Masafumi Akahira

A higher order approximation to a percentage point of the non–central t-distribution

A higher order approximation formula for a percentage point of the noncentral t–distribution with v degrees of freedom is given up to the order o(v-3), using the Cornish-Fisher expansion for the statistic based on a lin-ear combination of a normal random variable and a chi-random variable. The upper confidence limit and the confidence interval for the non–centrality parameter are given. Numerical results are also obtained.

Third order asymptotic efficiency of the sequential maximum likelihood estimation procedure

Under suitable regularity conditions, the third order asymptotic bounds for distributions of regular estimators are obtained. It is shown that the modified maximum likelihood estimation procedure combined with appropriate stopping rule is uniformly third order asymptotically efficient in the sense that its asymptotic distribution attains the bound uniformly in stopping rules up to the third order.

LOSS OF INFORMATION OF A STATISTIC FOR A FAMILY OF NON-REGULAR DISTRIBUTIONS

In the non-regular case, the asymptotic loss of amount of information (extended to as Rényi measure) associated with a statistic is discussed. It is shown that the second order asymptotic loss of information in reducing to a statistic consisting of extreme values and an asymptotically ancillary statistic vanishes. This result corresponds to the fact that the statistic is second order asymptotically sufficient in the sense of Akahira (1991, Metron, 49, 133–143). Some examples on truncated distributions are also given.

Discretized likelihood methods—asymptotic properties of discretized likelihood estimators (DLE's)

AbstractSuppose thatX1,X2, ...,Xn, ... is a sequence of i.i.d. random variables with a densityf(x, θ). Letcn be a maximum order of consistency. We consider a solution

Asymptotic optimality of estimators in non-regular cases

\hat \theta _n

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

of the discretized likelihood equation

On the asymptotic efficiency of estimators in an autoregressive process

\sum\limits_{i = 1}^n {\log f(X_i ,\hat \theta _n + rc_n^{ - 1} ) - } \sum\limits_{i = 1}^n {\log f(X_i ,\hat \theta _n ) = a_n (\hat \theta _n ,r)}