Statistical Inference of Odd Fréchet Inverse Lomax Distribution with Applications

Deanship of Scientific Research, King Abdulaziz University, Jeddah, Saudi Arabia Faculty of Graduate Studies for Statistical Research, Cairo University, Giza, Egypt College of Statistical & Actuarial Sciences, University of the Punjab, Lahore, Pakistan Quality Enhancement Cell, National College of Arts, Lahore, Pakistan Lahore College for Women University (LCWU), Lahore, Pakistan King Abdulaziz University, Jeddah, Saudi Arabia Valley High Institute for Management Finance and Information Systems, Obour, Qaliubia 11828, Egypt


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. e 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]. e model said belongs to an inverted family of distributions with more flexibility to analyze nonmonotone behavior of the real hazard rate function. e IL distribution is obtained by considering a random variable X as X = 1/Y, where Y has a Lomax distribution Y ∼ L(x; 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. e 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. e 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. us, 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: g(x; α, β) � αβ 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 OF-G family is used to boost the flexibility of the baseline distribution. Here, θ is the shape parameter, and g(x; ξ) considers a pdf of the baseline distribution. e random variable X with density (4) is denoted by e hazard rate function (hrf ) of the OF-G family is is article is planned as follows: in Section 2, 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. e analysis of real data sets is carried in Section 7, and conclusions are made in Section 8.

The Odd Fréchet Inverse Lomax Distribution
e cdf and pdf of the OFIL distribution are given by inserting (1) and (2) in (3) and (4) as follows: e survival function, hrf, and cumulative hazard rate function are defined as follows: Now, a random variable X that follows density (7) is denoted by X ∼ OFIL(x; θ, α, β). Some graphic features which include pdf and hrf plots of X ∼ OFIL(x; θ, α, β) are illustrated in Figures 1(a)-1(c). e impact of values of parameters on density and hazard curve is significant. Figure 1(a) shows density plots for specified values of parameters keeping one parameter fixed in each plot. e 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.

Quantile Function and Median.
e OFIL distribution can be easily simulated by inverting (6) as follows: if u ∼ U(0, 1), then the random variable X can be obtained from 2 Complexity  Complexity 3 where Q(u) is the quantile function of the OFIL distribution. e 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 Q(u). erefore, the density function of quantile Q(u) is Particularly, the median of the OFIL distribution is obtained from (9) considering u � 0.5. e median is defined as follows:

Skewness and Kurtosis.
e skewness and kurtosis measures based on quantile measures are given, respectively, as follows: Bowley's skewness measure is and Moors' kurtosis measure is 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. ese measures are less sensitive to outliers and defined by mathematical formulas. Moreover, they may exist for any distribution which does not have moments.
Plots of S 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).

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 where 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 where g(x; α(k + 1), β) denotes the pdf of the IL distribution given in equation (2) with parameters α(k + 1) and β.
Secondly, according to Haq and Elgarhy [16], the expansion of cdf [F(x)] h is given as follows: where

3.1.
Probability-Weighted Moments (PWMs). Probability-weighted moments are often considered to be superior to ordinary moments. ey are less sensitive to outliers and uniquely defined. ey are sometimes used when maximum likelihood estimates are unavailable or difficult to compute. ey may also be considered as initial values for maximum likelihood estimates. PWMs are denoted by τ r,s , which can be defined as e PWMs of OFIL are obtained by substituting (14) and (17) into (19) as follows: where g(x; ξ) and G(x; ξ) have pdf (1) and cdf (2), so we obtain τ r,s as follows: According to Hassan and Mohamed [12], the moments about the origin of the IL distribution are given as where .., c denotes Euler's constant and φ(r) � r k�1 1/k (see [17]).

Moments.
If X has pdf (16), then the r th moment can be obtained as Substituting (16) in (23) gives where τ r,k is the PWM of the IL distribution. e mean of X can be obtained using equation (22) by putting r � 1 and z � 0: where c denotes Euler's constant. e r th central moment μ r of X is derived as e variance (Var) of the OFIL distribution can be easily obtained as e moment-generating function (mgf ) is derived as e basic measures are obtained, as shown in Table 1, for particular values of parameters.
We can observe from Table 1 that (1) For lower values of θ and β < 1, skewness and kurtosis decrease as α increases (2) For higher values of θ > 1 and β > 1, skewness and kurtosis increase as α increases

Rènyi Entropy.
e Rènyi entropy is an important measure to find the amount of uncertainty in the data. e larger the uncertainty, the larger the value of Rènyi entropy.

