Estimation of the Inverse Weibull Parameters Under Ranked Set Sampling

In this paper we use the maximum likelihood (ML) and the modified maximum likelihood (MML) methods to estimate the unknown parameters of the inverse Weibull (IW) distribution as well as the corresponding approximate confidence intervals. The estimates of the unknown parameters are obtained based on two sampling schemes, namely, simple random sampling (SRS) and ranked set sampling (RSS). Comparison between the different proposed estimators is made through simulation via their mean square errors (MSE), Pitman nearness probability (PN) and confidence length.


Introduction
Ranked set sampling is recognized as a useful sampling technique for improving the precision and increasing the efficiency of estimation when the variable under consideration is expensive to measure or difficult to obtain but cheap and easy to rank. Ranked set sampling has been suggested by McIntyre (1952) in relation to estimating pasture yields. Takahasi and Wakimoto (1968) established the theory of RSS, they showed that the sample mean of RSS is an unbiased estimator for the population mean and is more efficient than the sample mean of SRS. According to RSS, we first select m 2 elements denoted by ( ) , ( = 1,2, . . , ) from the population at random. These elements are then randomly splitted into m sets of m units each.
On each set, we rank the m units by judgment or a supporting variable according to the characteristic of interest. We select the element with the smallest ranking, 1(1) , for measurement from the first set. From the second set we select the element with the second smallest ranking, 2 (2) . We continue in this way until we have ranked the elements in the mth set and selected the element with the largest ranking, ( ) , as in Figure 1. This complete procedure, called a cycle which is repeated independently k times to obtain a ranked set sample of size = (see Chen et al. (2004)). Marginally ( ) (ith order statistics of the ith random sample in a sample of size m) have the same distribution with pdf given by (see David and Nagaraja (2003)). The inverse Weibull distribution (it is also known as Fre'chet distribution) is more appropriate than the Weibull distribution for modeling a non-monotone and unimodal hazard rate functions. The probability density function (pdf) and cumulative distribution function (cdf) of the IW distribution are, respectively, given by and where λ is the scale parameter and θ is the shape parameter. Such model has been suggested as a model in the analysis of life testing data. Keller and Kamath (1982) introduced the IW distribution for modelling reliability data and failures of mechanical components subject to degradation. Estimation of the parameters of IW distribution in classical and Bayesian methods has been discussed in literature. Calabria and Pulcini (1989) discussed the statistical properties of the MLEs of the parameters and reliability for a complete sample. Erto (1989) used the Least Square method to obtain the estimators of the parameters and reliability. Calabria and Pulcini (1990) obtained the MLEs and the Least Square of the parameters. Calabria and Pulcini (1992) derived the Bayes estimators of the parameters and reliability. The hazard rate function of the IW distribution is The shape of the hazard rate function of the IW distribution can be decreasing, increasing or unimodal based on the value of the shape parameter. Many works have been suggested to estimate the unknown parameters of the IW distribution, see for example, Calabria and Pulcini (1994) The main objective of this study is to obtain the MLEs and MMLEs as well as the approximate confidence intervals for the scale and the shape parameters of the IW distribution based on SRS and RSS. We compare the performance of the different estimators by using a simulation study. The rest of the paper is organized as follows: In Section 2, the MLEs are obtained based on the SRS and RSS. The MMLEs under the SRS and RSS are derived in Section 3. In Section 4, a simulation study is conducted to evaluate the performance of the different estimators. A real data set is analyzed in Section 5 for illustrative purposes. Finally, the conclusion in Section 6.

Maximum likelihood estimation for IW distribution parameters
In this section, the ML estimation method is used to estimate the IW distribution parameters using SRS and RSS.

