نقش استقلال آماری در یافتن براوردگرهای بهینه

به طور طبیعی در انتخاب یک برآوردگر نقطه ای علاقه مند به انتخاب برآوردگری هستیم که برای تمام مقادیر فضای پارامتر، تابع مخاطره را مینیمم کند. در عمل با توجه به بزرگی رده  برآوردگرها چنین امکانی وجود ندارد. یکی از راه ها برای حل این مشکل (پیدا کردن برآوردگرهای بهینه)، محدود کردن رده  برآوردگرها و پیدا کردن بهترین برآوردگر در رده  محدود شده است. این محدودیت بر حسب این که خود را به رده  برآوردگرهای هم وردا یا نااریب محدود کنیم به ترتیب منجر به دو نوع برآوردگر بهینه یعنی بهترین برآوردگر هم وردا و برآوردگر نااریب با کمترین مخاطره می شود که در این مقاله نقش استقلال در ساده تر محاسبه کردن این برآوردگرها مورد بررسی قرار می گیرد. همچنین به استقلال تصادفی یک تابع ناوردا و هم وردا و مقایسه آن با قضیه باسو می پردازیم. 

the role of statistical independence in finding optimal estimators

naturally in choosing a point estimator we are interested in choosing an estimator thatminimizes the risk function for all parameter space values. in practice, this is not possibledue to the large number of estimators. one way to solve this problem (find optimalestimators) is to limit the range of estimators and find the best estimator in the finiterange. this limitation leads us to two types of optimal estimators, namely the bestequivariant estimator and the minimum risk unbiased estimator, respectively, in termsof limiting ourselves to the class of equivariant or unbiased estimators. in this paper, therole of independence in simplifying the calculation of these estimators is examined. wealso deal with the stochastic independence of an invariant function and its comparisonwith the basu’s theorem.to find the optimal estimators, the class of estimators can be limited. this limitationcan be applied to the class of equivariant or unbiased estimators, which leads to twotypes of optimal estimators, namely the best equivariant estimator and the minimum riskunbiased estimator, respectively. for this purpose, in addition to the rao-blackwell-lehmann-scheffé theorem (casella and berger, 2001), two other methods have beenproposed by sathe and varde (1969) and eaton and morris (1970) which can be usefulto achieve this goal. in these two methods, by limiting the class of equivariant orunbiased estimators, the estimator with the minimum risk is considered as the optimalestimator. independence can play a key role in making it easier to calculate the riskfunction of the best equivariant estimator and the minimum risk unbiased estimator. infact, the role of independence is to eliminate the conditional probability in calculatingthe risk function of the best equivariant estimator and the minimum risk unbiasedestimator, which in most cases results from basu’s theorem and the independence ofancillary statistic from complete sufficient statistics. similar to basu’s theorem, it canbe shown that in certain circumstances an invariant statistic and equivariant function areindependent of each other, which can play a role in eliminating the conditionalprobability by the independence of the maximum invariant statistic from the equivariantsufficient statistic. it is noteworthy that in this case, the completeness assumption of basu’s theorem has been replaced by equivariance assumption and the ancillarityassumption of basu’s theorem has been replaced by invariance.material and methodswe first introduce the definitions that are needed. in the second part, by limiting theclass of equivariant estimators, we create a type of optimal estimator called the bestequivariant estimator and show that in groups that act as transitive on the parameterspace, an invariant function is independent of the equivariant sufficient statistic. in thethird part, by limiting the class of unbiased estimators, we make a type of optimalestimator called minimum risk unbiased estimator, which in a special case where thesquare error loss function is the same as the minimum variance unbiased estimator,which in the fourth part, in addition to rao-blackwell-lehmann- scheffé theorem(casella and berger, 2001), introduce two other methods proposed by sathe and varde(1969) and eaton and morris (1970) which, with the help of independence, provide asimpler method for finding a minimum variance unbiased estimator.conclusionthe following conclusions were drawn from this research. in order to find the optimal estimators by limiting the class of estimators to theclass of equivariant or unbiased estimators, the independence of completesufficient statistics from ancillary statistics and applying basu’s theorem canbe a way to simplify calculations. in some statistical problems with the transitive transformation group, theequivariant function can be used instead of the complete sufficient statistic. inthis case, instead of using the basu’s theorem, the independence of an invariantfunction and the equivariant sufficient statistic can be inferred. hence, theassumption of completeness for establishing the basu’s theorem is replacedby the equivariance and having a transitive transformation group. having a transitive group, any invariant function is also ancillary, but theincompleteness of a sufficient statistic can also result in its independence froman invariant statistic. if the group of transitive transformations and completesufficient statistic is also equivariant, the case of basu’s theorem concludesthis view. the opposite is not always true and can be corrected in such a waythat if a complete statistic is also equivariant with a transitive group, it is notnecessary that each ancillary statistic be independent of it. rather, it is possibleto find an ancillary statistic that is also invariant, which is independent of thegiven equivariant complete sufficient statistic. by finding the condition thatan ancillary statistic is also invariant, these results can be extended, in whichcase basu’s theorem is the result. of course, this open problem needs furtherresearch and it is hoped that researchers will be diligent in generalizing it.