and β are determined. Then, the estimates of all methods and their mean square errors (MSEs) are documented in Tables 2–7.Table 2
Estimates and MSEs of the OFIL distribution for MLEs, LSEs, PCEs, and ADEs for set 1 (α = 2, θ = 2, and β = 0.5).
nMLEsLSEsPCEsADEsEstimatesMSEsEstimatesMSEsEstimatesMSEsEstimatesMSEs101.3741.1070.6191.2640.7731.3950.6871.5270.6402.2940.6902.2361.4212.2880.5942.4992.0961.1410.3950.9601.0311.0570.6670.922201.2871.1370.6051.2510.6581.2830.6471.4300.4152.2820.2252.1120.2682.1920.4542.3610.5970.9690.3800.9540.3980.9680.2120.871501.3490.4801.2440.5811.2860.5401.3080.6362.2390.1272.1150.0932.1760.1442.2200.1340.8640.1750.9160.1830.9030.2300.9180.2471001.3370.4721.2470.5691.3140.5021.2510.5782.1790.0702.2060.0812.1470.0572.2070.0760.8610.1620.8990.1630.8790.1890.9140.1982001.3420.4661.2490.5661.3290.4891.2530.5652.2030.0642.1880.0542.1090.0342.2010.0590.8480.1510.9010.1620.8570.1470.9020.1685001.3510.4501.2490.5641.3900.4821.2620.5462.1720.0392.1890.0472.1530.0342.1750.0360.8350.1330.8960.1580.8120.1420.8910.155Table 3
Estimates and MSEs of the OFIL distribution for MLEs, LSEs, PCEs, and ADEs for set 2 (α = 2, θ = 3, and β = 0.5).
nMLEsLSEsPCEsADEsEstimatesMSEsEstimatesMSEsEstimatesMSEsEstimatesMSEs101.3810.4461.2720.4891.3780.4821.4660.5793.5631.2553.1071.3543.0271.1463.2540.9440.8500.2450.8670.2060.8750.2690.9120.599201.4170.4121.2760.4611.3800.4421.4210.4293.4020.5533.1830.5153.1190.5203.2200.4160.7970.1460.8370.1310.8280.1630.7980.126501.4230.4051.3490.4391.4600.3151.3690.4093.2770.2043.1940.2213.0180.2433.2980.2770.8120.1390.8290.1180.7490.0740.8090.1031001.4370.3871.3560.4251.4750.2901.3690.4073.2760.1543.2010.1333.0650.0973.2400.1250.7930.1230.8110.1030.7370.0650.8020.0972001.4510.3571.3600.4181.5040.2581.3770.3963.2220.1003.2120.0913.0710.0573.2350.0960.7770.1080.8110.1020.7200.0540.7950.0925001.4760.3221.3780.3941.5170.2461.3890.3803.1920.0583.2100.0633.0880.0293.2070.0590.7560.0920.7940.0910.7090.0480.7870.086Table 4
Estimates and MSEs of the OFIL distribution for MLEs, LSEs, PCEs, and ADEs for set 3 (α=1,θ=3,andβ=0.5).
nMLEsLSEsPCEsADEsEstimatesMSEsEstimatesMSEsEstimatesMSEsEstimatesMSEs101.2240.2511.1830.0641.1540.1461.1480.1123.2560.7612.8321.0412.6830.8042.9410.610.5010.0770.490.0150.4780.0380.4840.045201.2030.1571.1190.0551.1780.0991.130.063.0970.3192.8830.3472.7410.3972.9250.2510.4510.0330.4470.0110.4320.0250.4470.014500.8230.0341.1300.0401.1650.0741.1170.0423.2440.2232.8840.1192.7980.1552.9320.1220.6690.0320.4380.0100.4280.0160.4480.0101000.8260.0331.1100.0321.1010.0391.0930.0273.2770.152.9640.0812.7360.1412.9840.0560.6650.0310.4468.906∗0.4599.868∗0.4557.406∗2000.8490.0271.0960.0251.1060.0371.0820.0243.2160.0892.8960.0462.8320.0632.9470.0330.6390.0230.4537.34∗0.4538.564∗0.4617.226∗5000.8520.0241.0870.0231.0640.0171.0990.0243.1600.042.9410.0252.8720.0422.9210.0240.6350.0210.4566.696∗0.4694.539∗0.4496.856∗∗Indicate that the value multiply 6.696 × 10−3.Table 5
Estimates and MSEs of the OFIL distribution for MLEs, LSEs, PCEs, and ADEs for set 4 (α = 2, θ = 2, and β = 1.5).
