Mohammad Fraiwan Al-Saleh

On the efficiency of monte carlo methods using steady state ranked simulated samples

Samawi (1999) showed that the efficiency of Monte Carlo methods of integrals estimation can be substantially improved by using ranked simulated samples (RSIS) in place of uniform simulated samples (USIS). However, in this paper it is shown that substantial improvement of efficiency can be achieved further by using the steady state ranked simulated sample (SRSIS). It appears that the modified Monte Carlo methods using SRSIS provide unbiased and more efficient estimators for the integrals. Some theoretical properties of SRSIS are given. A simulation study is conducted to compare the performance of the methods using SRSIS with respect to USIS, for some examples.

Improving the best linear unbiased estimator for the scale parameter of symmetric distributions by using the absolute value of ranked set samples

Ranked set sampling is a cost efficient sampling technique when actually measuring sampling units is difficult but ranking them is relatively easy. For a family of symmetric location-scale distributions with known location parameter, we consider a best linear unbiased estimator for the scale parameter. Instead of using original ranked set samples, we propose to use the absolute deviations of the ranked set samples from the location parameter. We demonstrate that this new estimator has smaller variance than the best linear unbiased estimator using original ranked set samples. Optimal allocation in the absolute value of ranked set samples is also discussed for the estimation of the scale parameter when the location parameter is known. Finally, we perform some sensitivity analyses for this new estimator when the location parameter is unknown but estimated using ranked set samples and when the ranking of sampling units is imperfect.

On multistage ranked set sampling for distribution and median estimation

A variation of ranked set sampling (RSS), multistage RSS (MSRSS), is investigated for the estimation of the distribution function and some of its quantiles, in particular the median. It is shown that this method is significantly more efficient than simple random sampling (SRS). The method becomes more and more effective as the number of stages r increases. Two estimators of the median based on MSRSS are proposed and compared to the sample median obtained by SRS.

Steady-state ranked set sampling and parametric estimation

Ranked set sampling (RSS) was first used to obtain a more efficient estimator of the population mean, as compared to the one based on simple random sampling. This technique is useful when judgment ordering of a simple random sample (SRS) of small size can be done easily and fairly accurately, but exact measurement of an observation is difficult and expensive. It is noted that, due to the complicated likelihood, parametric estimation with RSS is difficult. In this article, the notion of steady-state RSS is introduced, its relation to stratified sampling is established, and its possible use in parametric estimation is explored and put forward for further investigations.

On bivariate ranked set sampling for ratio and regression estimators

Abstract Relative performance of bivariate ranked set sampling (BVRSS), with respect to ranked set sampling (RSS) and simple random sampling (SRS), for estimating the population mean using ratio and regression methods is investigated. Properties of the proposed estimators are derived. Simulation study is conducted to gain insight in the behaviour of the relative efficiency of BVRSS relative to other sampling scheme. Also, real data that consist of heights and diameters of 399 trees are used to illustrate the procedure. The result of the simulation study and the real data illustration indicate that using BVRSS for ratio and regression estimation is more efficient than using a SRS and an RSS in most of the cases.

On the Similarity Structure of Order Statistics

In this article, we address the similarity structure between pairs of order statistics of an identically distributed independent random variables X1,…, Xn. The overlapping coefficient (Δ) of Weitzman (1970), is used to assess the degree of similarity or closeness between pairs of order statistics. It appears that the degree of the similarity between any of such pairs is independent of the parent distribution. Using this notion, it is shown that for i < j, the degree of similarity between distributions of the ith and the jth order statistics decreases as i and j sunder. Some possible biometric applicability of the value of Δ are explored. In particular, the use of this measure in estimation of the number of possible strata, subgroups or natural subdivisions in a population are suggested.

On the approximation of multiple integrals using multivariate ranked simulated sampling

Abstract The idea of using ranked simulated sampling approach (RSIS) to improve the well known Monte Carlo methods of integration, introduced by Samawi [H.M. Samawi, More efficient Monte Carlo methods obtained by using ranked set simulated samples, Commun. Stat. Simulat. 28 (3) (1999) 699–713], is extended to multivariate ranked simulated sampling approach (MVRSIS) for multiple integration problems. It is demonstrated that this approach provides unbiased estimators and improves the performance of some of the Monte Carlo methods of multiple integrals approximation. This, results in large saving in terms of cost and time needed to attain a certain level of accuracy. Two illustrations using simulation are used to compare the relative performance of this approach relative to multivariate uniform simulation.

Inference on Overlapping Coefficients in Two Exponential Populations Using Ranked Set Sampling

We consider using ranked set sampling methods to draw inference about the three well-known measures of overlap, namely Matusitas measure , Morisitas measure and Weitzmans measure . Two exponential populations with different means are considered. Due to the difficulties of calculating the precision or the bias of the resulting estimators of overlap measures, because there are no closed-form exact formulas for their variances and their exact sampling distributions, Monte Carlo evaluations are used. Confidence intervals for those measures are also constructed via the bootstrap method and Taylor series approximation.

Bivariate Sign Test for One-Sample Bivariate Location Model Using Ranked Set Sample

In this article, we introduce a bivariate sign test for the one-sample bivariate location model using a bivariate ranked set sample (BVRSS). We show that the proposed test is asymptotically more efficient than its counterpart sign test based on a bivariate simple random sample (BVSRS). The asymptotic null distribution and the non centrality parameter are derived. The asymptotic distribution of the vector of sample median as an estimator of the locations of the bivariate model is introduced. Theoretical and numerical comparisons of the asymptotic efficiency of the BVRSS sign test with respect to the BVSRS sign test are also given.

Estimation of the Proportion of Sterile Couples Using the Negative Binomial Distribution

A Sterile family is a couple who has no children by their deliberate choice or because they are biologically infertile. Couples who are childless by chance are not considered to be sterile. The object is to estimate the proportion of sterile couples in Jordan in- directly based on the 1994 population census, by separating the two types of childless couples into sterile and fertile couples. Three meth- ods of fitting a negative binomial distribution to the completed family size data obtained from 1994-population census are investigated. It appeared that the third method gives the best fit. Based on the fitted distribution, the proportion of sterile couples is estimated at 6.1% of all couples. This estimate is much lower than the corresponding esti- mate of sterile couples in the USA, which was estimated at 11%. The difference between the two can be due to some socio-cultural factors influencing the deliberate choice of couples to have no children. The method of estimation can be applied on other populations.