Mathematical Researches

fa بهینه سازی اندازه نمونه تصادفی در سانسور فزاینده نوع دو برمبنای معیار هزینه Optimization of Random Sample Size in Progressively Type II Censoring based on Cost Criterion آمار Stat علمی پژوهشی بنیادی S نمونه های سانسور شده تاکنون توسط پژوهشگران زیادی مورد مطالعه قرار گرفته اند. یکی از مهمترین روشهای سانسور، سانسور فزاینده نوع دو است. یکی از مسائلی که در بحث سانسورها مطرح است تعیین اندازه نمونه مناسب است. برای تعیین اندازه نمونه مناسب عوامل مختلفی تاثیرگذار هستند که از مهمترین عوامل می توان به معیار هزینه نمونه گیری اشاره کرد. در این مقاله، با فرض اینکه اندازه نمونه متغیری تصادفی از توزیع دوجمله ای باشد، به تعیین پارامتر بهینه توزیع اندازه نمونه در سانسور فزاینده نوع دو پرداخته می شود. این پارامتر بهینه طوری تعیین می شود که مقدار هزینه کل آزمایش از مقدار از قبل تعیین شده ای بیشتر نشود. در این مقاله توزیع نمایی برای توزیع مشاهدات در نظر گرفته شده است. برای ارزیابی نتایج بدست آمده مطالعه شبیه سازی نیز انجام شده است. در پایان نتیجه گیری از مقاله ارائه شده است.   Introduction &lrm;&lrm;Censored sample arises in a life-testing experiment whenever the experimenter does not observe the failure times of all units placed on a life-test&lrm;. &lrm;In medical or industrial studies&lrm;, &lrm;researchers have to treat the censored data because they usually do not have sufficient time to observe the lifetime of all subjects in the study&lrm;. &lrm;There are different types of censoring&lrm;. &lrm;The most common censoring schemes are type I and type  II censoring schemes&lrm;. Progressively type II censoring is also one of the most important methods of censoring. One of the most common questions any statistician gets asked is "How large a sample size do I need?"&lrm;. &lrm;Researchers are often surprised to find out that the answer depends on a number of&lrm; factors and they have to give the statistician some information before they can get an answer&lrm;. &lrm;So far different answers have been given to respond this question by considering different criteria&lrm;. &lrm;Cost criterion is one of the criteria that has always been of interest to researchers&lrm;. &lrm;So far&lrm;, &lrm;many researchers have used this criterion for determining the size of samples in different censoring methods&lrm;. &lrm; In some applications&lrm;, &lrm;such as clinical trials and quality control&lrm;, &lrm;it is almost impossible to have a fixed sample size all the time because some observations may be missing for various reasons&lrm;. &lrm;In other words&lrm;, &lrm;the sample size is a random variable&lrm;.  Material and methods In this paper, a cost function is introduced. Then, assuming that the sample size of progressively type II censoring is a random variable from the truncated binomial distribution, the optimal parameter of sample size distribution in progressively type II censoring, is determined. This optimal parameter is determined so that the introduced cost function does not exceed a pre-determined value, say <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image002.gif" style="width:17px; height:22px" >. In this article, the exponential distribution is considered for lifetimes of observations. A simulation study is also provided to evaluate the obtained results. Finally, the conclusion of the article is presented.    Results and discussion &lrm;We have computed the values of the expected cost function by considering three different censoring schemes&lrm;. &lrm;The results show that the expected cost function is an increasing function of m but a decreasing function of θ, &lrm;when other components are fixed&lrm;, &lrm;as we expected&lrm;.  Also, we can find that considering type II censoring leads to better results than other censoring schemes&lrm;. On the other hand, we can conclude that type II censoring provides the minimum cost among two other censoring schemes. In the sequel, by assuming an upper bound for the cost function, say <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image002.gif" style="width:17px; height:22px" >, the optimal parameter of sample size distribution is obtained.  Conclusion Determining the optimal sample size is one of the issues that has been studied by many researchers. In some cases, it is not possible for the sample size to be a fixed and pre-determined value. In other words, the sample size is a random variable. In this paper, assuming that the sample size of progressively type II censoring is a random variable from the truncated binomial distribution, the optimal parameter of the sample size distribution is determined. The criterion used in this research is the cost criterion. Next, the optimal parameter of the sample size distribution is determined so that the value of the cost function is less than the specified and predetermined value, say <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image002.gif" style="width:17px; height:22px" >. The results of the paper show that the type II censoring provides less values for the cost function. For all three censorsing schemes, the cost function is an increasing function of m but a decreasing function of θ, &lrm;when other components are fixed&lrm;, &lrm;as we expected&lrm;. As a result, the best case scenario is taking into account the type II censoring scheme, selecting smaller values for m, larger values for θ, and smaller values for the parameter of sample size distribution. اندازه نمونه تصادفی, بهینه‌سازی, تابع هزینه Random Sample Size, Optimaization, Cost Function 49 64 http://mmr.khu.ac.ir/browse.php?a_code=A-10-1115-1&slc_lang=fa&sid=1 Elham Basiri الهام بصیری elham_basiri2000@yahoo.com 10031947532846005461 10031947532846005461 Yes Kosar University of Bojnord دانشگاه کوثر بجنورد