Nao Ohyauchi

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.

A Bayesian view of the Hammersley–Chapman–Robbins-type inequality

From the Bayesian viewpoint, the information inequality applicable to the non-regular case is discussed. It is shown to construct an estimator which minimizes locally the variance of any estimator satisfying weaker conditions than the unbiasedness condition from which an information inequality is derived. The Hammersley–Chapman–Robbins inequality is also obtained as a special case of the inequality. An example is also given.

Second-order asymptotic loss of the MLE of a truncation parameter for a truncated exponential family of distributions

ABSTRACT For a truncated exponential family of distributions with a truncation parameter γ and a natural parameter θ as a nuisance parameter, the stochastic expansions of bias-adjusted maximum likelihood estimators and of γ when θ is known and when θ is unknown, respectively, are derived. The second-order asymptotic loss of relative to is also obtained through their asymptotic variances. Further, some examples are given.

COMPARISON OF THE BAYES RISKS OF ESTIMATORS FOR A FAMILY OF TRUNCATED NORMAL DISTRIBUTIONS

ABSTRACT As a typical non-regular case, we consider a family of symmetrically truncated normal distributions with a location parameter θ. The information inequalities for the Bayes risk of any estimator of θ are asymptotically given up to the higher order. The asymptotic lower bounds for the Bayes risk are shown to be sharp up to the higher order. The comparison of the Bayes risks of the mid-range, the maximum likelihood estimator and the maximum probability estimator is done, and, from the viewpoint of the asymptotic concentration probability, that of the estimators is also provided.

Information inequalities for the bayes risk for a family of non-regular distributions

For a family of non-regular distributions with a location parameter including the uniform and truncated distributions, the stochastic expansion of the Bayes estimator is given and the asymptotic lower bound for the Bayes risk is obtained and shown to be sharp. Some examples are also given.

Comparison of risks of estimators under the LINEX loss for a family of truncated distributions

In most cases, we use a symmetric loss such as the quadratic loss in a usual estimation problem. But, in the non-regular case when the regularity conditions do not necessarily hold, it seems to be more reasonable to choose an asymmetric loss than the symmetric one. In this paper, we consider the Bayes estimation under the linear exponential (LINEX) loss which is regarded as a typical example of asymmetric loss. We also compare the Bayes risks of estimators under the LINEX loss for a family of truncated distributions and a location parameter family of truncated distributions.

The Asymptotic Bound by the Kiefer-Type Information Inequality and Its Attainment

In non-regular cases when the regularity conditions does not hold, the Chapman–Robbins (1951) inequality for the variance of unbiased estimators is well known, but the lower bound by the inequality is not attainable. In this article, we extend the Kiefer-type information inequality applicable to the non-regular case to the asymptotic situation, and we apply it to the case of a family of truncated distributions, in which the lower bound by the Kiefer-type inequality derived from an appropriate prior distribution is attained by the asymptotically unbiased estimator. It also follows from the completeness of the sufficient statistic that the lower bound is asymptotically best. Some examples are also given.

A Higher Order Approximation to a Percentage Point of the Distribution of a Noncentral t-Statistic Without the Normality Assumption

Noncentral distributions appear in two sample problems and are often used in several fields, for example, in biostatistics. A higher order approximation for a percentage point of the noncentral t-distribution under normality is given by Akahira (1995) and is also shown to be numerically better than others. In this article, without the normality assumption, we obtain a higher order approximation to a percentage point of the distribution of a noncentral t-statistic, in a similar way to Akahira (1995) where the statistic based on a linear combination of a normal random variable and a chi-statistic takes an important role. Its application to the confidence limit and the confidence interval for a noncentrality parameter are also given. Further, a numerical comparison of the higher order approximation with the limiting normal distribution is done and the former one is shown to be more accurate. As a result of the numerical calculation, the higher order approximation seems to be useful in practical situations, when the size of sample is not so small.