NEW LIFE-TIME CONTINUOUS PROBABILITY DISTRIBUTION WITH FLEXIBLE FAILURE RATE FUNCTION

Recently, the area of distribution theory has been receiving increased interest in generating or defining new classes of continuous probability distributions by way of extending the existing distributions. The new generated distribution is expected to be more flexible and have wider acceptability in modeling and predicting real world data sets. In this research, we proposed and study an extension of Ikum distribution using Zubai G-Family (2018) of distribution called Z-Ikum distribution. Expression of some basic structural properties of the new distribution such as cdf, pdf, quantile functions, moments, moment generating function, characteristics function and order statistics was derived. Survival function, hazard rate function, commutative hazard rate function and reversed hazard rate function was also discussed. Plots of the hazard rate function showincrease, decrease and bathtub shapes.Estimation of the proposed distribution parameters was carried out using MLE method. Performance of the parameter estimation was also evaluated via simulation studies. Result of the simulation studies indicates that our estimator is consistent. Three life data sets were used to evaluate the performance of our proposed distribution over some existing distributions. Result of the empherical study revealed that our proposed distribution performwell in modeling real life data than the competing distributions.

different types of generalized distributions developed and applied to various phenomena (Olalekan et al., 2021).Some well-known methods in the early days for generating univariate continuous distributions include methods based on differential equations developed by Pearson (1895), methods of translation developed by Johnson (1949), and the methods based on quantile functions developed by Tukey (1960).The interest in developing new methods for generating new or more flexible distributions continues to be active in the modern decades.Lee et al. (2013) indicated that the majority of methods developed after 1980s are the methods of 'combination' for the reason that these new methods are based on the idea of combining two existing distributions or by adding additional parameters to an existing distribution to generate a new family of distributions.As a result,many new families of distributions have been developed and studied by researchers.An obvious reason for generalizing a standard distribution is because the generalized form provides larger flexibility in modeling real data.The Ikum distribution which is the inverse form of the Kumaraswamy distribution is obtained using transformation  =  −1 by Abd AL Fattah et al, (2017) has been used in modeling lifetime data.However, in many applied instances, the IKum distribution fails to give adequate fits to lifetime data such as the life cycle of machines, human mortality and biomedical data which show non-monotone failure rates.

Zubair G Family of distribution
A family of life distributions, called the Zubair-G family was introduced by Zubair (2018).The benefit of using this family is that its cdf has a closed form solution and capable of data modeling with monotonic and non-monotonic failure rates.TheCDF and PDF of the new family defined by Zubair (2018) for random Variable  is given in (3) and ( 4) respectively.

𝐹(𝑥
Where  is vector of the baseline distribution parameter,  is the parameter of Zubair G-family (; ) and (; )are pdf and cdf of the baseline distribution respectively.

Statistical Properties of Z-Ikum distribution
This section studies the statistical properties of the proposed distributions such as the quintile functions,order statistics and moments.Also reliability analysis of the proposed distribution is discussed in details.

Quantile Function
The quantile function Z-Ikum distribution is obtain by inverting (5) as given in ( 12) 325 To obtained, the first quartile, the median, and the third quartile, we replace u with 0.25, 0.5 and 0.75 in (12) respectively. .

Moments
The r th non-central moment of the IKum random variable is derived.By definition 8), we have Using power series where

Parameters Estimation and Simulation Studies of Z-Ikum Distribution
In this section, estimators are developed for estimating the parameters of Z-Ikum distribution using the well known method of maximum likelihood estimate (MLE).

Simulation Studies
The performance of the maximum likelihood estimates for theZ-Ikum distribution parameters was evaluated using Monte Carlo simulation for a three parameter combinations.Different sample sizes (n = 25, 50, and 75)and some selected parameter values ( = 0.10,  = 1.50,  = 0.6)were used to perform the simulation.Result of the simulation is presented in the table below.

Model Comparison and Selection Criteria
In this case, we will consider the generally well known criteria such as Akaike Information Critareion (AIC),the Bayesian Information Criterion (BIC), the Consistant Akaike Information Cretarion (CAIC) and Hannan-Quinn Information Criterion (HQIC) to compare Z-Ikum distributionwith some existing distributions using three sets of real life data.

RESULTS AND DISCUSSION
A three parameter distribution called Zubair-Ikum (Z-Ikum) distribution is proposed in this research.The proposed distribution is an extension of Ikum distribution using the Zubair G-Family (2018) of continuous probability distribution.Some Structural properties such as Quantile functions, moments, moment generating functions, characteristic functions, order statistics of the new distributions was derived.Survival function, hazard function, reversed hazard rate function and a cumulative hazard rate function was also obtained.From the hazard rate plot its evident that Z-Ikkum distribution has increase, decrease,

Manu et al., FJS
proof.Figures1 and 2below displayed the plots of the pdf and cdf of the Z-Ikum distributionfor some selected parameter values respectively.

Figure 1 :
Figure 1: Plot of of Z-Ikum PDF

Figure 3 :
Figure 3: Plot of of Z-Ikum hr Figure 4: Plot of of Z-Ikum hr The plot of the hazard rate function of Z-Ikum distribution for specific sets of parameter values exhibits Bath-Tub shape, right skewed, increasing, and decreasingshape.Reverse Hazard Rate Function ofZ-Ikum distribution Using (35) the Reverses hazard rate functions of Z-Ikum distribution is obtained as in (36) () = () () .663