Maximum likelihood estimation under SRS
Let 1 , 2 , … , be a simple random sampling of size n from the IW distribution, then from the pdf in (1) the log likelihood function is The maximum likelihood estimators of λ and θ can be obtained by differentiate (3) with respect to λ and θ and equating the results to zero as follows From equation (4), we can obtain the MLE of λ as Substituting the value of ̂ given by (6) in (5), we can obtain the MLE of the parameter θ. Now we use the large sample approximation to construct the approximate confidence intervals for the parameters λ and θ. The approximate inverse of observed information matrix of the unknown parameters is given by Thus, the approximate confidence intervals of λ and θ, are respectively, given by where (̂) and (̂), respectively, are the main diagonal elements in (7) and 2 ⁄ is the upper 2 ⁄ percentile of a standard normal distribution. 2(2) , … , ( ) be a RSS of size n from the IW distribution with pdf (1) and cdf (2), then the likelihood function ignoring the constant term can be written as

Maximum likelihood estimation under RSS
The log-likelihood function is given by From (9), the likelihood equations can be written as follows It is to be noted that the likelihood equations in this case cannot be solved explicitly, so the MLEs of λ and θ can be obtained by using any numerical technique. The elements of the observed information matrix with respect to λ and θ are as follow:

Modified maximum likelihood estimation for IW distribution parameters
In this section, the MMLs of the unknown parameters of the IW distribution are obtained using SRS and RSS.

Modified maximum likelihood estimation under SRS
It is seen that from (5) the ML equation of θ cannot be obtained in explicit form, therefore, the MMLEs which have explicit form are obtained. Let = − , then follows the extreme value distribution with pdf and cdf given, respectively, by (see Johnson et al. (1994)). and where = − ⁄ is the location parameter and = 1⁄ is the scale parameter. From (12)  Using these linear approximations, the modified likelihood equations can be written as and The solutions of equations (14) and (15) are .

Modified maximum likelihood estimation under RSS
Using the same approach in subsection (3.1), we can obtain the MMLEs under RSS. From (12) and (13)  The solutions of these equations are ̂− = +̂ and The elements of the observed information matrix with respect to μ and σ are as follows

Simulation study
In this section, a simulation study is carried out using MATHCAD program to compare the performance of the ML and MML estimators based on SRS and RSS for IW distribution. The simulation study is conducted by choosing 1000 random samples of different samples size that were generated from IW distribution.
Using where ̂1 and ̂2 are the estimators of . Based on PN probability criteria, we can say that ̂1 is better than ̂2 if > 0.5 (Pitman (1937)). All of the calculation in this section were done using MATHCAD program version 2007 and based on 1000 replications. Proceedings of the simulation are described as follow: Step 1: Suppose U has a uniform (0,1) distribution, then = (− ( )⁄ ) −1⁄ follows the IW distribution. 1000 simple random samples of sizes 20,30,50,75 and 100 were generated from IW distribution.
Step 3: Equations (4) and (5) were solved to obtain the ML estimates based on SRS and the MML estimates based on SRS are obtained by solving equations (14) and (15). Equations (10) were solved to obtain the ML estimates based on RSS and the MML estimates based on RSS are obtained by solving equations (17).

Numerical example
In this section, we use a data set of Dumonceaux and Antle (1973) to show the applicability of the proposed estimators. The data set represents the maximum flood levels of the Susquehenna River at Harrisburg, Pennsylvenia over 20 four-year periods   Maswadah (2003) analyzed these data and showed that the IW distribution provide a good fit to the data. Also, Maswadah (2003) obtained the MLE of λ and θ from the complete data set as ̂= 0.0119 and ̂= 4.3138. Here, we select a random sample of size 15 by using SRS and RSS, sampling was done with replacement. In RSS method, five matrices 3 × 3 are drawn and then applying the technique presented in Figure 1 Table 7 displays the different estimates of the unknown parameters under SRS and RSS as well as the corresponding confidence intervals bounds. From table 7, it is seen that for parameter λ the MML method under SRS perform better than other methods in terms of confidence interval length, while for parameter θ the ML method under RSS perform better than other methods in terms of confidence length (CL).

Conclusion
In this paper, we have considered the estimation problem of the unknown parameters of the IW distribution based on SRS and RSS. We used the ML and MML methods of estimation to estimate the unknown parameters. It is noted that the ML estimators cannot be obtained in closed forms, so, the MMLEs have been presented. We used a simulation study to compare the performance of the different estimators in terms of their MSEs, confidence length and PN probability. It is noted that the MMLEs based on RSS perform very well relative to their MSE, confidence length and PN probability. This indicates that the estimation based on RSS method is more efficient than estimation based on SRS.