A New Generalization of Quadratic Hazard Rate Distribution

For the first time, a five-parameter distribution, called the kumaraswamy quadratic hazard rate distribution is defined and studied. The new distribution contains as special models some well-known distributions discussed in lifetime literature, such as the Linear failure rate, Exponential and Rayleigh distributions, among several others. We obtain the moments, moment generating and quantile functions. We discuss the method of maximum likelihood to estimate the model parameters and determine the observed information matrix. A real data sets illustrate the importance and flexibility of the proposed models.


Introduction
The quadratic hazard rate distribution () QHR distribution was introduced by Bain (1974).This distribution generalizes several well known distributions.Among these distributions are the linear fialure (hazard) rate, exponential and Rayleigh distributions.Also, the QHRD may have an increasing (decreasing) hazard function or a bathtub shaped hazard function or an upsidedown bathtub shaped hazard function.This property enables this distribution to be used in many applications in several areas, such as reliability, life testing, survival analysis and others.

A random variable
X is said to have the quadratic hazard rate distribution () QHRD with three parameters ,,  and  , if it has the cumulative distribution function    This restriction on the parameter space is made to be insure that the hazard function with the following form is positive, see Bain (1974)  The Kumaraswamy () w K distribution is not very common among statisticians and has been little explored in the literature.The cdf and pdf of the Kumaraswamy distribution are defined by where >0 a and >0 b are shape parameters, and the probability density function which can be unimodal, increasing, decreasing or constant, depending on the parameter values.It does not seem to be very familiar to statisticians and has not been investigated systematically in much detail before, nor has its relative interchangeability with the beta distribution been widely appreciated.However, in a very recent paper, Jones (2009) explored the background and genesis of this distribution and, more importantly, made clear some similarities and differences between the beta and w K distributions.However, the beta distribution has the following advantages over the w K distribution: simpler formulae for moments and moment generating function (mgf), a one-parameter sub-family of symmetric distributions, simpler moment estimation and more ways of generating the distribution by means of physical processes.
In this note, we combine the works of Kumaraswamy (1980) and Cordeiro and Castro (2011) to derive some mathematical properties of a new model, called the Kumaraswamy quadratic hazard rate () KQHR distribution, which stems from the following general construction: if G denotes the baseline cumulative function of a random variable, then a generalized class of distributions can be defined by where >0 a and >0 b are two additional shape parameters.The Kw G  distribution can be used quite effectively even if the data are censored.Correspondingly, its density function is distributions has a very simple form The density family (1.6) has many of the same properties of the class of beta-G distributions (see Eugene et al. (2002)), but has some advantages in terms of tractability, since it does not involve any special function such as the beta function.A physical interpretation of the w K  G distribution given by (1.5) and (1.6) (for a and b positive integers) is as follows.Suppose a system is made of b independent components and that each component is made up of a independent subcomponents.Suppose the system fails if any of the b components fails and that each component fails if all of the a subcomponents fail.Let 12 , ,..., j j ja X X X denote the life times of the subcomponents with in the th j component, = 1,..., j b with common (cdf) G .Let j X denote the lifetime of the th j component, = 1,..., j b and let X denote the lifetime of the entire system.Then the (cdf) of X is given by So, it follows that the w K  G distribution given by (1.5) and (1.6) is precisely the time to failure distribution of the entire system.
The rest of the article is organized as follows.In Section 2, we define the cumulative, density and hazard functions of the KQHR distribution and some special cases.In Section 3. includes th r moment , moment generating function .The distribution of the order statistics are proposed in Section4.Least squares and weighted least squares estimators introduced in Section 5. Finally, maximum likelihood estimation is performed in Section 6.

Kumaraswamy Quadratic Hazard Rate Distribution
In this section we studied the kumaraswamy quadratic hazard rate () KQHR distribution and the sub-models of this distribution.Now using (1.1) and (1.2) in (1.5) we have the cdf of Kumaraswamy quadratic hazard rate distribution The corresponding probability density function (pdf) of the kumaraswamy quadratic hazard rate distribution is given by ) ) Graph of (2.4) for various values of a,b,α,β and θ is given in appendix-II and respectively.It is important to note that the units for () KQHR hx is the probability of failure per unit of time, distance or cycles.These failure rates are defined with different choices of parameters.The cumulative hazard function of the Kumaraswamy quadratic hazard rate distribution is denoted by ()

KQHR
Hx and is defined as It is important to note that the units for ()

KQHR
Hx is the cumulative probability of failure per unit of time, distance or cycles.we can show that .For all choice of parameters the distribution has the decreasing patterns of cumulative instantaneous failure rates.

