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MISEIC 2018

Surabaya, July 21, 2018

 

 

Parameter Estimation for the Lomax Distribution Using the E-Bayesian Method

 

 

Aldila Fitrilia*¹, Ida Fithriani² and Siti Nurrohmah³

 

^{1, 2} Department of Mathematics, Faculty of Mathematics and Natural Sciences Universitas Indonesia, INDONESIA.

(E-mail: aldilafitrilia@gmail.com, ida.fithriani@gmail.com)

³ Department of Mathematics, Faculty of Mathematics and Natural Sciences Universitas Indonesia, INDONESIA.

(E-mail: snurrohmah@sci.ui.ac.id)

 

ABSTRACT

Statistics is the study of how to collect, process, analyze, present data and make conclusions based on the analysis of the data. One of the analyzes used in the branch of statistics is the analysis of life time data. Life time data analysis is one of the most useful statistical techniques for analyzing testing on the survival or reliability of a component or individual. Life time data analysis is called survival analysis. Survival analysis is a statistical analysis used to investigate the life time of an object or an individual in a special case. Survival analysis has been applied in many subjects, that is in biology, medicine, industry, and others. In survival analysis, survival data is needed which includes the survival time and status of the survival time of the object under study.

Survival data is data collected from a collection of duration of time between the point of time when an object entered the study to the point where the object is experiencing an event (Borovkova, 2002). Survival data can be either censored or uncensored data. To obtain uncensored data takes a long time and a large cost to do research until all objects fail so that uncensored data is rarely used in research with large number of objects. Therefore, to saving time and cost, censored data is used. Censored data is data that some of the information is incomplete, which is caused by various reasons such as the object of observation to dropout or the observed events did not occur during the time of the study. Censored data can be either right, left, or interval censored data. The right censored data can be either right censored data type I or right censored data type II. The data is said to be the right censored data type I if the observation is stopped until the specified time. The data is said to be right censored data type II if the observation is stopped until some object (r object) of all the studied objects (n) has failed so there are still some  objects is survived (n-r object) (Lee & Wang, 2003). In this study, we will find the parameter estimator from Lomax distribution on one of censored data, that is right censored data type II.

Lomax distribution is also called Pareto type II distribution. The Lomax distribution was first introduced by K. S. Lomax (1954). Lomax distribution is a probability distribution used in business, economics, and actuarial sciences. The Lomax distribution has two parameters, that is the shape parameter () and the scale parameter (). Probability density function of Lomax distribution is . Since the parameters of  and  are not known, it is necessary to estimate the parameters. Parameter estimation can be point estimation and interval estimation. Methods for finding point estimates are the method of moments, least square methods, maximum likelihood method and Bayes method. The Bayes method is a method that uses or incorporates subjective (past) knowledge of the parameters to be assessed with information obtained from the sample data. The preceding information is also called the prior information obtained from the distribution of these parameters. Parameters in the Bayes method are considered random variables. This causes the parameter to have a certain distribution. The distribution of these parameters is called the prior distribution. The information from the data is summarized in the likelihood function. Combine prior information and information from the data will produce posterior information (Walpole, 1993). After obtaining the posterior distribution, then searched the parameter estimator of the Lomax distribution on the right censored data type II.

In this study, the parameter will be estimate is the shape parameter with the assumption of scale parameters () has been known. The parameter estimation method used in this study is E-Bayesian estimation method. E-Bayesian estimation method is used to estimate failure rate, otherwise this method is suitable for censored data with small sample sizes and high reliability (Han, 2008). Estimated parameters using the Bayesian approach, which to obtain E-bayesian estimate of  (i.e. expectation of the Bayes estimate of ) is obtained by calculating the mean of the Bayes estimator. This study will use prior Gamma as conjugate prior distribution from Lomax distribution and Loss function will be used in this study is balanced squared error loss function (BSELF). Thus, the main purpose of this study is to find the parameter estimator of the Lomax distribution on the right censored data type II using the E-Bayesian method.

The final result of this study, we get the probability density function from Lomax distribution on the right censored data type II and the parameter estimator from Lomax distribution on the right censored data type II using the E-Bayesian method. Beside that, we will run simulation to get deep analysis about characteristic of parameter estimator of Lomax distribution, but the result is still in progress.

Keywords: conjugate prior, E-Bayesian method, Lomax distribution, right censored survival data type II, survival analysis

Acknowledgment: We would like to thank the DRPM Universitas Indonesia for supporting in our research with Hibah PITTA UI 2018.

conjugate prior; E-Bayesian method; Lomax distribution; right censored survival data type II; survival analysis