nMLEsLSEsPCEsADEsEstimatesMSEsEstimatesMSEsEstimatesMSEsEstimatesMSEs101.9180.2431.8260.3681.9110.3411.8350.1392.430.8131.9640.641.8910.5772.0590.3931.7970.7521.7470.1351.6690.3911.9271.449201.9180.2291.8250.0621.9310.2591.8390.0662.1620.1912.010.2051.9360.2561.9890.1321.7420.6111.7380.0871.560.1411.7330.148501.8210.0431.7880.062.060.0751.8020.0612.1150.082.050.0741.9220.1032.0480.0731.6810.0431.7270.0791.4930.0731.7210.0881001.8240.0391.7880.0552.110.071.8160.0522.0650.0292.0580.051.9170.0532.0320.0291.680.0411.7220.0691.4490.0521.710.0722001.8350.0291.790.0542.1060.0631.8140.0462.0550.0152.0390.0231.9210.0412.0330.0141.6690.0311.7210.0681.450.0431.6950.0595001.8310.0321.7950.052.1150.0621.8320.0392.0276.331∗2.0239.657∗1.9340.0122.036.73∗1.6740.0331.7170.0631.4320.0371.6760.048∗Indicate that the value multiply 6.331 × 10−3.Table 6
Estimates and MSEs of the OFIL distribution for MLEs, LSEs, PCEs, and ADEs for set 5 (α=3,θ=2,andβ=1.5).
nMLEsLSEsPCEsADEsEstimatesMSEsEstimatesMSEsEstimatesMSEsEstimatesMSEs102.3490.9552.2931.5322.3921.3122.3091.1532.4440.6781.9910.6431.9380.6342.0660.3982.2832.0652.1170.4352.0060.3462.1010.84202.4040.5572.3360.6272.5280.852.3990.7352.1560.2232.050.2121.9870.2772.0570.1522.1320.4022.110.4271.9080.341.9770.339502.4560.4342.410.5562.540.3622.4430.4922.0930.081.9970.0761.9280.1362.0130.092.0350.3942.0710.4071.8760.2132.0350.3241002.5380.3122.5220.4362.570.3372.450.4922.1030.0472.0860.0521.960.0572.0790.0552.0320.3752.0380.4011.8470.1962.010.1892002.6260.2142.6260.4132.6190.3262.4820.4812.0920.0232.0610.0221.9840.0252.0530.0242.0230.2072.0190.3931.8060.1961.9280.2695002.7930.0542.6840.3792.5390.3132.5260.4132.0860.0122.080.0141.9940.012.0640.0132.0110.1272.0180.3861.7550.1892.0240.253Table 7
Estimates and MSEs of the OFIL distribution for MLEs, LSEs, PCEs, and ADEs for set 6 (α=0.5,θ=2,andβ=1.5).
