Ken-ichi Koike

Second-order asymptotic comparison of the MLE and MCLE for a two-sided truncated exponential family of distributions

ABSTRACT For a one-sided truncated exponential family of distributions with a natural parameter θ and a truncation parameter γ as a nuisance parameter, it is shown by Akahira (2013) that the second-order asymptotic loss of a bias-adjusted maximum likelihood estimator (MLE) of θ for unknown γ relative to the MLE of θ for known γ is given and and the maximum conditional likelihood estimator (MCLE) are second-order asymptotically equivalent. In this paper, in a similar way to Akahira (2013), for a two-sided truncated exponential family of distributions with a natural parameter θ and two truncation parameters γ and ν as nuisance ones, the stochastic expansions of the MLE of θ for known γ and ν and the MLE and the MCLE of θ for unknown γ and ν are derived, their second-order asymptotic means and variances are given, a bias-adjusted MLE and are shown to be second-order asymptotically equivalent, and the second-order asymptotic losses of and relative to are also obtained. Further, some examples including an upper-truncated Pareto case are given.

Unbiased estimation for sequential multinomial sampling plans

For sequential multinomial sampling, a sufficient condition for the stopping rule to be closed is obtained. And, by the application of the Rao-Blackwell method,an unbiased estimator based on the sufficient statistics is given for some functions of unknown parameters. The results are applied to some sequential stopping rules.

Sequential Interval Estimation of a Location Parameter with the Fixed Width in the Uniform Distribution with an Unknown Scale Parameter

Abstract We consider a sequential interval estimation with fixed width of a location parameter θ of a sequence of uniform random variables with unknown scale ξ. A stopping rule is proposed, and its asymptotic properties are investigated. Numerical evaluations are also done. Further, the exact distribution of the size of sample is given.

Sequential Interval Estimation of a Location Parameter with Fixed Width in the Nonregular Case

Abstract For a location-scale parameter family of distributions with a finite support, a sequential confidence interval with a fixed width is obtained for the location parameter, and its asymptotic consistency and efficiency are shown. Some comparisons with the Chow-Robbins procedure are also done.

ON THE INEQUALITY OF KSHIRSAGAR

ABSTRACT A lower bound for the variance of unbiased estimators which is due to Kshirsagar is studied. The bound is free from regularity assumptions. It is also shown that the bound is an improvement of Bhattacharyya bound in regular case.

Completeness for sequential sampling plans

In the sequential multinomial sampling case,a sufficient condition for a non-randomized sequential procedure to be complete is given,and also a necessary and sufficient condition for a randomized sequential procedure to be complete is obtained.

Sequential Estimation Procedures for End Points of Support in a Non-Regular Distribution

In this article, we consider sequential estimation of the end points of the support based on the extreme values when the underlying distribution has a bound support. Some sequential fixed-width confidence intervals are proposed. Stopping rules based on the range are proposed and the estimation procedures based on them are shown to be asymptotically efficient. The results of numerical simulations are presented. Moreover, the sequential point estimation problem is considered under squared loss plus cost of sampling.

Sequential Point Estimation of the Location Parameter in the Location-Scale Family of Non-Regular Distributions

Abstract In this paper, we consider sequential estimation of the location parameter based on the midrange in the presence of an unknown scale parameter when the underlying distribution has a bounded support. The estimation is done under squared loss plus cost of sampling. Stopping rules based on the range are proposed and are shown to be asymptotically efficient. The risks of the sequential procedures are compared with the Robbins sequential estimation procedure based on the sample mean. The former are shown to be asymptotically more efficient than the latter in the sense of the sample size when the density function changes sharply at the end points of the support. Koike (2007) observed a similar asymptotic superiority of the sequential estimation procedure based on the midrange in the sequential interval estimation procedure under the same condition.

A lower bound for the bayes risk in the sequential case

A lower bound for the Bayes risk in the sequential case is given under the regularity conditions. A related result to the minimax risk is also discussed. Further, some examples are given for the exponential and Poisson distributions.

An Integral Bhattacharyya Type Bound for the Bayes Risk

Bhattacharyya type integral inequalities for the integrated risk for estimators are given extending the work of Borovkov and Sakhanienko (1980). As an application, an asymptotic approximation of the lower bound for locally minimax risk is given.