A Novel Method for Developing Efficient Probability Distributions with Applications to Engineering and Life Science Data

Department of Statistics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan Department of Mathematics, College of Science, Jazan University, Jazan, Saudi Arabia Institute of Numerical Sciences, Kohat University of Science & Technology, Kohat, Pakistan Department of Mathematics, Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, Kuala Nerus 21030, Terengganu, Malaysia


Introduction
In the last few years, the literature of distribution theory has become rich due to the induction of additional parameters in the existing distribution. e inclusion of an extra parameter has shown greater flexibility compared to competitive models. e inclusion of a new parameter can be performed either using the available generator or by developing a new technique for generating new improved distribution compared to classical baseline distribution. Azzalini [1] proposed a modified form of the normal distribution by inserting an extra parameter, known as skew normal distribution, which indicated greater flexibility over normal distribution. Mudholkar and Srivastava [2] introduced exponentiated Weibull distribution by introducing a shape parameter in two-parameter Weibull distribution. Its cumulative distribution function is as follows: is model provides greater flexibility compared to the base line distribution. Note that, for β � 1, the exponentiated Weibull distribution and base line distribution coincide. Later on, various researchers have introduced different forms of exponentiated distributions; see, for example, the work of Gupta et al. [3]. Marshall and Olkin [4] introduced another technique to add an extra parameter to a probability distribution. Eugene et al. [5] suggested the beta-generated technique and applied this method to beta distribution and proposed beta-generated distribution by incorporating an extra parameter in beta distribution. Alzaatreh et al. [6] proposed a new technique and produced T-X class of continuous probability models by interchanging the probability density function of beta distribution with a probability density function, g(t), of a continuous random variable and used a function W (F (x)) which fulfills some particular conditions. Recently, Aljarrah et al. [7] introduced T-X class of distributions using quantile functions. For more details about new techniques to produce probability distributions, see the works of Lee et al. [8] and Jones [9]. Al-Aqtash et al. [10] proposed a new class of models using the logit function as a baseline and obtained the particular case referred to as Gumbel-Weibull distribution. Alzaatreh et al. [11] studied the gamma-X class of distributions and recommended the particular case using the normal distribution as a baseline distribution. Abid and Abdulrazak [12] introduced truncated Frechet-G class of distributions. Korkmaz and Genc [13] presented a generalized two-sided class of probability distributions. Alzaghal et al. [14] worked on the T-X class of distributions. Aldeni et al. [15] used the quantile function of generalized lambda distribution and introduced a new family. For more details, see the works of Cordeiro et al. [16], Alzaatreh et al. [17], and Nasir et al. [18]. e more recent modified Weibull distributions are introduced by Abid and Abdulrazak [12], Korkmaz and Genc [13], Aldeni et al. [15], Cordeiro et al. [16], and Pe and Jurek [19].
Pearson [20] used the system of differential equation technique and produced new probability distributions. Burr [21] also proposed a new method by using the differential equation method, which may take on a wide variety of forms of the continuous distributions. Since 1980, methodologies of suggesting new models moved to the inclusion of extra parameters to an existing family of distributions to increase the level of flexibility. ese include Weibull-G presented by Bourguignon et al. [22], Garhy-G proposed by Elgarhy et al. [23], Kumaraswamy (Kw-G) proposed by Cordeiro and de Castro [24], Type II half-logistic-G by Hassan et al. [25], exponentiated extended-G suggested by Elgarhy et al. [26], the Kumaraswamy-Weibull introduce by Hassan and Elgarhy [27], exponentiated Weibull by Hassan and Elgarhy [28], odd Frechet-G introduced by Haq and Elgarhy [29], and Muth-G by Almarashi and Elgarhy [30]. For a short review, one can study the work of Kotz and Vicari [31]. Recently, Mahdavi and Kundu [32] developed a new technique for proposing probability distributions which is referred as alpha power transformation (APT) technique, defined by the cumulative distribution function (CDF) as follows: e core purpose of introducing new family of distributions is to overcome the difficulties that are present in the existing probability distributions. In this study, we suggest a new method for obtaining new continuous probability distributions. Frechet distribution is used as a submodel to have a new probability distribution which is referred as modified Frechet (MF) distribution. Our proposed family of distributions also models monotonic and nonmonotonic hazard rate function and provides increased flexibility as compared to the already available probability distribution in the literature.

The Proposed Class of Distributions
e proposed class of probability distributions is termed as modified Frechet class (MFC) of distributions. e cumulative distribution function (CDF) and probability density function (PDF) of the suggested class of distributions are given by the following expressions: where F (x) and f (x) are the CDF and PDF of the baseline distribution and α is the shape parameter. e method in (3) is used to produce a new model referred as modified Frechet (MF) distribution with the aim to attain more flexibility in modeling life time data. e derivation of MF distribution is given in Section 2.1.

e Proposed Distribution.
e CDF of Frechet distribution is as follows: where β is the shape parameter. is portion of the manuscript is concerned with introducing a subclass of MF class of distributions using the cumulative distribution function of Frechet distribution. e resultant distribution is what we call modified Frechet (MF) distribution.

