Estimation of 2-and 3-parameter Burr Type XII distributions using EM algorithm

The Burr Type XII distribution is one of the systems of continuous distributions and is widely known because the distribution includes the characteristics of various well known distributions such as Weibull and gamma distributions. Maximum likelihood estimation (MLE) has been a common method in estimating model parameters. An alternative method that is the expectation-maximization (EM) algorithm is presented in this paper to estimate the twoand three-parameter Burr Type XII distributions in the presence of complete and censored data. Furthermore, simulation study is conducted to compare the efficiency and accuracy of MLE and EM algorithm approaches. The result indicates that EM algorithm is more efficient and accurate than those estimates obtained via MLE approach.


INTRODUCTION
The word 'Burr' was introduced by [1] in 1942 when a few forms of cumulative distribution function were suggested to fit the data.Burr Type XII was used in many fields because of the potentiality in practical situations.In his guide to the Dagum distribution (2007), [2] stated that in economics, the Burr Type XII distribution is known as the Singh-Maddala distribution.Burr Type XII distribution has at least two unknown parameter.[3] derived the probability density function of a six-parameter generalized Burr Type XII distribution and obtained cumulative distribution function meanwhile [4] introduced properties of seven parameters Burr Type XII distribution.
In statistics, estimation process is very important to find the approximate value of unknown parameters.Several methods have been used to estimate the parameters of the Burr Type XII distribution such as Maximum Likelihood Estimation (MLE), Least Squares Estimation (LSE) and Bayesian Estimation.The parameter estimation process involves the presence of complete and censored data.
Nowadays, researchers used censored data to estimate the parameter.Different mechanisms can lead to different type of censored data such as right-censored, leftcensored and randomly censored.The usage of censored data gives the possibility to compute the estimation method to fit a model to censored data.This paper aims to estimate the parametersc and k for 2-parameter and c, k and s for 3-parameter of Burr Type XII distribution with complete and censored data using two methods which include MLE and EM algorithm approaches.Then, the estimated parameters from both methods were compared based on bias and mean square error (MSE).

2-Parameter Burr Type XII distribution
The probability density function (pdf) of the standard Burr Type XII distribution for 2-parameter is written in the form of and cumulative distribution function (cdf) for 0  t with both c and k are shape parameters.

Maximum likelihood estimation (MLE)
MLE is the most popular method of parameter estimation.The likelihood function of the censored data is given by There are r failures at times With f(t) and F(t) given by equations ( 1) and ( 2) respectively, the logarithm of the likelihood function becomes Differentiate equation ( 4) with respect to c and k and equate each result to zero, then Two equations are solved simultaneously to obtain the estimates of c and k. 7) exhibits no explicit solutions to solve the equations analytically, and the maximization is performed through the mathematical approach that is the Newton-Raphson method to obtain the approximate solutions.According to [6], the definition is given by

EM Algorithm
According to [5], let denotes the observed data where Then, the probability density function of 2-parameter Burr Type XII distribution when given From f(t) in equation ( 1), the complete data log-likelihood function of the Burr Type XII distribution is expressed as The Q-function of 2-parameter Burr Type XII distribution of censored data is obtained as Using numerical integral and apply Taylor series, equation (13) is solved as follows.

3-Parameter Burr Type XII distribution
The probability density function (pdf) of the standard Burr Type XII distribution for 3-parameter is written in the form of for 0  t with both c and k are shape parameters and s is scale parameter .

Maximum Likelihood Estimation (MLE)
The likelihood function for the censored data is given by There are r failures at times . Equations ( 12) and (13) become Differentiate equation ( 17) with respect to c, k and s and equate each result to zero, the equation becomes 0 Three equations are solved simultaneously to obtain the estimates of c, k and s.
The Newton-Raphson method is used to obtain the approximate solutions for equation (19) as follows.

EM Algorithm
(26) f (t) as given in equation (11) can be expressed as The Q-function of 3-parameter Burr Type XII distribution for multiple censored data is obtained as )] , , ( Equation (28) is solved using numerical integral and applies Taylor series as follows.

Efficiency and accuracy
The bias is calculated by the difference between the expected value of an estimator and the true value of the estimator in order to ascertain the accuracy of the estimators in the model.Let  ˆis the estimator of the parameter, and then biasedness is calculated as   The Burr Type XII distribution will be written as Burr XII (c, k) for 2-parameter and Burr XII (c, k, s) for 3parameter.The estimated parameter of 2-parameter Burr Type XII distribution is investigated by varying the k values in Table 1 and c values in Table 2.The discussion of the estimated parameter will be given to Burr XII (1, 2) from Table 1 since the explanations were the same for other parameter values.The sample size is standardized to 200 and the result for the MLE and EM algorithm showed the estimated parameter for c and k are approximately close to the true parameter of 1 and 2 for uncensored data.With respect to the bias and MSE of parameter c and k, the EM algorithm outperforms the MLE with bias is -0.0764 for c parameter and 0.0317 for the k parameter; with MSE is 0.0058 and 0.0010 for c and k parameter respectively which is smaller than the bias and MSE of MLE with bias 0.1241 and -0.0702 for c and k parameter respectively and MSE is 0.0154 for c and 0.0049 for k parameter.In the presence of 30% censoring level, the accuracy of c ˆand k ˆdropped with the value 1.1816 and 1.7132 respectively.Table 1 showed that when the percentage of incomplete data (censored) is increased, the value of estimated parameter c and k is also increased.However, the EM algorithm still gives the smaller values of bias and MSE than the MLE.This is also same for several values of c and k.
The estimated parameter of Burr Type XII distribution will be discussed only to the example of Burr XII . Ismail et al. / Malaysian Journal of Fundamental and Applied Sciences Vol.10, No.2 (2014) 74-81 square error (MSE) is a way of measurement through the average of the square error.It provides better quality measurement for the estimator since it accesses on both the variances and bias term.
complete sample sizes were generated and the curve showed that the simulated data were well fitted to the estimated 2-parameter Burr Type XII survival function in Figure (a).
Figure (b) and (c) show that the complete simulated data undergone some amount of censoring and each of them contains 10% and 30% of the censored data respectively.

Table 3 3 Table 4 1 Table 5
Comparison of the estimators, bias, MSE (parentheses) and negative log-likelihood for multiple data sets of 3parameter Burr Type XII distribution with true value of c = 2, k = 3 and s = 1, Comparison of the estimators, bias, MSE (parentheses) and negative log-likelihood for multiple data sets of 3parameter Burr Type XII distribution with true value of c = 2, k = 1, 5 and s = Comparison of the estimators, bias, MSE (parentheses) and negative log-likelihood for multiple data sets of 3parameter Burr Type XII distribution with true value of c = 0.5, 2, k = 2 and s =2 Malaysian Journal of Fundamental and Applied Sciences Vol.10, No.2 (2014) 74-81 N.H.Ismail et al. / Malaysian Journal of Fundamental and Applied Sciences Vol.10, No.2 (2014) 74-81 Ismail et al. / Malaysian Journal of Fundamental and Applied Sciences Vol.10, No.2 (2014) 74-81