برآورد واريانس توزيع نرمال با استفاده از نمونه ‌گيري مجموعۀ رتبه ‌دار

Estimation of Variance of Normal Distribution using Ranked Set Sampling

نمونه‌گيري يكي از مهم‌ترين بخش‌هاي علم آمار است. در هر تحقيق، پژوهشگر در پي يافتن روش مناسب براي جمع‌ آوري نمونه و اطلاعات مربوط به آن است كه كارا و كم هزينه باشد. در شرايطي كه اندازه‌گيري واحدهاي جامعه مشكل يا پرهزينه باشد، اما بتوان واحدهاي جامعه را به‌سادگي و با كم‌ترين هزينه رتبه‌بندي كرد، روش نمونه‌گيري مجموعۀرتبه‌دار مورد استفاده قرار مي‌گيرد. در اين مقاله ابتدا روش نمونه‌گيري مجموعۀ رتبه‌دار معرفي مي‌‌شود. سپس چند روش برآورد واريانس توزيع نرمال با تركيب برآوردگرهاي بين‌گروهي و درون‌گروهي نااريب ارائه مي‌شود. در نهايت برآوردگرهاي ارائه شده با استفاده از پژوهش‌هاي شبيه‌سازي با يكديگر مقايسه مي‌شوند.

In some biological, environmental or ecological studies, there are situations in which obtaining exact measurements of sample units are much harder than ranking them in a set of small size without referring to their precise values. In these situations, ranked set sampling (RSS), proposed by McIntyre (1952), can be regarded as an alternative to the usual simple random sampling (SRS) to draw a more representative sample from the population of interest than what is possible in SRS. To draw a ranked set sample, one first draws n simple random samples, each of size n, from the population of interest and ranks them in an increasing magnitude. The ranking process is done without measuring sample units and therefore it need not to be accurate. One then identifies the ith sample unit from the ith sample for actual quantification (for i=1, …, n). Finally, he repeats this process m times (cycle) if he/she is required to obtain a sample of size mn. Since a ranked set sample contains information from both measured sample units and their corresponding ranks, one intuitively expects that statistical inference based on RSS to be more accurate than what is possible to obtain based on SRS. This paper is concerned with problem of estimating variance of the normal distribution in RSS. Several methods of estimation of variance of the normal distribution are described and compared via a Monte Carlo simulation study. Material and methods All simulation studies in this paper have been done using R statistical software version R-3.3.1 Results and discussion In this paper, we consider estimation of the normal variance based on a ranked set sample with single (multiple) cycle(s) and propose different unbiased estimators for each case. Our simulation results indicate that the mean square error (MSE) of each estimator is decreased as the values of n or m increases while the other parameters are kept fixed. It is also found that the estimator based on combining variance estimators of within and between ranking classes has typically better performance than the others. Conclusion The following results can be obtained based on our simulation study: