Estimation of the Bivariate Generalized. Lomax Distribution Parameters. Based on Censored Samples

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 9, 2014, no. 6, HIKARI Ltd, Estimation of the Bivariate Generalized Lomax Distribution Parameters Based on Censored Samples A.F. Attia*, S. A. Shaban* and Y. M. Amer** * Institute of Statistical Studies and Research Cairo University, Giza, Egypt ** Institute of Statistical Studies and Research Cairo University, Giza, Egypt Copyright 2014 A.F. Attia, S. A. Shaban and Y. M. Amer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Recently it is observed that the generalized exponential distribution can be used quite effectively to analyze lifetime data in one dimension. This paper extends Marshall and Olkin's bivariate exponential model to the Generalized Bivariate Lomax (BGL) Distribution. The cumulative distribution function, the probability density function and the Marginal distribution of the BGL distribution are reached. The maximum likelihood estimation procedure is derived for the estimation of the BGL parameters based on Censored Samples when all parameters are unknown and also obtain the observed Fisher information matrix. A special case of the distribution of the BGL distribution is reached in a closed form when one of the parameters is known. Simulation study was analyzed, and a numerical comparison is made between the proposed estimation procedure of the BGL distribution and Block-Basu estimation technique. Keywords: Generalized Lomax distribution; maximum likelihood estimators; Fisher information matrix; Marshall - Olkin bivariate distribution

2 258 A.F. Attia, S. A. Shaban and Y. M. Amer 1- Introduction The Lomax (1954), or Pareto II, distribution has been quite widely applied in a variety of contexts. Although introduced originally for modelling business failure data, the Lomax distribution has been used for reliability modelling and life testing (e.g., Hassan and Al-Ghamdi, 2009), and applied to income and wealth distribution data (Harris, 1968, Atkinson and Harrison, 1978), firm size (Corbelini et al., 2007) and queuing problems. It has also found application in the biological sciences and even for modelling the distribution of the sizes of computer files on servers (Holland et al., 2006). Some authors, such as Bryson (1974), have suggested the use of this distribution as an alternative to the exponential distribution when the data are heavy-tailed. Let X be a random variable with the following CDF as follows; [ ] The distribution of this form is said to be a generalized Lomax distribution with parameters and will be denoted by GL. The PDF and the hazard function of GL will be [ ] The main objective of this paper is to extend Marshall and Olkin's bivariate exponential model to the Generalized Bivariate Lomax (BGL) Distribution. We mainly compute the cumulative distribution function, the probability density function and the Marginal distribution of the BGL distribution. Derive the maximum likelihood estimation procedure for the estimation of the BGL parameters based on Censored Samples when all parameters are unknown and also obtain the observed Fisher information matrix. This paper is organized as follows; in Section 2 we present the bivariate generalized Lomax distribution, the cumulative, the probability and Marginal density functions of the proposed Bivariate Generalized Lomax (BGL) distribution are derived; in Section 3 we derive the maximum likelihood estimation procedure of the unknown parameters Based on Censored Samples; simulation results and conclusions are given in Section Bivariate Generalized Lomax Distribution In this Section, we reach the joint cdf, pdf and the conditional pdf of the Generalized Lomax Distribution.

3 Estimation of the bivariate generalized Lomax distribution parameters 259 Let Ui follows a univariate Generalized Lomax distribution and all three distributions are mutually independent. Define and then, the bivariate vector has a bivariate generalized Lomax distribution. Marshall & Olkin (1967), gave the "Stress Model" as an example of how this bivariate vector occur in practice; suppose that a system has two components, each is subjected to individual independent stress say U 1 and U 2 and the system has an overall stress U 3 transmitted to both the components equally, and independently. The observed stress for each of the two components are and. Gupta and Kundu (2007) gave the example of the "Maintenance Model when two components are maintained independently and also they have to pass an overall maintenance; when the lifetime of the individual component is increased by U i amount due to the independent maintenance or by U 3 due to the overall maintenance; then the increased lifetimes of the two component are and respectively. The following results give the joint cdf, joint pdf and the conditional pdf of the Bivariate Generalized Lomax (BGL) distribution. Theorem 2.1: If, then the joint CDF of for,, is [ ] [ ] [ ( ) ] The proof stems from Equation (1) above and the independence property of the two variables. Corollary 2.1: The joint CDF of the can also be written as ( ) Theorem 2.2: If, then the joint CDF of for,, is { (5) [ ] [ ] [ ]

4 260 A.F. Attia, S. A. Shaban and Y. M. Amer [ ] [ ] [ ] [ ] The BGL distribution [Equation 5] has both an absolute continuous part and a singular part, similar to Marshall and Olkin's bivariate exponential model. The function may be considered to be a density function for the BGL distribution if it is understood that the first two terms are densities with respect to two-dimensional Lebesgue measure and the third term is a density function with respect to one dimensional Lebesgue measure (Bemis, Bain and Higgins, 1972). It is well known that although in one dimension the practical use of a distribution with this property is usually pathological, but they do arise quite naturally in higher dimension. In case of BGL distribution, the presence of a singular part means that if X and Y are BGL distribution, then X = Y has a positive probability. In many practical situations it may happen that X and Y both are continuous random variables, but X = Y has a positive probability (Marshall and Olkin, 1967). The following theorem will provide explicitly the absolute continuous part and the singular part of the BGL distribution function. Theorem 2.3: If, then [ ] and [ ] [ ] [ ] [ ] here and are the singular and the absolute continuous parts respectively.

