Linex Unbiasedness in a Prediction Problem

A statistical prediction problem under LINEX loss function is considered. Some results about LINEX-unbiased predictor are derived and the best LINEX-unbiased predictor is given. We also show that the best risk-unbiased predictor is equal to the best equivariant predictor in the location family.     

Information Criteria for Small Diffusions via the Theory of Malliavin–Watanabe

Information criteria based on the expected Kullback–Leibler information are presented by means of the asymptotic expansions derived with the Malliavin calculus. We consider the evaluation problem of statistical models for diffusion processes with small noise. The correction terms are essentially different from the ones for ergodic diffusion models presented in Uchida and Yoshida [34, 35].     

同源密度函数方差估计值

本文提出同源密度函数方差估计值。它是依据每个随机变量函数方差的近似公式由条件死亡概率方差和同源生存率方差估计值推导出来的。其数值、置信限平均宽度和经验覆盖在各种极端临床条件下均与Greenwood密度函数方差估计值相等或相近 ,而计算大大简化。由此我们认为同源密度函数方差估计值可以取代Greenwood估计值     

A third-order optimum property of the maximum likelihood estimator

Let θ⁽ⁿ⁾ denote the maximum likelihood estimator of a vector parameter, based on an i.i.d. sample of size n. The class of estimators θ⁽ⁿ⁾ + n^?1q(θ⁽ⁿ⁾), with q running through a class of sufficiently smooth functions, is essentially complete in the following sense: For any estimator T⁽ⁿ⁾ there exists q such that the risk of θ⁽ⁿ⁾ + n^?1q(θ⁽ⁿ⁾) exceeds the risk of T⁽ⁿ⁾ by an amount of order o(n^?1) at most, simultaneously for all loss functions which are bounded, symmetric, and neg-unimodal. If $\text{q}{}^1$ is chosen such that $\text{θ}{}^{\mathrm{(n)}}\text{+ n}{}^{\mathrm{?1}}\text{q}{}^1\text{(θ}{}^{\mathrm{(n)}}\text{)}$ is unbiased up to $\text{o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$, then this estimator minimizes the risk up to an amount of order o(n^?1) in the class of all estimators which are unbiased up to $\text{o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$.The results are obtained under the assumption that T⁽ⁿ⁾ admits a stochastic expansion, and that either the distributions have—roughly speaking—densities with respect to the lebesgue measure, or the loss functions are sufficiently smooth.     

A definition of estimator efficiency ink-parameter case

Summary The paper introduces a new definition of efficiency in the multiparameter case (θ₁,...,θ_k) when the variance-covariance matrix of the vector estimator (t ₁, ...t _k) exists. The definition is also applicable to the asymptotically unbiased estimators. The basic idea is that, as we want in general to estimate some function g(θ₁,...θ_k) of the parameters, efficiency of the vector estimator shall be defined as the smallest efficiency of the estimatorg(t ₁, ...t _k),g being regular. It is shown that this definition is asymptotically equivalent to the one obtained by any linear combination of the estimators, as it happens, naturally, for quantile estimation in the location-dispersion case. This efficiency is larger than Cramér efficiency which is, thus, not attained, apart from a very exceptional case. Finally, a lower bound for the asymptotic variance is obtained.     

Two testing problems relating the real and complex multivariate normal distributions

The first problem considered is that of testing for the reality of the covariance matrix of a p-dimensional complex normal distribution, while the second is that of testing that a 2p-dimensional real normal distribution has a p-dimensional complex structure. Both problems are reduced by invariance to their maximal invariant statistics, and the null and non-null distributions of these are obtained. Complete classes of unbiased, invariant tests are described for both problems, the locally most powerful invariant tests are obtained, and the admissibility of the likelihood ratio tests is established.     

On monotonicity of the modified likelihood ratio test for the equality of two covariances

For testing the hypothesis of equality of two covariances (Σ₁ and Σ₂) of two p-dimensional multivariate normal populations, it is shown that the power function of the modified likelihood ratio test increases as λ₁ increases from one and λ_r decreases from one where λ₁ > … > λ_r > 0 are the distinct characteristic roots of Σ₁Σ₂^?1, r ≤ p. As a by-product we get the unbiased result already established by Sugiura and Nagao (1968).     

关联实验数据的4种x2形式及其性质

研究了关联实验数据的四种因子化形式的x2表达式及其极小化性质,同时讨论了极小化估计值的有偏性与无偏性.利用简化的R值测量定量地检验了相关的理论结果.     

δ修正Cramér-von Mises检验的非无偏性

在简单假设下,证明了对任意给定水平α∈(0,1)和样本容量n,δ修正Cramér-von Mises检验的非无偏性.