nMLEsLSEsPCEsADEsEstimatesMSEsEstimatesMSEsEstimatesMSEsEstimatesMSEs100.6740.1050.7130.1120.6860.1050.8330.3282.0210.281.780.5911.8071.2351.8320.3140.4050.0810.3540.0890.3570.0920.3760.134200.6050.0490.660.0370.6610.0750.6470.0582.0580.131.7720.1621.6620.291.8970.1390.440.0530.3340.0410.3610.0750.3970.058500.5830.030.6610.030.6630.0410.6430.031.9860.0381.8110.0691.6190.1781.8560.0550.420.0210.3150.0380.3450.0430.3450.0381000.5570.0120.6450.0240.6520.0390.6330.0221.9380.0311.7750.0651.6180.1621.8620.0470.4330.0140.3260.0330.3270.040.3410.032000.5464.68∗0.6460.0230.6830.0340.6310.021.9520.0121.7970.0511.6590.1241.8160.0420.4349.358∗0.3230.0330.3390.0380.3360.0295000.5514.337∗0.6460.0220.6830.0370.6350.021.9240.0121.7830.0511.6590.1221.8160.040.4268.235∗0.3530.0320.3450.0330.3320.029∗Indicate that the value multiply 8.235 × 10−3.In Table8, ranks are assigned to different methods of estimation based on MSEs of the corresponding method for specified combinations of the parameters. From Table 8, we can conclude that the ML estimation method outperforms all other estimation methods (overall score is 73.5). Therefore, based on our simulation study, we can conclude that the ML estimation method is the best for the OFIL distribution.Table 8
Partial and overall ranks of all the methods of estimation for some combinations of parameters.
ParametersnMLLSPCAD(α=2,θ=2,β=0.5)101.53.51.53.5202314501234100132420023145001.541.53(α=2,θ=3,β=0.5)101324202.52.5415023.513.51003.53.51220043125002.5412.5(α=1,θ=3,β=0.5)1041.531.5203.51.53.51.55031.541.5100324120042315004312(α=2,θ=2,β=1.5)104321203142501243100134220013.53.5250013.53.54(α=2.5,θ=2,β=1.5)103.53.51.51.5202341502413100142.52.520013245001423(α=0.5,θ=2.0,β=0.5)101234201243501342100134220013425001342∑Ranks73.510395.590Overall rank1432


## 7. Applications

In this section, we give two examples to illustrate the performance of the proposed model. To show the better performance of the OFIL distribution and numerical calculations, we use R software. For comparison purpose, we consider the following distributions:(i)
Marshall–Olkin inverse Lomax (MOIL) distribution [13]:(52)fx=θαβ1+β/x−α−1x21−1−θ1−1+β/x−α2,θ,α,β>0,x>0.(ii)
Alpha-power Weibull (APW) distribution [26]:(53)fx=logαα−1λβxβ−1e−λxβα1−e−λxβ,λ,α,β>0,x>0.(iii)
Alpha-power inverted exponential (APIE) distribution [27]:(54)fx=logαα−1λx2e−λ/xαe−λ/x,λ,α>0,x>0.(iv)
Alpha-power transformed IL (APTIL) distribution [14]:(55)fx=logαα−1abx21+bx−a−1α1+b/x−a,α,a,b>0,x>0.(v)
Weibull IL (WIL) distribution [12]:(56)fx=abλβ−λbxλb−11+xβ−λb+11−1+xβ−λ−b+1e−a1+β/xλ−1−b,a,b,λ,β>0,x>0.The data sets are reported in Table9. The first data set consists of 48 observations of maximum annual flood discharges of North Saskatchewan in units of 1000 cubic feet per second of the North Saskatchewan River at Edmonton over a period of 47 years [28]. The second data set consists of the monthly actual tax revenue in Egypt from January 2006 to November 2010 [29]. The third data set consists of a sample of 30 failure times of the air-conditioning system of an airplane [30].Table 9
Maximum annual flood discharges, actual taxes, and failure times of the air-conditioning data sets.