5 Estimation of the bivariate generalized Lomax distribution parameters 261 Proof To find from, we compute { which may be obtained as and { Once ρ and are determined, can be obtained by subtraction. Alternatively, probabilistic arguments are also can be provided as follows: Let { } { } Then Therefore, ( ) ( ) ( ) Moreover for z as defined before, ( ) [ ] And accordingly ( ), can be obtained by subtraction. Clearly, [ ] is the singular part as its mixed second partial derivative is zero when, and ( ) is the absolute continuous part as its mixed partial derivative is a density function. Corollary 2.1: The joint pdf of X and Y can be written as follows for ;

6 262 A.F. Attia, S. A. Shaban and Y. M. Amer where and { [ ] Clearly, here and are the absolute continuous part and singular part respectively. The following theorem gives the marginal probability density functions of X and Y. Theorem 2.4: If, the marginal pdf of is given by [ ] [ ] Proof First we derive using the fact that Using the expressions of,, and [Equation 5 ] given in Theorem (2.2), we can get of the form (9). Proceeding similarly, we can derive as given in (9), which completes the proof of the theorem. 3- Maximum Likelihood Estimation of the parameters for BGL distribution The univariate random censoring scheme given by Hanagal [1992] is used for estimating the bivariate life time distribution, which takes into account that individuals do not enter at the same time the study and a withdrawal of an individual will censor both life times of the components which in the sequel will be called implants, because the model was developed and applied in the framework of teeth implants for upper and lower jaws.

7 Estimation of the bivariate generalized Lomax distribution parameters 263 Suppose that there are n independent pairs of components, for example, paired kidneys, lungs, eyes, ears in an individual under study and i -th pair of the components have life times and censoring time. The life times associated with i -th pair of the components is given by { For i = 1, 2,..., n There are six different types of events which might occur with respect to,. These are the following: 1. Type 1: 2. Type 2: 3. Type 3: 4. Type 4: 5. Type 5: 6. Type 6: The likelihood of the sample of size n after discarding factors which do not contain any of the parameters of interest is given as follows Where ( ) ( ) ( ) ( )( )( ) ( ) = [ ] [ ] [ ] [ ] [ ] [ ] [ ]

8 264 A.F. Attia, S. A. Shaban and Y. M. Amer ( ) [] [ ] ( [ ] ) ( ) [] [ ] ( [ ] ) [ ] where, be the numbers of realizations falling in the range corresponding to,,, and respectively.,, and are density functions with respect to the Lebesque measure on R 2, while,, and are density functions with respect to the Lebesque measure on R. Let the range of variability corresponding to be denoted by, j=1, 2,., 6.and are disjoint sets and letting,, and then log-likelihood function can be written as follows: ( ) ( ) ( ) where,, and ( ) ( )

9 Estimation of the bivariate generalized Lomax distribution parameters 265 [ ] The following likelihood equations are obtained by equating the partial derivatives of with respect to and θ to zero: Then ( ( ) ) ( ( ) ) ( ( ) ) ( ( ) ) ( ( ) ) The equations 14,15 and 16 are solved for the maximum likelihood estimates ( ) using the Newton-Raphson procedure. The observed Fisher information matrix is given by { } where the second order partial derivatives of the log-likelihood function are given by: ( ( ( )) ( ) ) ( ( ( )) ( ) ) ( ( ( )) ( ) ) ( ( ( )) ( ) ) ( ( ( )) ( ) ) ( ( ( )) ( ) ) ( ( ( )) ) ( ) where ( ) is the ML-estimator of the parameter.

10 266 A.F. Attia, S. A. Shaban and Y. M. Amer The quantity ( ) has an asymptotic multivariate normal distribution with mean vector zero and observed variance-covariance matrix. 4- Simulation Study and Conclusions The sample data are generated based on following algorithms: Step 1: Generate using the Generalized Lomax distribution white parameters. Step 2: Take and and, therefore follows a bivariate Generalized Lomax distribution of Marshall- Olkin type. Step 3: Generate using the exponential distribution with failure rate, where are the censoring times. The estimates are obtained by taking the mean of the 1000 maximum likelihood estimates and the mean of the 1000 standard deviations from the 1000 samples of size n = 20, 35,50 and 100. The estimates of the standard deviation of the maximum likelihood estimates of are obtained by taking square root of the diagonal elements of the inverse of the observed Fisher information matrix. The estimates are close to the true parameter values and the standard errors decrease as the sample size increases. The estimates for both methods are obtained by taking the mean of the 1000 maximum likelihood estimates and the mean of the 1000 standard deviations from the 1000 samples of size n = 20, 35,50 and 100. The estimates of the standard deviation of the maximum likelihood estimates of are obtained by taking square root of the diagonal elements of the inverse of the observed Fisher information matrix. Parameters n n n

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