Inverse Power Akash Probability Distribution with Applications

This paper introduces an inverse power Akash distribution as a generalization of the Akash distribution to provide better fits than the Akash distribution and some of its known extensions. The fundamental properties of the proposed distribution such as the shapes of the distribution, moments, mean, variance, coefficient of variation, skewness, kurtosis, moment generating function, quantile function, Rényi entropy, stochastic ordering and the distribution of order statistics have been derived. The proposed distribution is observed to be a heavy-tailed distribution and can also be used to model data with upside-down bathtub shape for its hazard rate function. The maximum likelihood estimators of the unknown parameters of the proposed distribution have been obtained. Two numerical examples are given to demonstrate the applicability of the proposed distribution and for the two real data sets, the proposed distribution is found to be superior in its ability to sufficiently model heavy-tailed data than Akash, inverse Akash and power Akash distributions respectively.


Introduction
In many fields of life, statistical models are formulated to analyze real-life problems and these models are commonly built using probability distributions. Several kinds of probability distributions have been used by various researchers to model different kinds of real life problems. One of such distributions is the Akash distribution developed by 2 2 θ + respectively.
The mathematical and statistical properties of the Akash distribution were equally studied by [1] and this distribution was applied in modelling datasets from the engineering and medical fields. In addition, it was shown that the Akash distribution performed better than the well-known exponential and Lindley distributions in statistical modelling. Another related study by [2] and [3] used several datasets to substantiate the claim that the Akash distribution outperforms the exponential, Lindley and Shanker distributions respectively in modelling real-life phenomena.
In recent times, several extensions and generalizations of the Akash distribution have been given in the literature. The essence of such generalizations as pointed out by [4] is to provide better fits to data and obtain more flexible models. In this regard, [5] proposed a quasi Akash distribution and defined its probability density function (pdf) as ( ) A two-parameter Akash distribution due to [8] has its pdf given by ( ) ( ) Obviously, none of the generalizations of the Akash distribution reviewed in this paper allow for modelling data with heavy tails. By heavy tails, we mean tails that decay more slowly than the tails of the normal distribution. Heavy-tailed distributions do not have all their power moments finite and quite often they do not have finite variance. As we may be aware, many datasets are heavy tailed, especially those arising from finance, actuarial science, insurance etc. The aim of this article is to introduce a heavy-tailed version of the Akash distribution, called the inverse power Akash distribution (IPAD).

Derivation of the Inverse Power Akash Distribution
Given the distribution of the Akash random variable X defined in (1). Let us assume that another random Y is related to X by the inverse power function ( ) 1 .
The derivation of the pdf of the inverse power Akash distribution entails finding the distribution of the random variable Y. One way to determine the pdf of Y is to assume that As stated in [9], the probability density function of a continuous random variable ( ) Similarly, the cumulative density function (cdf) of the inverse power Akash distribution (IPAD) is derived as follows Integrating Eq. (7) by parts leads to ( ) Further simplification of (8)  Interestingly, the inverse Akash distribution offers more flexibility than the Akash distribution since it can be used to model datasets with heavy-tailed data as well as bathtub and upside-down bathtub shapes for its failure rate. It may be noted that the inverse power Akash distribution proposed in this work has polynomial tails for all values of α and θ, and so can be referred to as a heavy-tailed distribution. In this distribution, α is called the heavy tailed parameter, which controls the rate of decay of the upper tail. The larger the value of α, the less heavy the rate of decay of the upper tail. The smaller the value of α, the more heavy the tail becomes.

Investigation of the Proposed Inverse Power Akash Distribution for Proper Density Function
According to [9], a function ( ) Case 2. The property that total area under the curve of ( ) f y is unity implies that ( ) and . y −α = ω Notably, the mode of the proposed distribution is the value of y −α for which ( ) φ ω = One is then required to find the zeroes of ( ) 0. φ ω = To this end, we write ( ) 0 φ ω = as Defining 2 3 u a = ω + and 2 3 u a ω = − in (11), one obtains where ( ) ( ) The solution of (12) is obtained using the formula After the value of u has been obtained, it has to be substituted into the relation 2 3 u a ω = − to get the value of ω. Finally, the value of y is obtained from the relation , y −α = ω and this value of y is the mode of the inverse power Akash random variable.