Data set I19.885, 20.940, 21.820, 23.700, 24.888, 25.460, 25.760, 26.720, 27.500, 28.100, 28.600, 30.200, 30.380, 31.500, 32.600, 32.680, 34.400, 35.347, 35.700, 38.100, 39.020, 39.200, 40.000, 40.400, 40.400, 42.250, 44.020, 44.730, 44.900, 46.300, 50.330, 51.442, 57.220, 58.700, 58.800, 61.200, 61.740, 65.440, 65.597, 66.000, 74.100, 75.800, 84.100, 106.600, 109.700, 121.970, 121.970, 185.560Data set II5.9, 20.4, 14.9, 16.2, 17.2, 7.8, 6.1, 9.2, 10.2, 9.6, 13.3, 8.5, 21.6, 18.5, 5.1, 6.7, 17.0, 8.6, 9.7, 39.2, 35.7, 15.7, 9.7, 10.0, 4.1, 36.0, 8.5, 8.0, 9.2, 26.2, 21.9, 16.7, 21.3, 35.4, 14.3, 8.5, 10.6, 19.1, 20.5, 7.1, 7.7, 18.1, 16.5, 11.9, 7.0, 8.6, 12.5, 10.3, 11.2, 6.1, 8.4, 11.0, 11.6, 11.9, 5.2, 6.8, 8.9, 7.1, 10.8Data set III23, 261, 87, 7, 120, 14, 62, 47, 225, 71, 246, 21, 42, 20, 5, 12, 120, 11, 3, 14, 71, 11, 14, 11, 16, 90, 1, 16, 52, 95.Some descriptive measures for all three data sets are given in Table10. TTT plots and box plots for three data sets are given in Figures 3 and 4, respectively. From both TTT plots, we see a concave curve, indicating that the hrf behind the data is possibly increasing. This specificity also belongs to the hrf for the OFIL distribution for some values, justifying its consideration for these data sets. Further details on the TTT plots can be found in [31].Table 10
Brief description of data sets I, II, and III.
nMeanMedianStandard deviationSkewnessKurtosisData set I4851.5040.4032.376842.0685857.950652Data set II5913.4910.608.0514961.6082965.256002Data set III3059.6022.0071.884771.6936 054.966655Figure 3
TTT plots for the data sets.
(a)(b)(c)Figure 4
Box plots for the data sets.
(a)(b)(c)TTT plots and box plots for all three data sets are given in Figures3 and 4.From Figure3, it is revealed that the TTT plots show an increasing trend of the hazard rate function for data sets I and II, whereas data set III shows a decreasing trend.From Figure4, box plots show that data sets are positively skewed, and some outliers also exist.The parameters are estimated using the MLE approach. Furthermore, for appropriate model selection, we used the following goodness-of-fit measures: log-likelihoodl^, Akaike information criterion (AIC), Bayesian information criterion (BIC), Anderson–Darling statistic (A∗) and its p-value, and Cramer–von Mises statistic (W∗) and its p value. The lower the values of these criteria, the better the fit. We also provide the value for the Kolmogorov–Smirnov (KS) statistic and its p value.MLEs and goodness-of-fit measures are presented in Tables11–13 for data sets I, II, and III, respectively. The likelihood equations are plotted in Figures 5(c), 6(c), and 7(c), respectively. These figures confirm the MLEs of the OFIL distribution. The goodness-of-fit measures are least for the OFIL distribution when compared with other fitted models for all three cases.Table 11
ML estimates along with goodness of fit for data set I.
ModelMLE−logLAICBICA∗W∗KSOFILθ^ = 2.52944215.112436.224441.8380.14619 (0.9989)0.02085 (0.996)0.06974 (0.9737)α^ = 0.89524β^ = 41.2755APTILα^ = 0.00044216.492438.984444.5970.27286 (0.9569)0.03900 (0.9400)0.07224 (0.9636)a^ = 709.021b^ = 0.14638WILa^ = 7.83735216.536441.072448.5570.27011 (0.9586)0.03803 (0.9430)0.07251 (0.9624)b^ = 0.11073λ^ = 2359.93β^ = 0.40060MOILθ^ = 0.03115216.599439.198444.8120.21208 (0.9868)0.02745 (0.9847)0.07569 (0.9462)α^ = 1132.43β^ = 0.12629APTWα^ = 6.31 ∗ 1011217.398440.796446.4100.47026 (0.7765)0.06620 (0.7774)0.09011 (0.8306)β^ = 0.63123λ^ = 0.33636APIEα^ = 5.102 ∗ 1010236.609477.218480.9606.46850 (0.0006)1.22190 (0.0007)0.28348 (0.0009)λ^ = 1.62081Table 12
ML estimates along with goodness of fit for data set II.
