We will consider a stochastic expansion described by random variables whose covariance matrix is asymptotically degenerate. Though the conventional approach with Bhattacharya-Ghosh's transform requires the nondegeneracy of the covariance matrix, it is known that this method still works even in degenerate cases with the help of the so-called global approach. In this paper, we explain this fact and also mention, as an example, the third order asymptotic expansion of the maximum likelihood estimator for the O-U process.

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We consider an efficient estimation of an unknown parameter appearing in both the drift and the diffusion coefficient for a d-dimensional dynamical system with small noise. Asymptotic properties of an M-estimator obtained from an approximate quadratic martingale estimating function are stated. The sample path is observed at equidistant times k/n, k = 0,1,…, n. The type of asymptotics considered is when a small dispersion parameter ε goes to 0 and n goes to ∞ simultaneously.

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In this paper we consider the time-varying parameter model. Since there is no uniformly most powerful test for the constancy of parameters, the locally best invariant (LBI) test has often been considered in the literature, including the study by Nabeya and Tanaka (1988). We show the existence of the limiting distribution of the point optimal invariant (POI) test statistic when we can derive the limiting distribution of the LBI test statistic. We prove that the limiting characteristic function of the POI test statistic can be expressed using that of the LBI test statistic.

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In this paper, based on the assumption that the four-factor and higher-order interactions are to be negligible, we consider a balanced fractional 2^m factorial design derived from a simple array such that all the main effects are estimable, i.e., resolution R^*({1}|3). In this situation, using the algebraic structure of the triangular multidimensional partially balanced association scheme and a matrix equation, we can get designs of four types of resolutions: the first is of resolution R({1}|3), the second is of resolution R({0,1}|3), the third is of resolution R({1,2}|3), i.e., resolution VI, and the last is of resolution R({0,1,2}|3), i.e., resolution VI. This paper gives the characterization of designs mentioned above, and also it gives optimal designs with respect to the generalized A-optimality criterion for 6 ≤ m ≤ 8 when the number of assemblies is less than the number of non-negligible factorial effects.

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This paper uses the concept of characterized model for bivariate extensions of univariate life distributions based on mean residual life properties. Different bivariate distributions can be generated from different choices of marginal distributions. The retention of univariate IMRL, DMRL, NBUE, NWUE, HNBUE and HNWUE class properties in the bivariate setup has been ensured along with results of importance for reliability analysis. A characterization of the exponential, Lomax and finite range distributions has been obtained in this process.

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As a typical non-normal case, we consider a family of elliptically symmetric distributions. Then, the moment parameter and its consistent estimator are presented. Also, the asymptotic expectation and the asymptotic variance of the consistent estimator of the general moment parameter are given. Besides, the numerical results obtained by Monte Carlo simulation for some selected parameters are provided.

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We consider a sequential point estimation of the ratio of two exponential scale parameters. For a fully sequential sampling scheme, second order approximations are obtained to the expected sample size and the risk of the sequential procedure. We also propose a bias-corrected procedure to reduce the risk.

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In the multivariate mixed linear model or multivariate components of variance model with equal replications, this paper addresses the problem of predicting the sum of the regression mean and the random effects. When the feasible best linear unbiased predictors or empirical Bayes predictors are used, this prediction problem reduces to the estimation of the ratio of two covariance matrices. We propose scale equivariant shrinkage estimators for the ratio of the two covariance matrices. Their dominance properties over the usual estimators including the unbiased one are established, and further domination results are shown by using information of order restriction between the two covariance matrices. It is also demonstrated that the empirical Bayes predictors that employ these improved estimators of the ratio of the two covariance matrices have uniformly smaller risks than the crude Efron-Morris type estimator in the context of estimation of a mean matrix in a fixed effects linear regression model where the components are unknown parameters.

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