Four Parameters Kumaraswamy Reciprocal Family Of Distributions

In this paper, kumaraswamy reciprocal family of distributions is introduced as a new continues model with some of approximation to other probabilistic models as reciprocal, beta, uniform, power function, exponential , negative exponential, weibull, rayleigh and pareto distribution. Some fundamental distributional properties, force of mortality, mills ratio, bowley skewness, moors kurtosis, reversed hazard function, integrated hazard function, mean residual life, probability weighted moments, bonferroni and lorenz curves, laplace-stieltjes transform of this new distribution with the maximum likelihood method of the parameter estimation are studied. Finally, four real data sets originally presented are used to illustrate the proposed estimators.


Introduction
The reciprocal distribution is a continuous probability distribution, gets its name because the density function is proportional to -1 x . It is a right-skewed distribution bounded between min (a) and max (b), and it is looks like a triangle (min, mode, max), if the min and the max are almost equal. Also, it is an example of an inverse distribution where the reciprocal of a random variable with a reciprocal distribution itself has a reciprocal distribution.
The probability density function (pdf) of the reciprocal distribution is The reciprocal distribution is of considerable importance in numerical analysis as a computer's arithmetic operations transform mantissas with initial arbitrary distributions to the reciprocal distribution as a limiting distribution. Also, it is widely used as an uninformed prior distribution in Bayesian inference for scale parameters, Hamming (1970) .
In the last few years, many new generated families are formed by adding additional one or more shape parameters to develop new models and increase its flexibility, such as Quasi Lindley distribution (Rama and Mishra, 2013), Kumaraswamy Weibull-generated family of distributions (Hassan, and Elgarhy, 2016), Kumaraswamy sushila distribution (Shawki and Elgarhy, 2017), L-Quadratic distribution (Salma, 2018), TAS distribution (Salma and Arwa, 2018), and others. The Kumaraswamy distribution is one of them, it has the following density function with two additional shape parameters and     , It is well known in general that, a generalized model is having more flexibility than the base model and it is favored by data analysts in analyzing data. Therefore, the main objective of this paper to propose a new continuous kumaraswamy reciprocal family and explain its flexible behavior, and how it is fitted for various continuous data sets in different applied fields. Section (2) identifies the four parameters kumaraswamy reciprocal family with some of its approximation to other probabilistic models and studding its reliability measurements behavior. Section (3) contains the properties of the kumaraswamy reciprocal family and the Maximum Likelihood (ML) method of the parameter estimation. Section (4) presents the practical applications to illustrate the proposed model.

The Four Parameters Kumaraswamy Reciprocal Family
The new distribution will abbreviate by  under different values of the shape parameters ( , ) and the scale parameters ( , ). It is noted that the PDF of the new distribution takes many shapes, and the plot of the CDF indicates increasing cumulative distribution function.

Approximation to other probabilistic models
Some of approximation to other probabilistic models are presented in Table.1 as some special cases of

Maximum likelihood estimation
The reliability function (RF) of the four parameters The force of mortality (FM) of the four parameters The reversed hazard function (RHF) of the four parameters The integrated hazard function (IHF) of the four parameters The mills ratio (MR) of the four parameters

Quantile Function
The quantile function

Median and Random Numbers
From Eq.(3), if u equal to 0.5 then the median of can be generated.

Bowley Skewness and Moors Kurtosis
The Bowley Skewness (BS), and Moors Kurtosis (MK) of the four parameters

Mean Residual Life
Theorem 1: The mean residual life of a r.v X with pdf (1), is given by denotes the complementary incomplete beta function which can be evaluated in MathCad 15. Using Eq.(4), the mean residual life can be expressed as

Moments about Origin
, then r th moments of X is given by By using the partial derivative which are the mean and the variance of the base reciprocal distribution , respectively.

Incomplete Moments
, then r th incomplete moments of X is given by Proof: r th incomplete moments is defined as denotes the complementary incomplete beta function, which can be evaluated in MathCad 15. Using Eq. (7), the incomplete moments can be expressed as

Bonferroni and Lorenz Curves
Suppose that is a count response variable that follows the EWGD in Equation (2.2) and is associated with a set of predictors. We wish to fit the response variable by using the predictors. Suppose we have a -1 row vector of predictors ( 1, , , , ) , and using Eq. (7) ( ) Proof: The moment generating function of a r.v X is defined as

Estimation of Parameters
Let n X , ... , 2 X , 1 X be a random sample of size n, distributed as Then the log likelihood function (l) is given by . By taking the partial derivatives of Eq.8 with respect to the four unknown parameters ( ) , the four normal equations are obtained as The solution of the above normal equations cannot be obtained in closed form, so MathCad 15 package can be used to get the MLE of the unknown parameters. Therefore, the elements of the Fisher information matrix (FI) for the MLE can be obtained as the expectations of the negative of the second partial derivatives, and the asymptotic variance-covariance matrix ( )  V for the MLE is defined as the inverse of the Fisher's information matrix i.e., FI
Some measures of goodness of fit are used, including Kolomgrof-Smirnof test (K_S), Akaike information criterion (AIC), consistent Akaike information criterion (CAIC), Bayesian information criterion (BIC), Anderson Darling (AD) and Cramrvon Mises (CVM) statistics for assessing that the data sets follow the   Table.3. Also, moments about origin, incomplete moments, probability weighted moments, Laplace-Stieltjes transform and mean residual life at the mission time t of the 4-kumaraswamy reciprocal family are calculated and represented in Table.4. In addition, mean, median, 1 st quartile, 3 rd quartile, bowley skewness (BS), moors kurtosis (MK) and mean residual life (MRL) of the four kumaraswamy reciprocal family are calculated and represented in Table.5.   and estimate the true parameters (a,b,α,β) well with relatively small MSE and reduce towards zero, with short confidence intervals. Moreover, the mean residual life decreases when the mission time ( ) increases as shown from the results in Table.3. In general, the proposed model and the asymptotic approximation work well under the situation and provides a better fit for the real data. So, the ) , b, KRF(a,   very useful in many fields and has many benefits especially in practical life. Therefore, it can be applied to several realistic data and chosen as a suitable model for lifetime data.

Conclusions
This paper presented a new distribution named the four parameters kumaraswamy reciprocal family abbreviate by . The maximum likelihood method is applied for estimating the model parameters. Nine special models namely, reciprocal, beta, uniform, power function, exponential, negative exponential, weibull, rayleigh and pareto distribution are provided. Further, the derived properties of the ) , b, KRF(a,   are valid to the selected model, such as, quantile function, median, random numbers, bowley skewness, moors kurtosis, reliability function, force of mortality, reversed hazard function, integrated hazard function, mills ratio, mean residual life, moments about origin, incomplete moments, probability weighted moments, bonferroni and lorenz curves, moment generating function, characteristic function, laplace-stieltjes transform with estimation of parameters. It also explained the behavior of the reliability function, force of mortality, mills ratio, reversed hazard function, integrated hazard function, bonferroni and lorenz curves. Four real life applications have also presented for explaining the better fit and the benefit of the observed model in many practical life. The better fit and the benefit of the new model is illustrated by four real data sets and the results of the applications nicely exhibit the fact that the ) , b, KRF(a,   provides a better fit than others sub models in many. And it performs better and a suitable in many situations and can be applied to several realistic data.