ModelMLE−log LAICBICA∗W∗KSOFILθ^ = 5.09654188.297382.594388.8270.23206 (0.9790)0.03594 (0.9541)0.06401 (0.9690)α^ = 0.22799β^ = 185.518APTILα^ = 0.00060189.036384.072390.3050.34829 (0.8976)0.06196 (0.8036)0.06549 (0.9620)a^ = 656.496b^ = 0.04073WILa^ = 9.34279189.833387.666395.9760.47459 (0.7722)0.08094 (0.6882)0.08074 (0.8365)b^ = 0.25068λ^ = 41.1497β^ = 3.28888MOILθ^ = 0.01297189.899385.798392.0310.32147 (0.9209)0.05019 (0.8769)0.08001 (0.8445)α^ = 9.07896β^ = 6.74118APTWα^ = 3.94 ∗ 1011189.422384.844391.0770.53069 (0.7150)0.08913 (0.6425)0.08714 (0.7615)β^ = 0.62236λ^ = 0.79610APIEα^ = 0.00062189.030382.060386.2150.34695 (0.8988)0.06188 (0.8041)0.06530 (0.9629)λ^ = 26.6433Table 13
ML estimates along with goodness of fit for data set III.
ModelMLE−log LAICBICA∗W∗KSOFILθ^ = 1.82147152.235310.470314.6740.43739 (0.8098)0.06749 (0.7708)0.12544 (0.7327)α^ = 0.21103β^ = 430.774APTILθ^ = 2.50484152.974311.848315.8520.45508 (0.8019)0.07796 (0.7735)0.14875 (0.8700)α^ = 2.00719β^ = 7.87555WILa^ = 3.19731154.471316.942322.5470.71912 (0.5418)0.12031 (0.4966)0.13307 (0.6628)b^ = 0.04466λ^ = 43.8688β^ = 30.6925MOILθ^ = 10.0722152.957311.915316.1190.53144 (0.7136)0.07532 (0.723)0.13013 (0.6899)α^ = 40.6762β^ = 0.06644APTWα^ = 84.2552153.147312.294316.4970.49192 (0.7537)0.08907 (0.7410)0.14207 (0.7255)β^ = 0.91642λ^ = 3.32433APIEα^ = 101.393153.372310.744313.5470.59654 (0.6499)0.08915 (0.6441)0.12508 (0.7259)λ^ = 3.93214Figure 5
(a) Plots of the estimated pdf and cdf for data set I. (b) Plots of the estimated sf and PP plot for data set I. (c) Log-likelihood plots for data set I.
(a)(b)(c)Figure 6
(a) Plots of the estimated pdf and cdf for data set II. (b) Plots of the estimated sf and PP plot for data set II. (c) Log-likelihood plots for data set II.
(a)(b)(c)Figure 7
(a) Plots of the estimated pdf and cdf for data set III. (b) Plots of the estimated sf and PP plot for data set III. (c) Log-likelihood plots for data set III.
(a)(b)(c)The empirical and fitted pdfs and empirical cdfs are shown in Figures5(a), 6(a), and 7(a), respectively, for three data sets. The empirical and fitted sfs and empirical PP plot are shown in Figures 5(b), 6(b), and 7(b), respectively.Based on accuracy measures and these plots, it can be seen that the OFIL distribution provides a better fit than other models for all three data sets.


## 8. Conclusion

In this paper, we introduced a new, three-parameter lifetime distribution called the odd Fréchet inverse Lomax distribution. To complete the practical aspect, we provided the main mathematical properties accompanied with characterizations based on truncated moments and hrf of the proposed distribution, mixture representations for the probability density and cumulative distribution functions, quantile function, Bowley skewness and Moors kurtosis, ordinary moments, and probability-weighted moments, The parameters of the OFIL distribution were estimated using ML, LS, PC, and AD methods of estimation. The simulation study confirmed that the MLE method is the best for this distribution. Three applications on practical data sets showed that the OFIL distribution provides a better fit than several serious competitors, validating its potential in terms of applicability.



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*Source: 4658596-2020-07-22.xml*