Lehmann Type II Frechet Poisson Distribution: Properties, Inference and Applications as a Life Time Distribution

A new generalization of the Frechet distribution called Lehmann Type II Frechet Poisson distribution is defined and studied. Various structural mathematical properties of the proposed model including ordinary moments, incomplete moments, generating functions, order statistics, Renyi entropy, stochastic ordering, Bonferroni and Lorenz curve, mean and median deviation, stress-strength parameter are investigated. The maximum likelihood method is used to estimate the model parameters. We examine the performance of the maximum likelihood method by means of a numerical simulation study. The new distribution is applied for modeling three real data sets to illustrate empirically its flexibility and tractability in modeling life time data.


Introduction
Frechet distribution which is also known as Inverse Weibull distribution belong to the class of Type II extreme value distribution was developed by Frechet (1924) is a very useful distribution for modeling life time data. The Frechet distribution is one of the important distributions in extreme value theory and has several applications which include: floods, horse racing, accelerated life testing, earthquakes, sea waves, rainfall and wind speeds. For more studies on the properties and applications of Frechet distribution, see Kotz and Nadarajah (2000), also Harlow (2002). This distribution can be used to analyse life time data that exhibits decreasing increasing or constant failure rate. However, models with complex hazard rate shapes such as bathtub, unimodal and other shapes are often encountered in real life time data analysis which may include mortality studies, reliability analysis etc., which the Frechet distribution may not provide a reasonable parametric fit when used for modeling complex phenomenon. Several modifications have been made to improve its parametric fits, some of which are: Beta-Exponential Frechet was developed and studied by Mead et al. (2017). The properties of Transmuted Frechet was investigated by Mahmoud and Mandour (2013), Transmuted Exponentiated Frechet was studied by Elbatal et al. (2014), Krishna et al. (2013) developed and studied Marshall-Olkin Frechet distribution, gamma extended Frechet distribution was studied by Silva et al. (2013) and the exponentiated Frechet distribution was studied by Nadarajah and Kotz (2003). The Odd Lindley Frechet Distribution was studied by Korkmaz et al. (2017), alpha power transformed Frechet was studied by Suleman et al. (2019) and Mead and Abd-Eltawab (2014) studied Kumaraswamy Frechet. Afify et al. (2016a) studied Weibull Frechet, Kumaraswamy Marshall-Olkin Frechet distribution was developed by Afify et al. (2016b), Kumaraswamy transmuted Marshall-Olkin Frechet was studied by Yousof et al. (2016), Beta Transmuted Frechet distribution was developed and studied by Afify et al. (2016c). Yousof et al. (2018b) developed and studied the Topp Leone Generated Frechet distribution and Odd log-logistic Frechet was studied by Yousof et al. (2018a). distribution was derived and studied by Barreto-Souza and Cribari-Neto (2009). The Kumaraswamy Lindley-Poisson distribution which generalises the Lindley-Poisson distribution was studied by Pararai et al. (2015), Mohamed and Rezk (2019) developed and studied the properties and applications of the extended Poisson-Frechet distribution.
Suppose that a random variable X follows a Frechet distribution, having a cumulative distribution function ( ) and probability density function ( ) , respectively given as: And ( ; , ) = (2) Where > 0and > 0. is a scale parameter and is a shape parameter.

Frechet Poisson Distribution
Suppose that the failure time of each subsystem has the Frechet model defined by and in (1) and (2). Given , let denote the failure time of the subsystem which are independently and identically distributed random variable from Frechet distribution. Taking to be distributed according to the truncated Poisson random variable with probability mass function ( ) Suppose that the failure time of each subsystem has the Frechet distribution defined by the cdf given in equation (1).

= min { }
Unconditional cdf of given is The equation (4) above is the exponentiated Frechet distribution.
So, the unconditional cdf of X (for > 0) is given by The density function is given by Where , > 0and > 0. is a scale parameter, and are shape parameters

