LENGTH BIASED BETA-PARETO DISTRIBUTION AND ITS STRUCTURAL PROPERTIES WITH APPLICATION

The concept of length biased distribution can be em ployed in development of proper models for lifetime data. Its method is adjusting the original probability de nsity function from real data and the expectation o f th se data. This adjustment can bring about to correct co nclusions of models. Therefore, we introduced the L ength Biased Beta-Pareto (LBBP) distribution, so-called a new generalized of Pareto distribution in this art icle. The distribution is more flexible and has some interest ing properties such as hazard rate, Renyi and Shann on entropies and other types of distribution. There ar e several sub-models include in the length biased P areto, arcsine, log-beta, exponential and beta-Pareto dist ributions. We apply maximum likelihood estimation t o estimate parameters of the distribution. We illustr ate the superiority of the LBBP distribution to Nor wegian fire claims data. The LBBP distribution seems to be th most appropriate model for this data set, sinc e t provides a significantly better fit than the length biased Pareto and the beta-Pareto distributions. W e hope that the LBBP distribution is an alternative distributio n can be used in lifetime data analysis and other f ields.


INTRODUCTION
The problem of determining proper model for interested information are one thing that important for data analysts. One major benefit of the weighted distribution theory provides a unifying approach for these problems was proposed by Rao (1965). It is ability of fitting skewed data that will not be properly fitted by existing distributions. Many authors including the concept of weighted distribution for example; Patil and Rao (1978) examine some general models leading to weighted distributions with weight functions not necessarily bounded by unity and studied length biased (size biased) sampling with applications to wildlife populations and human families. Characteristics of many length biased distributions, preservation stability results and comparisons for weighted and length biased distributions were presented; (Khattree, 1989;Oluyede and George, 2002).
Probability weighted moment inequalities and variability orderings for weighted and unweighted reliability measures and related functions were presented by Oluyede (2006). Also, stochastic comparisons and moment inequalities were given. More recently, Das and Roy (2011) developed the length biased form of the weighted Weibull distribution and discussed various properties of it. The result of this distribution suggested a good fitted to consecutive years data. Further, Ahmed et al. (2013) presented the class of size biased generalized gamma distribution and derived Shannon entropy which is the measure of the uncertainty in this distribution. Alkarni (2012) introduced lifetime class with decreasing failure rate by compounding truncated logarithmic distribution with any proper continuous lifetime distribution.
Statistical models and methods for lifetime data analysis are extensively used in many fields, including the biomedical sciences, engineering and management; (Lawless, 2011). The Pareto distribution is broadly used to modelling a diverse range of lifetime observable phenomena. Many transformations and generalization of the Pareto distribution have been proposed in order to get more flexible models; (Newman, 2005;Ali and Nadarajah, 2006;Manas, 2011;Nassar and Nada, 2013). Moreover, Johnson et al. (1995) and Nadarajah (2005) discussed other types of Pareto distribution more than the Probability Density Function (PDF) of Pareto density. Different methods may be used to introduce a shape parameter to the Pareto model.

JMSS
Recently, The Beta-Pareto (BP) distribution was introduced by Akinsete et al. (2008). It is a continuous distribution that is found to be unimodal and reduces to some existing distributions that are known in the literature. A random variable (r.v.) X has a BP distribution if its PDF with parameters α, β, θ and γ as follows Equation (1): Where: It is easy to prove the mean of PDF in (1) follows form Equation (2): The rth moments of X is given by Equation (3): The BP distribution can be applied to some phenomena events flood data sets, it provides a significantly better fit than the Pareto, Weibull and generalized Pareto distributions.
In this work, we propose a new distribution which is called the Length Biased Beta-Pareto (LBBP) distribution. The article is outlined as follows: In Section 2, we introduce the LBBP distribution and illustrate some sub-models of distribution and provide plots of the density function. We investigate some mathematical properties are the hazard rate of distribution and devoted to the discussion on the rth moments. Rényi and Shannon entropies are discussed in this section. Maximum Likelihood Estimation (MLE) is addressed in Section 3 and we provide application of the LBBP distribution to real data set in this section and discussion in Section 4. Finally, we offer some concluding remarks on the main results and their significance.

Length Biased Beta-Pareto Distribution
This section, we derived the shape of the PDF for the LBBP distribution and consider its some sub-models.

Definition 1:
Let X be a nonnegative r.v. with PDF f(x) and assuming that E(X)<∞. Then the PDF f L (x) for a length biased distribution of X can be obtained by:

Proof:
Substitute (1) and (2) in Definition 1, then PDF for LBBP r.v. can be obtained: . If X has PDF as (4), we shall write X~LBBP(α,β,θ,γ). The CDF of LBBP r.v., denoted as F L (x), can be rewrite as F L (x) = 1-F * L (x), then, F * L (x) for PDF in (4) is: By setting y = (t/θ) -γ , 0<y<1, the above integration becomes: We display the graphs of the LBBP distribution in some parameters values of α, β and γ in Fig. 1. Due to the fact that, γ and β are direct variation, but θ reverse variation with behavior kurtosis in these graphs.
We consider some sub-models of the LBBP distribution in the following corollaries.

