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

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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).