Lehman Type II Frechet Poisson Distribution
In this sub-section, we present the Lehman Type II Frechet-Poisson ( ) distribution, and derive some of its properties which include: , , hazard function (ℎ( )), reversed hazard function ( ( )), quantile function and sub-models.
The Lehman type II distribution is a hybrid of the generalised exponentiated distribution developed by Cordeiro et al. (2013). Given ( ) to be an arbitrary baseline in the interval (0,1). The cdf ( ), called the Exponentiated-G ( )distribution has the Where > 0 and > 0 are two additional shape parameters which exhibits tractable properties especially for simulations, since the quantile function takes a simple form given by Where ( ) is the baseline quantile function. The two extra shape parameters can control both tail weight and entropy of EG distribution. The expression in (7) can be splitted into two generalised distribution called the Lehman type I and the Lehman type II distribution by respectively taking = 1 and = 1. The distribution function of Lehman type I and Lehman type II are given respectively by and Where ( ) is the baseline distribution. For the purpose of this study ( )is the cdf of distribution. Thus, the goal of this study is to develop another generalization of the Frechet Poisson distribution called the Lehman Type II Frechet Poisson distribution with a wider scope of applications that may be used in modeling real life time data which may include applications in medicine, reliability, aeronautical engineering, weather forecasting and other extreme conditions with a better fit than the Frechet Poisson distribution.
Plots of the of LFP distribution are given below in figure 1 and figure 2 for arbitrary values of , , , .

Survival and the Hazard Function
The survival function of the LFP distribution is given by: for > 0, > 0, > 0, > 0 > 0. The graph of the survival function for various values of the parameters , , , and is given in figure 3.
Putting equations (12) and (13) Putting equations (11) and (12)   The graph of the hazard function in figures 4 and 5 for different values of the parameters exhibits various shapes such as monotonically decreasing, increasing, increasing-decreasing and upside down bathtub shapes. This feature indicates the flexibility of distribution and its suitability in modeling monotonic and non-monotonic hazard behaviour which are often encountered in real life situations.

Some Sub-models of the LFP Distribution
In this sub-section, we give the sub-models of LFP distribution for selected values of the parameters , , and are presented  When = 1, we obtain the Lehmann Type II Inverted Weibull Poisson distribution.

 When
= 1, we obtain the Lehmann Type II Inverse exponential Poisson distribution which is given in equation (49).
 When = 1, we obtain the Frechet Poisson distribution which is given in equation (50).
 When = = 1, we obtain the Inverted Weibull Poisson distribution which is given in equation (51).
 When = = 1, we obtain the Inverse exponential Poisson distribution which is given in equation (52).
 When = = 1, we obtain the Lehmann Type II Inverted Weibull distribution.
 When = = 1, we obtain the Lehmann Type II Inverse exponential distribution  When = 1, we obtain the Lehmann Type II Frechet distribution.

Expansion of the Density Function
Considering the binomial series expansion given by Thus we have: Then we have, Applying equation (20) to equation (19), we obtain Where, Finally we have, , ( ; ( + 1), )

Quantile Function
The quantile function of the LFP distribution is obtained by solving the equation ( ) = , where 0 < < 1. Then we obtain Classical measures of skewness and kurtosis may be difficult to obtain due to non-existence of higher moments in several heavy tailed distributions. When such a situation occurs, the quantile measures can be considered. The Bowley ( )skewness; Kenny and Keeping (1962) is one of the foremost measures of skewness that is based on quantile of a distribution. It is given by Consequently, the coefficient of Kurtosis can be obtained using Moor's (1988) approach to estimating kurtosis which is based on octiles of a distribution and is given by It is of noteworthy that the two measures are more robust to outliers.

Moments
In this section, we obtain the moment of the distribution. Momentplays important role in statistical analysis, most especially in determining the structural properties of a distribution such as skewness, kurtosis, dispersion, mean etc.
Theorem 1. Let a random variable follows the Lehmann type II Frechet Poisson distribution, the moment of distribution is given by Proof: let be a random variable from distribution, the moment is given by Substitute for ( ) in equation (25), we have By letting Taking, = ( + 1) , = − { ( + 1)} and putting it in equation (27), we have It then follows that the moment of distribution is given as Table 2 given below represents the first four moments, Variance( ), the Coefficient of Variation ( ), Coefficient of Skewness( ) and Coefficient of Kurtosis ( )for arbitrary values of the parameters of distribution taking a fixed value of = 3.0 and = 5.5 for Table 2 and for Table 3, we fixed = 0.1, = 0.5.  It can be observed from Table 2 and Table 3 that the distribution can be to model data that skewed to the right (positively skewed) or left (negatively skewed) with various degree of kurtosis.

Moment Generating Function
Moment generating function is a very useful function that can be used to describe certain properties of the distribution. The moment generating function of LFP distribution is given in the following theorem.
Theorem 2. Let follows the LFP distribution, the moment generating function, ( ) is Proof: The moment generating function of a random variable X is given by Where ( ) is given in equation (21). Using series expansion in (20), we have Using ( ) given in equation (29) in equation (31), we have It could be observed from the series expansion of (32) that moments are the coefficient of ! .

