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

在本研究中，我们考虑了一类具有有限支撑的位置-尺度分布族作为非正则分布。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。接下来，考虑到成本，我们构造了一个位置的序贯点估计过程，如下所示。当样本量为n时，分别用M_n和R_n表示中值和极差。我们定义停止规则为τ：=min {n <$n_0：n^3 <$AR^2_n/（2a^2d）}，其中2a是分布的支撑宽度，n_0是满足一定条件的初始样本容量，A是某个常数。我们用M_n估计位置参数。当样本容量为n时，定义渐近必要的最小样本容量为n^*，当样本容量为n时，定义风险为r_n。然后我们有以下内容。(I)lim _<d→0+>τ/n^*=1，（ii）lim_<d→0+>E（τ/n^*）=1，（iii）lim_<d→0+>r_τ/r_n.= 1.因此，这表明该过程是渐近有效的。这个停止规则也以概率1为界，而著名的罗宾斯程序（1965）可能不是。Koike（2007）也观察到，当密度在支持的端点急剧变化时，在相同的假设下，基于中值的序贯区间估计程序对位置的序贯估计程序具有类似的渐近优越性。请注意，在Akahira和Takeuchi（1995）中可以找到非序列情况下位置族的类似结果。

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).