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].

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. Let X: Ω ⟶ (0, ∞) be distributed as equation (7), and e random variable (r. v.) X follows OFIL distribution if and only if the function η is of the form Proof. Let X be a random variable with density equation (7); then, and finally, As Complexity Conversely, given q 1 (x), q 2 (x), andη(x), we show that the random variable X has OFIL distribution.
Here, (7): which can be simplified to Let X be distributed as equation (7), and let q 1 (x) be as Proposition 1. e pdf of X be (6) if and only if there exist functions q 2 (x) and η(x) defined in Proposition 1 (equations (34) and (35)) satisfying the differential equation

Characterization Based on the Hazard Function.
It is known that the hrf, h F , of a twice differentiable distribution function, F, satisfies the first-order differential equation is may be the only characterization available in terms of hrf for many univariate continuous distributions.

Proposition 2.
e pdf of the OFIL distribution is (7) if and only if its hrf satisfies the following differential equation: Proof. If X has pdf (7), then clearly (42) holds. Now, if (42) holds, then

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

ML Estimation.
Using ML estimation technique, let X 1 , . . ., X n indicate observed values from the OFIL distribution, and then log-likelihood function, say ℓ, can be written as e ML equations of the OFIL distribution are obtained by taking the partial derivative with respect to parameters α, θ, and β. We have Equating zℓ/zα, zℓ/zθ, and zℓ/zβ with zeros and solving simultaneously, we obtain the ML estimators of α, θ, and β.

Ordinary LS Estimators (LSEs)
. Suppose X 1 , X 2 , . . . , X n be a random sample (r.s.) of size n from OFIL distribution with a corresponding ordered sample which is X (1) , X (2) , . . . , X (n) . e mean and variance of OFIL are independent of the unknown parameters, given by where F(X (i) ) is the cdf of i th order statistic X (i) . e LS estimators can be obtained by minimizing the following sum of square of errors: 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: with respect to α, θ, and β (see [24]).

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 with respect to α, θ, and β.

Simulation Study
is 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 8 Complexity been made through simulation using 5000 simulated samples. We took samples of size n � 10, 20, 50, 100, 200, and 500 from the OFIL distribution with different variations of parameters: and set 6 � (α � 0.5, θ � 2, β � 1.5). e MLEs, LSEs, PCEs, and ADEs of α, θ, and β are determined. en, the estimates of all methods and their mean square errors (MSEs) are documented in Tables 2-7.
In Table 8, 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).
erefore, based on our simulation study, we can conclude that the ML estimation method is the best for the OFIL distribution.

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]: (ii) Alpha-power Weibull (APW) distribution [26]: (iii) Alpha-power inverted exponential (APIE) distribution [27]: (iv) Alpha-power transformed IL (APTIL) distribution [14]:  10 Complexity (v) Weibull IL (WIL) distribution [12]: e data sets are reported in Table 9. e 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]. e second data set consists of the monthly actual tax revenue in Egypt from January 2006 to November 2010 [29]. e third data set consists of a sample of 30 failure times of the air-conditioning system of an airplane [30].
Some descriptive measures for all three data sets are given in Table 10. 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. is specificity also belongs to the hrf for the OFIL distribution for some values, justifying

Complexity 11
its consideration for these data sets. Further details on the TTT plots can be found in [31]. TTT plots and box plots for all three data sets are given in Figures 3 and 4.
From Figure 3, 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.
MLEs and goodness-of-fit measures are presented in Tables 11-13 for data sets I, II, and III, respectively. e likelihood equations are plotted in Figures 5(c), 6(c), and 7(c), respectively. ese figures confirm the MLEs of the OFIL distribution. e goodness-of-fit measures are least for the OFIL distribution when compared with other fitted models for all three cases. e empirical and fitted pdfs and empirical cdfs are shown in Figures 5(a), 6(a), and 7(a), respectively, for three data sets. e empirical and fitted sfs and empirical        Complexity 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.

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, e parameters of the OFIL distribution were estimated using ML, LS, PC, and AD methods of estimation. e simulation study confirmed that the MLE method is the best for this distribution.
ree 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.