Incomplete Moment
The incomplete moment can be used to estimate the mean deviation, median deviation and the measures of inequalities such as the Bonferroni and Lorenz curves. The incomplete moment of Lehmann type II Frechet Poisson distribution is given in the following theorem.

Renyi Entropy
The Renyi Entropy measures the uncertainty in a distribution as defined by Renyi (1961). The Renyi entropy of distributionis given in the following theorem.

Stress-strength Parameter
Suppose and are two continuous and independent random variables, where ~( , , , ) and ~( , , , ), then an expression for the stress-strength parameter can be obtained using the relation given by Using the pdf and the cdf of in the expression above, the strength-stress parameter, Ќ, can be obtained as

Order Statistics
Suppose that , , . . . , is a random sample of size from a continuous pdf, ( ). Let : < : < . . . : represent the corresponding order statistics. If , , . . . , is a random sample from distribution, it then follows from equations (11) and (12) that the pdf of the order statistic, say = : is given by And ; , ( + 1) is the Frechet with parameters > 0and ( + 1) > 0. Thus, we can define the distribution of the order statistics as a linear combination of the Frechet distribution.

Stochastic Ordering
In this section, we examine the stochastic and reliability properties of LFP distribution. Stochastic ordering has applications in many field of study such as survival analysis, insurance, actuarial and management sciences, finance, reliability and survival analysis Shaked and Shanthikumar (2007

Maximum Likelihood Estimation
The log-likelihood function ɷ ⁄ = ɷ ⁄ of the LPF distribution is given by The partial derivatives of the log-likelihood function with respect to the model parameters ( , , , )yield the score vector and are obtained as The equations (44), (45), (46) and (47) are non-normal equations which cannot be solved by setting the above partial derivatives to zero, therefore the parameters , , , must be found using the iterative methods. The maximum likelihood estimate of the parameters, denoted by ɷ is obtained by solving the nonlinear equation , , , = 0, using a numerical method such as Newton-Raphson procedure, Trapezoidal techniques etc. The Fisher information is given by І(ɷ) = , × = − , , , = 1,2,3,4 can be numerically obtained by using R or MATLLAB software. For the purpose of this study we make use of Adequacy model in R, the Fisher information matrix І(ɷ) can be approximated by For a given set of observations, the matrix given in equation (48) is obtained after convergence of the Newton-Raphson procedure in R or MATLAB.
The multivariate Normal distribution 0, ((ɷ ) ) , with mean vector 0 = (0,0,0,0) , can be used to construct the confidence interval and the confidence regions for the model parameters and for the hazard and the survival functions. The approximate 100(1 − )% two-sided confidence intervals for , , are given by: Respectively, where І (ɷ ), І (ɷ ), І (ɷ ) and І (ɷ ) are diagonal elements of І (ɷ ), and is the upper percentile of the distribution of the standard normal.

Application
In this section, we demonstrate the applicability and flexibility of the LFP distribution in modeling real life data using three life data sets. The method of maximum likelihood is used to estimate the model parameters; also, we carried out a Monte Carlo simulation for different parameter values coupled with different sample sizes.

Monte Carlo Simulation
A simulation study is carried out in order to test the performance of the MLEs for estimating LFP model parameters. We consider two different sets of parameters = 0.4, = 0.6, = 0.3, = 0.5 and = 0.5, = 1.6, = 0.5, = 0.5. For each parameter combination, we simulate data from model with different sample sizes = 50, = 100, = 150 = 200, taking from a population size = 1000. Table 4 list the Absolute bias (AB), standard error (SE) and the mean square error (MSE). According to the simulation result the mean square error decay to zero as the sample size increases as expected.

Conclusion
This work examined the flexibility, tractability and applicability of Lehmann Type II Frechet Poisson distribution. Some structural properties of the newly developed distribution are derived and population parameters are obtained using maximum likelihood estimation method. Simulation study and three real life data were used to illustrate the model usefulness in modeling life data. Among other competing models considered the Lehmann Type II Frechet Poisson distribution provides the best fit. We recommend that further studies should be carried by using different estimation methods such as moment method, least square method etc. and compare the performance of the estimation techniques.

Conflicts of interest
The authors want to declare that there is no conflict of interest during and after the preparation of the manuscript.

Acknowledgement
The authors wish to express their sincere appreciations to the anonymous referees for their suggestions, comments and contributions that help us to improve more on the work.