COMPUTATION APPROACHES FOR PARAMETER ESTIMATION OF WEIBULL DISTRIBUTION

This paper examines the estimation comparison of two methods for Weibull parameters, one is the maximum likelihood method and the other is the least squares method. A numerical simulation study is carried out to understand performance of the two methods. Based on sample root mean square errors, we make a comparison between the two computation approaches. We find that the least squares method significantly outperforms the maximum likelihood when the sample size is small. Key WordsWeibull distribution, Maximum likelihood method, Least squares method, Sample root mean square error, Simulation


INTRODUCTION
In the field of applied mechanics, one of distribution functions of wide applicability is the Weibull distribution. The cumulative Weibull distribution function can be given by Its corresponding probability density is as where  and  are positive constants. This is the well-known ''Weibull'' family of distributions, named after the Swedish engineer Waloddi (1887-1979) who popularized its application for reliability analysis, especially for metallurgical failure modes.
The constant  is called the scale parameter, because it scales the x variable. On the other hand, the constant  is called the shape parameter, because it decides the shape of the rate function 1 ) ( . If  is less than 1 the rate is decreasing with x . Whereas if  is greater than 1 the rate is increasing with x . And if  =1 the rate is constant, in which case the Weibull distribution becomes the exponential distribution. The three different-type shapes of the rate functions for the Weibull family of distributions can be referred to Hoyland and Rausand [1]. Recently, Yang & Nie [2], Dai et.al. [3], and Shi et. [4] presented an advanced algorithm for maximum likelihood estimation of three parameter Weibull distribution.
The maximum likelihood method and the least squares method are usually used to estimate the population parameters of a distribution. Existing in literature, computational comparison on the two estimation methods is not found. This would motivate us to study the computation approach for the Weibull parameters. The objective of this paper is to perform a comparative efficiency on estimating parameters of the Weibull distribution between the two methods. A numerical simulation study is carried out to investigate behavior of each estimation method. Based on sample root mean square errors, we determine which method provides superior estimates of the parameters of the Weibull distribution. According to simulation results, some useful information is summarized to engineers and scientists.

MAXIMUM LIKELIHOOD ESTIMATION
Suppose that n X X X ,..., , 2 1 are independent and identically distributed Weibull random variables each having probability density function ) (x f expressed in equation (2), where the parameters are assumed unknown. To estimate the parameters  and  , the maximum likelihood method is employed. The likelihood function of n X X X ,..., , 2 1 can be constructed from equation (2) as Taking natural logarithmic transformation, we obtain Differentiating ) , ( ln   L with respect to  and  respectively yields Equating to zero shows that the maximum likelihood estimates ) , or equivalently, (10) Equations (9)-(10) can then be solved numerically for ˆ, which will also determine  subsequently. For detailed process of the maximum likelihood method, the interested reader can refer to [5]. However, rather than pursuing the maximum likelihood method any further, let us consider a second method in the next section, which is not only computationally easier but also, as indicated by a simulation study, produces more accurate estimates of ) , (   when the sample size n is small enough.

LEAST SQUARES ESTIMATION
Let n X X X ,..., , 2 1 be a random sample from the Weibull distribution of equation (2). From (1), its distribution given by Now let If we were able to approximate the quantities ))) Subsequently, we could choose  and  to minimize the sum of the square errors. That is,  and  are chosen to minimize Following Hogg et al. [6] and Roussas [7], we define ) ln ( x y . To apply the foregoing, we need to determine values i y that approximate ))), . We now present one approach for doing this. Using the fact that (16) And then approximate . Therefore, this approach calls for using (14) and (15)

THE SIMULATION STUDY
In order to investigate behavior of ) , (   and ) , (   , a numerical simulation study is carried out. In the simulation experiment we set three Weibull distributions on X with different combinations of parameters ) , (   , in which the corresponding probability density function, distribution function, mean, variance and type of failure rate are illustrated in Table 1 below.
Decreasing rate

Increasing rate
For each Weibull distribution assumed on X , we choose a random sample ) ,..., , ( . Subsequently, we compute the sample mean and the sample root mean square errors for 1 S and 2 S in terms of the following formulas: (23) All simulation runs are performed by using a PC Pentium 4 with Matlab® 7.0.4 code. In this paper, we simulate three types of Weibull distribution shown in Table 1, in which one is decreasing failure rate, one is constant failure rate, and one is increasing failure rate. The sample means and the sample root mean square errors of ) , . In other words, the least squares method outperforms the maximum likelihood method when the underlying Weibull distribution has decreasing rate or constant rate with small sample size n; however, the maximum likelihood method dominates the least squares method if the underlying Weibull distribution has increasing rate with sufficiently large sample size n.

CONCLUSIONS
This paper explores behavior of the maximum likelihood method and the least squares method used to estimate Weibull parameters ) , . Through a numerical simulation study, we realize performance of these two estimation method. Based upon the criterion of sample root mean square errors, we found that the least squares method is superior to the maximum likelihood method when the sample size is small enough. Alternatively, the maximum likelihood method performs better if the sample size is large enough. We summarize the above results in the following Table 3. . This may be due to the fact that the maximum likelihood estimating equations (equations (7) and (8)