برآورد واریانس توزیع نرمال با استفاده از نمونه‌گیری مجموعۀ رتبه‌دار

نمونه‌گیری یکی از مهم‌ترین بخش‌های علم آمار است. در هر تحقیق، پژوهشگر در پی یافتن روش مناسب برای جمع‌آوری نمونه و اطلاعات مربوط به آن است که کارا و کم هزینه باشد. در شرایطی که اندازه‌گیری واحدهای جامعه مشکل یا پرهزینه باشد، اما بتوان واحدهای جامعه را به‌سادگی و با کم‌ترین هزینه رتبه‌بندی کرد، روش نمونه‌گیری مجموعۀرتبه‌دار مورد استفاده قرار می‌گیرد. در این مقاله ابتدا روش نمونه‌گیری مجموعۀ رتبه‌دار معرفی می‌‌شود. سپس چند روش برآورد واریانس توزیع نرمال با ترکیب برآوردگرهای بین‌گروهی و درون‌گروهی نااریب ارائه می‌شود. در نهایت برآوردگرهای ارائه شده با استفاده از پژوهش‌های شبیه‌سازی با یکدیگر مقایسه می‌شوند.

estimation of variance of normal distribution using ranked set sampling

introduction    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 methodsall simulation studies in this paper have been done using r statistical software version r-3.3.1results and discussionin 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.conclusionthe following results can be obtained based on our simulation study:if there is a single cycle in rss, then the proposed estimator in the case of single cycle beats stokes-modified unbiased estimator.in the multiple cycle case in rss, the estimator based on combining variance estimators of within and between ranking classes is the best one. ./files/site1/files/51/%d9%85%d9%87%d8%af%d9%88%db%8c_%d9%85%d9%86%d8%b4_%d8%a7%db%8c%d8%b1%d8%a7%d9%86%d9%be%d9%86%d8%a7%d9%87.pdf