Asymptotic behaviour of the proposed distribution
The asymptotic behaviour of a distribution is investigated by evaluating the limit of the probability density function (pdf) as the observed values of the random variable tends to zero and infinity respectively. As orchestrated by [10], if   The asymptotics of the pdf, cdf and the upper tails of the pdf are polynomials (i.e., heavy upper tails), while its lower tails decay exponentially. The mode moves more to the right and the pdf becomes more peaked with increasing values of θ.

Quantile function of the Inverse power Akash distribution
The th p quantile of the IPAD is the value of y that satisfies the equation On solving (14) for y, we get Undoubtedly, the quantile function can be useful for random number generation, estimation based on percentiles and quantile regression methods. For random number generation, one can solve (15) for p y for p a uniform random number between 0 and 1.
The solving must be performed numerically, for example, using uniroot in the R software [11].

Moments of the inverse power Akash distribution
The moments of distributions are used to describe some of the most important features of a model like dispersion, skewness and kurtosis. Consequently, the th r raw moment of the IPAD is given by ( ) ( ) The th r central moment of the IPAD is given by In particular, the second, third and fourth central moments of the inverse power Akash distribution are Notably, the first raw moment ( )

Coefficient of variation of the inverse power Akash distribution
The ratio of the standard deviation to the mean gives the coefficient of variation (cv) of the inverse power Akash distribution (IPAD) as

Skewness and Kurtosis of the inverse power Akash distribution
The skewness (sk) of the inverse power Akash distribution (IPAD) is given by Also, the kurtosis (kur) of the inverse power Akash distribution (IPAD) is

Moments generating function of the inverse power Akash distribution
In addition to moments, many of the interesting features of a statistical distribution can also be obtained through its moment generating function (mgf). Let Y denote a random variable having the inverse power Akash distribution (IPAD) with parameters θ and α, then its moment generating function (mgf) is

Reliability Analyses
In this section, we present the survival, hazard rate, reversed hazard, cumulative hazard and odds functions of the inverse power Akash distribution, which are useful in reliability analysis.

Survival function for the inverse power Akash distribution
Let Y be a continuous random variable having the inverse power Akash distribution with parameters θ and α, then the survival function of Y is defined to be The survival function is also known as the reliability function and it indicates the probability of surviving an age y or becoming older than y. The study of ( ) S y is at the heart of survival analysis and reliability theory. It is important in describing systems of components, that is, in calculating systems' reliability.

Hazard rate function of the inverse power Akash distribution
The hazard rate function of a statistical distribution is obtained mathematically as the 14 ratio of the probability density function ( ) f y to the survival function ( ).
S y Thus, the hazard rate function for the inverse power Akash distribution is defined as

Reversed hazard rate for the inverse power Akash distribution
The reversed hazard rate refers to the ratio of the probability density function (pdf) to the cumulative distribution function (cdf). It extends the concept of hazard rate to a reverse time direction and is given by The reversed hazard ( ) R h y describes the probability of an immediate past failure, given that the unit has already failed at time y, described by ( ).
h y

Cumulative hazard rate function for the inverse power Akash distribution
The cumulative hazard rate (chr) of the probability density function (pdf) is defined as

Odds function for the inverse power Akash distribution
The odds function of the inverse power Akash distribution is defined as

Rényi Entropy of the Inverse Power Akash Distribution
Entropy provides tool for quantifying the amount of information (or uncertainty) contained in a random sample regarding its parent population. A large value of entropy implies that there is greater uncertainty in the data. The concept of entropy is important in different areas such as physics, probability and statistics, communication theory, economics and so on. The Rényi, Shannon and Tsallis entropy, among others, are some different forms of entropy. In this paper, the widely used R ńyi entropy was considered. Consequently, the Rényi entropy is defined by [12] as Using the fact that ( )

Stochastic Ordering of the Inverse Power Akash Distribution
Stochastic ordering of positive continuous random variables is an important tool for judging the comparative behaviour of random variables. The different types of stochastic orderings which are useful in ordering random variables include the usual stochastic order, the hazard rate order, the mean residual life order, and the likelihood ratio order for the random variables under a restricted parameter space. Suppose X and Y are independent random variables with cumulative distribution functions The inverse power Akash distribution is ordered with respect to the strongest "likelihood ratio" ordering. To show the flexibility of the IPAD, its likelihood ratio is defined as Next, the log of the likelihood ratio in (39) is θ ≥ θ or ( ) 1 2 or for . θ = θ

