由概似函數建構離散型分配的信賴區間

Abstract

延伸Chen(2004,處理連續行分配的問題)的觀念，我們利用概似函數的頂端區域來建構離散型分配參數的信賴區間。有幾個地方值得我們注意。 首先，這個區間顯示在具有相同信賴係數的區間中所包含x的點數最少。因為期望長度是一般用來比較信賴區間的準則，而實際上在建構平均長度一致最短的信賴區間目前並未有令人滿意的結果。利用樣本數量的方式來衡量離散型分配的信賴區間不失為一個好方法。第二，傳統區間估計方法得到的信賴區間最被批評的一點是：它可能不包含欲估之參數(即最大概似估計量)。而由概似函數所建構的信賴區間則具有此一性質。另外，現有的信賴區間的建構並未利用到所有參數的資訊。此種區間具有最大概似估計量的性質，如不變性和充分性。由於貝氏型態的事後分配信賴區間也具有最短長度之最佳性，但貝氏過於依賴先驗分配，且並未具有不變性的特性。

Generalizing from Chen (2004) of dealing for continuous distributions, we introduce a likelihood-based confidence set for the discrete distributions. Besides some properties extended from maximum likelihood estimator, four additional properties are of special interest. First, this set is shown to have volume for the sample falling in the confidence set the smallest among classes of 100(1-α)% confidence sets. With the fact that expected-length is a popularly used criteria for comparing confidence intervals and, actually, there is no satisfactory results for constructing the optimal one, minimizing the volume, in terms of sample x, is a good criterion for evaluating confidence interval for discrete distributions. Second, the likelihood-based confidence sets include maximum likelihood estimate, the most plausible parameter value after an observation has been made, whereas the traditional frequentist approaches of confidence set are criticized for that may not include it. Third, the construction of these confidence sets are based on Fisher's likelihood principle which ask that any statistical procedure should depend upon the likelihood function whereas the existed confidence sets do not fulfill this desirability. Fourth, properties behaving for maximum likelihood estimator such as invariance and sufficiency have been carried over to these approaches. The property of invariance is interesting for the fact that the Bayesian highest posterior density confidence set has also an optimality of shortest width, however, this Bayesian interval is criticized for depending on the prior density and for not having the desired property of invariance.