Definition 1.
A random variable X is said to have MF distribution with two parameters α and β if its PDF is given as follows: Its CDF is given by e hazard rate function of MF distribution is as follows: ese four functions have been plotted in Figures 1 and 2 .
is a decreasing function and vice versa.

Lemma 2. For α < 1, if f(x) is log-convex and decreasing function, then h MF (x) is a decreasing function.
Proof. If the second-order differential of f(x) exists and Taking second-order derivative of equation (10), we obtain When 0 ≤ α < 1 and β > 0, then ); then, the quantile function is as follows: where U is uniformly distributed random variable. e quantile function of the MF model is given as

Median.
Median of MF distribution is obtained by substituting u � 1/2 in equation (13), that is, 2.1.3. Mode. Mode of MF is obtained by solving the following equation for x.
Mode of the distribution satisfies the above equation.

r th Moment of MF Distribution.
Let X ∼ MF(α, β); and the r th moment of X is as follows: Using x − β � y and then e − αy � z in (16), the expression will take the following form: where m � (− r/β). Again substituting log z � u in (17) and after some simplification, the expression becomes   Journal of Mathematics Using series notation e − e u � ∞ r�0 ((− e u ) k /k!) in (18), we obtain where b > 0.

Moment Generating Function.
Let X ∼ MF(α, β); then, the moment generating function is given by Using series e tx � ∞ r�0 (t r x r /r!) in (20) and simplifying, we have where m � (− r/β) and b > 0.

Order Statistics.
Let X 1 , X 2 , . . . , X n be a random sample taken from MF distribution, and let X (1) ≤ X (2) ≤ · · · ≤ X (n) denote the order statistics. en, the probability density function of X i: n is given by Substitute PDF and CDF of MF in equation (22), and we obtain distribution of i th order statistic as

Lemma 4. e mean waiting time, say μ(t), of MF distribution is given by
Proof. By definition, the mean waiting time of MF distribution is (41) e final expression of mean waiting time of MF distribution is obtained. By substituting and □ 2.7. Parameters' Estimation. Let X 1 , X 2 , . . . , X n be a random sample of size n selected from MF(α, β); then, the loglikelihood function of MF distribution is given as Differentiating equation (44) with respect to α and β and equating them to 0, we obtain Solving (45) and (46) together, we get the estimates of α and β.
e Newton-Raphson method or the bisection method is used to get solution of the above equations as an analytical solution which is not possible. e maximum likelihood estimators (MLE) are asymptotically normally distributed, that is, where Σ is variance covariance matrix and can be obtained by inverting the observed Fisher information matrix F given below: e second derivative of equations (45) and (46) with respect to α and β yields (48)-(50) given as and Asymptotic (1 − ζ)100% confidence intervals of the parameters of the proposed distribution can be obtained as where Z ζ is the upper ζ th percentile of the standard normal distribution.

Simulations' Studies
In order to measure the performance of MLE of the parameters of MF distribution, their mean square error (MSE) and bias are calculated using simulation study. We consider W � 100 samples of sizes n � 50, 70, and 100 generated from the MF model. Bias and MSE are calculated using the expressions: Simulation results have been obtained for different values of α and β.
e MSEs and bias are presented in Table 1.
e consistency behavior of MLE can be easily verified from these results as the MSEs and bias of the estimates decrease for all parameter combinations with increasing sample size. Hence, we conclude that MLE procedure executes very well in estimating the parameters of MF distribution.

Dataset 1.
e performance of the suggested model is assessed using two datasets. e first dataset is taken from the work of Gross and Clark [37] which consists of 20 observations of patients receiving an analgesic and is given as follows: In order to compare the MF model with other models, some standard model selection criteria such as Akaike's information criteria (AIC), consistent Akaike's information criteria (CAIC), Bayesian information criterion (BIC), Hannan-Quinn information criteria (HQIC), Kolmogorov-Smirnov (K-S), and P value are used. Tables 2 and 3 demonstrate results based on dataset 1 and 2, respectively.     It is evident from the results in Tables 2 and 3 that the proposed MF distribution executes well as compared to other competitive distributions. Figures 3 and 4 represent various graphs of MF distribution for dataset 1 and dataset 2.

Conclusion
In this paper, a new method for deriving new continuous probability distributions has been offered which we called modified Frechet Class (MFC) of distributions. Also, a new probability model has been proposed using MFC. We called it modified Frechet (MF) distribution. Several statistical properties of the said distribution were derived and investigated for MF distribution. e MLE method was adopted to estimate the parameters of the proposed distribution. Simulation results showed that these estimates were consistent. In order to check the performance of the AFF model, two real datasets. e results based on these datasets revealed promising performance of the suggested model compared to some other distributions existing in the literature.

Data Availability
e data used in this paper are freely available upon citing this article.