Statistical Properties
In this section we study the statistical properties of the kumaraswamy quadratic hazard rate distribution.Specifically quantile, moments and moment generating function .Moments are necessary and important in any statistical analysis, especially in applications.It can be used to study the most important features and characteristics of a distribution (e.g., tendency, dispersion, skewness and kurtosis).

Quantile and Random Number Generation
The quantile q x of the KQHR ( , , , , ) ab    is real solution of the following equation The random number generation as x of the KQHR ( , , , , ) ab    is defined by the following relation = 0,where (0,1)

Moments
The following theorems give the    ,then the th r moment of X is given by the following ( 1) then the using the binomial of )) 33 )) but the expansion of Based on Theorem (3.1) the measures of variation, skewness and kurtosis of the KQHR ( , , , , , ) ab     distribution can be obtained according to the following relation   

Moment Generating Function
In this subsection we derived the moment generating function (mgf) of kumaraswamy quadratic hazard rate distribution.

Order Statistics
Moments of order statistics play an important role in quality control testing and reliability, where a practitioner needs to predict the failure of future items based on the times of a few early failures.These predictors are often based on moments of order statistics.We now derive an explicit expression for the density function of the ni    out  of  n system which consists of n independent and identically components.Then the pdf of ( : ) in X ,1 in  is given by ) )) 33 )) 33 ( ) 1 and the pdf of the smallest order statistic (1) X is given by 1 23 ( 21:

Least Squares and Weighted Least Squares Estimators
In this section we provide the regression based method estimators of the unknown parameters of the kumaraswamy quadratic hazard rate distribution, which was originally suggested by Swain, Venkatraman and Wilson (1988) to estimate the parameters of beta distributions.It can be used some other cases also.Suppose ( 1) and ( ), ( ) = ; for < , ( 1) ( 2) see Johnson, Kotz and Balakrishnan (1995).Using the expectations and the variances, two variants of the least squares methods can be used.( , , , , ) = ( ) , 1

Method 1 (Least Squares Estimators). Obtain the estimators by minimizing
with respect to the unknown parameters.Therefore in case of KQHR distribution the least squares estimators of , , ,    a and b , say ,, To minimize equation (5.2) with respect to ,  ,, a  and b and, we differentiate with respect to these parameters, which leads to the following equations. )) )) )) The estimates of the parameters are obtained by equating the above equations to zero.Although the proposed estimators cannot be expressed in closed form, they can be obtained through the use of an appropriate numerical solution algorithm.Method 2 (Weighted Least Squares Estimators).The weighted least squares estimators can be obtained by minimizing ()


with respect to the unknown parameters only.

Maximum Likelihood Estimators
In this section we consider the maximum likelihood estimators (MLE's) of KQHR distribution.Let = ( , , , , ) ,    then the log likelihood function can be written as Differentiating L with respect to each parameter ,,    , a and b and setting the result equals to zero, we obtain maximum likelihood estimates.The partial derivatives of L with respect to each parameter or the score function is given by )) 33 1 ( 1) = 0, ) 3 ) 3 ) 3 31 )) 33 1 ( 1) = 0, ) 3 3 1 1 By solving this nonlinear system of equations (6.2) -(6.6), these solutions will yield the ML estimators for  , ,  , a and b . For the five parameters kumaraswamy quadratic hazard rate distribution KQHR ( , , , , , ) a b x    pdf all the second order derivatives exist.Thus we have the inverse dispersion matrix is given by ,.
By solving this inverse dispersion matrix these solutions will yield asymptotic variance and covariances of these ML estimators for ,

2 (
from (3.7) into (3.6)we have 2,..., in denotes the ordered sample.The proposed method uses the distribution of () () i GY .For a sample of size n , we have and estimate the parameters ,,    , a and b of kumaraswamy quadratic hazard rate distribution, let 1 ,..., x n x be a random sample of size n from KQHR ( ; , , , , ) x a b

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4 New Generalization of Quadratic Hazard Rate DistributionPak.j.stat.oper.res.Vol.IX No.4 Breast Cancer Survival" is used to obtain fit The quadratic hazard rate distribution () QHR distribution.The ML estimates of the parameters of The quadratic hazard rate distribution () QHR distribution are given in the table 2. We have also given the AIC and negative of log likelihood function to decide about the suitability of the model.