Corollary 1:
If α = β = 1 it is easy to show that the LBBP distribution reduces to the Length Biased Pareto (LBP) distribution (Patil and Rao, 1978) with PDF:

Other Types of LBBP Distribution
Let X has the LBBP distribution. The PDF of LBBP type II, III and IV are, respectively, given by: ( ) Since, the PDF of LBBP distribution in (4) is originated (Eugene et al., 2002):

Hazard Rate
Hazard rate (or failure rate) and survival function are extensively apply in many fields. For instance; the Science Publications JMSS trivariate hazard rate function of trivariate liftime distribution was presented by Wahyudi et al. (2011).
By definition, the hazard rate of X is given by: where, f(x) and F(x) are PDF and CDF of X, respectively. Using (4) and (5), the hazard rate of the LBBP distribution may be expressed as: We display some graphs of hazard rate for the LBBP distribution in Fig. 2. It is noted that by setting α = β = 1 in h(x), we have the hazard rate of the LBP distribution.

Moments of LBBP Distribution
The result of this section gives the rth moments of the LBBP distribution. Some of the most important mathematical properties of distribution can be studied through rth moments.

Definition 2:
Let X be a nonnegative r.v. with the rth moments, E(X r )<∞. Then the E L (X r ) for length biased distribution of X can be obtained by: , r 1,2,3,... E X + = =

Rényi and Shannon Entropies
Statistical entropy is a probabilistic measure of ignorance about the outcome of a random experiment and is a measure of a reduction in that uncertainty. Entropy of X with PDF f(x) is a measure of variation of the uncertainty (Nadarajah and Kotz, 2006). (12):

Rényi entropy is defined by Equation
where, ξ>0 and ξ≠1 for the PDF of LBBP distribution in (4) is given by: The Rényi entropy can be written as Equation (13): Shannon entropy can be obtained as E[-log f(X)]. It is the special case of (12) for ξ→1, is given by: Limiting ξ→1 in (13) and using L'Hospital's rule, the Shannon entropy can be expanded as: where, Ψ(z) = Γ′(z)/Γ(z) is a digamma function.

Parameters Estimation
The estimation of parameters for the LBBP distribution via the MLE will be discussed. Let X 1 ,...,X n be a random sample from X~LBBP(α,β,θ,γ) the likelihood function is given by: With corresponding log-likelihood function Equation (14): The maximum likelihood estimate of θ is the firstorder statistic x (1) . The first order conditions for finding the optimal values of the parameters obtained by differentiating in (14) with respect to α, β and γ give rise to the following differential Equation (15 and 16): and Equation (17): The maximum likelihood estimates α , β and γ , respectively, are taken by solving iteratively (15)-(17) to zero. These differential equations are not in close form, a numerical method can be employed to obtain the expectations of them.

Application of LBBP Distribution
In this section, the LBBP distribution is fitted to a real data set, we consider the data set in the field of insurance which has received extensive attention in the actuarial literature. This data set is one among the twenty sets of Norwegian fire claims (in millions of Norwegian krones) is presented in Fernandez (2013). The parameter estimation for Norwegian fire claims data and K-S statistics are shown in Table 2. In this case, since the values of the K-S statistics are smaller for the LBBP distribution compared to those values of the LBP and BP distributions.

DISCUSSION
The LBBP distribution is consequence of length biased distribution method which is a new generalized of Pareto distribution. In this study, the LBBP distribution found that it provides a significantly better fit than the LBP and BP distributions which are some sub-models of the LBBP distribution. As well as the research of Mahmoudi (2011) found the beta generalized Pareto distribution using the exceedances of flood peaks of the Wheaton River near Carcross in Yukon Territory, Canada provides a better fit than sub-models of it. Furthermore, the result of this study consistent with the findings of Das and Roy (2011), the length biased Weibull distribution provided fit to data of June rainfall in Tezpur Assam, India better than its sub-models.

CONCLUSION
We proposed the Length Biased Beta-Pareto (LBBP) distribution. We found some well-known sub-models such as; LBP, arcsine, log-beta, exponential, BP distributions and LBBP type II, III and IV. Rényi and Shannon entropies and hazard rate are provided. We derive the rth moments and apply MLE to estimate parameters of the distribution. An application to a real data set shows that the fit of the LBBP distribution is best fit to the data with highest p-value. We hope the LBBP distribution may attract extensive applications in lifetime data analysis and other fields. The future research may consider in parameter estimation using Bayesian or other approaches. In addition, a new mixture between the LBBP and BP distributions will be developed.