Maximum Likelihood Estimators of the Inverse Power Akash Distribution
In estimation theory, the method of maximum likelihood have been used more often to find the parameters of statistical distributions due to the fact that it possesses the consistency, asymptotic efficiency and invariance properties. Thus, to obtain the maximum likelihood estimators of parameters of the inverse power Akash distribution, let 1 2 , , ..., n Y Y Y constitute a random sample of size n from this distribution and define the likelihood function of the random sample as ( ) Taking the natural log of (44), one obtains the log-likelihood function of the random sample as The maximum likelihood estimates of α and θ can is obtained by solving the nonlinear system of equations (48) and (49). It is usually more convenient to use nonlinear optimization algorithms such as quasi-Newton algorithm to numerically maximize the log-likelihood function. The R package provides nonlinear optimization for solving such problems. In what follows, their corresponding confidence intervals cannot be constructed explicitly. Thus, there is need to find the approximate confidence intervals of α and θ. To do this, one is required to obtain the asymptotic distribution of the maximum likelihood estimators of α and θ. In this regard, we first obtain the second-order partial derivatives, which are required in order to determine the Fisher information matrix.

The Asymptotic Distribution and Approximate Confidence Interval
To solve (54), one is required to find the expressions for To resolve the integrals in (56), we recall from [13] that Again, we obtain the following expectation Finally, we obtain the expectation , In [13], According to [14], the asymptotic distribution of ( ) In practice, the distribution of ( ) , α − α θ − θ can be approximated by a bivariate normal distribution with zero means and covariance matrix ( ) say for n sufficiently large. This approximation can be used to construct confidence intervals and test of hypotheses. For example, ( ) 100 1 − γ confidence intervals for θ and α are respectively given by 11

Numerical Examples
In this section, two real-life data sets are used to illustrate the importance and flexibility of the inverse power Akash distribution proposed in this work. The first and second data sets, which appeared in the work [15], are provided in Tables 1 and 2 respectively.   [15].
The inverse power Akash distribution is fitted the to the two data sets by using the method of maximum likelihood and the results are compared with the other competitive models namely, Akash (A), Inverse Akash (IA) and Power Akash (PA) distributions respectively.
Next, some criteria like the Akaike information criterion (AIC), Bayesian information criterion (BIC), and Consistent Akaike information criterion (CAIC) are used to verify which of the aforementioned distributions fits the research data better. The formulae for computing the vales of AIC, BIC and CAIC are respectively given by where l denotes the log-likelihood function evaluated at the maximum likelihood

Inverse Power Akash Probability Distribution with Applications
estimates, k is the number of model parameters, n is the sample size. For calculation of the analytical measures, the optimum () R-function with the argument method= "BFGS".
A distribution is said to provide the best fit to the data if among all the distributions under consideration, it corresponds to minimum values of AIC, BIC, CAIC and the loglikelihood respectively. The maximum likelihood estimates with the standard error of the fitted models and the corresponding model selection criteria for data sets 1 and 2 are presented in Tables 3 and 4.  Based on the results displayed in Tables 3 and 4 respectively, it is evident that the IPA distribution has the smallest AIC, BIC, CAIC and log-likelihood values among all competing models, and so it could be chosen as the best model among all the distributions which have been fitted to the two data sets.

Conclusion
This paper introduced a new two-parameter heavy-tailed distribution called the inverse power Akash distribution and derived some of its properties like moments, mean, variance, mode, coefficient of variation, skewness, kurtosis, moment generating function, quantile function, R ńyi entropy, stochastic ordering and the distributions of order statistics. In addition, some functions commonly used in reliability analysis, such as survival, hazard, reversed, cumulative hazard and odds functions respectively have been derived. The model parameters were estimated by using the maximum likelihood estimation procedure. Finally, the proposed model was fitted to two real-life data sets and was compared with the estimates from other extensions of the Akash distribution. The proposed distribution was found provide a better fit than some other competition distributions considered in this study. It is hoped that the proposed distribution will serve as an alternative model to other models available for modelling heavy-tailed data in many areas such as finance, insurance and economics.