Efficiencies of sequential estimation procedures by information inequalities in non-regular estimation

在本研究中，我们将有限支承的位置尺度分布族视为非正则分布。首先，当尺度未知时，我们构造了位置参数的序贯区间估计过程。其次，在考虑成本的情况下，我们构造了一个位置的序贯点估计程序，取值为每次采样的成本。当样本大小为n时，分别用M_n和R_n表示中值和极差。停止规则由τ：=Min{n≧n_0：n^3≧AR^2_n/(2a^2d)}定义，其中2a是分布的支撑点宽度，n_0是满足一定条件的初始样本量，A是某个常数。我们用M_n估计位置参数，当ξ已知时，用n^*定义渐近必要的最小样本量，当样本量为n时，用r_n定义风险。(I)Lim_&lt；d→0+&gt；τ/n^*=1，(Ii)Lim_&lt；d→0+&gt；E(τ/n^*)=1，(Iii)Lim_&lt；d→0+&gt；r_τ/r_n.=1。这个停止规则也是以概率1为界的，而著名的Robbins过程(1965)可能不是。Koike(2007)还观察到，在相同的假设条件下，当支撑点的密度急剧变化时，基于中值区间的序贯估计方法具有类似的渐近优势。注意，在非连续情况下的位置族的类似结果可以在Akahira和Takeuchi(1995)中找到。

In this research, we considered a location-scale family of distributions with the finite support as a non-regular distribution. At first, we construct a sequential interval estimation procedure of the location parameter when the scale is unknown. Next, taking the cost into account, we construct a sequential point estimation procedure of the location as follows.Put d as the cost per sampling. Denote the midrange and the range by M_n and R_n when the sample size is n, respectively. We define the stopping rule by τ: =min {n≧n_0: n^3 ≧AR^2_n/ (2a^2d)}, where 2a is the width of the support of the distribution, n_0 is the initial sample size satisfying a certain condition and A is some constant. We estimate the location parameter by M_n. Define the asymptotically necessary minimum sample size by n^* when ξ is known, and the risk by r_n when the sample size is n. Then we have the following. (I) lim _<d→0+>τ/n^*=1, (ii) lim_<d→0+>E (τ/n^*)=1, (iii) lim_<d→0+>r_τ/r_n.=1.Therefore this shows that the procedure is asymptotically efficient. This stopping rule is also bounded with probability 1 while the well-known Robbins' procedure (1965) may not. And also Koike (2007) observed a similar asymptotic superiority of the sequential estimation procedure based on the midrange in the sequential interval estimation procedure for the location under the same assumptions when the density changes steeply at the end points of the support. Note that similar results for the location family in the non-sequential case can be found in Akahira and Takeuchi (1995).