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凹损失下球对称分布位置参数的极小极大估计AI 解读

For $p \geqslant 4$ and one observation $X$ on a $p$-dimensional spherically symmetric distribution, minimax estimators of $\theta$ whose risks are smaller than the risk of $X$ (the best invariant estimator) are found when the loss is a nondecreasing concave function of quadratic loss. For $n$ observations $X_1, X_2, \cdots, X_n$, we have classes of minimax estimators which are better than the usual procedures, such as the best invariant estimator, $\bar{X}$, or a maximum…

配对关系丢失的随机样本中相关系数的估计AI 解读

Inference about the correlation coefficient $\rho$ in a bivariate normal distribution is considered when observations from the distribution are available only in the form of a broken random sample. In other words, a random sample of $n$ pairs is drawn from the distribution but the observed data are only the first components of the $n$ pairs and, separately, some unknown permutation of the second components of the $n$ pairs. Under these conditions, the estimation of $\rho$ is,…

多个协方差矩阵及多元正态总体相等的似然比检验无偏性AI 解读

Random samples of size $N_\alpha$ are taken from the $p$-variate normal populations $N_p(\mu^{(\alpha)}, \Sigma^{(\alpha)}), 1 \leqslant \alpha \leqslant k$, with $\mu^{(\alpha)}$ and $\Sigma^{(\alpha)}$ unknown. Bartlett's modification of the likelihood ratio test (LRT) for the hypothesis $H_1:\Sigma^{(1)} = \cdots = \Sigma^{(k)}$ rejects $H_1$ for large values of $|S|^n/\Pi|S^{(\alpha)}|^{n_\alpha}$, where $S = \Sigma S^{(\alpha)}, n_\alpha = N_\alpha - 1, n = \Sigma n_i$,…

非参数回归中窗口估计的一致性AI 解读

Stone proved that nearest neighbor estimates of a nonparametric regression function are "universally" consistent. We show that the same holds for window estimates, and we obtain a rate of convergence under some restrictions.

Let $(X, Y)$ be an $\mathbb{R}^d \times \mathbb{R}$-valued random vector and let $(X_1, Y_1), \cdots, (X_n, Y_n)$ be a random sample drawn from its distribution. We study the consistency properties of the kernel estimate $m_n(x)$ of the regression function $m(x) = E\{Y\mid X = x\}$ that is defined by $m_n(x) = \Sigma^n_{i=1} Y_ik((X_i - x)/h_n)/\Sigma^n_{i=1}k((X_i - x)/h_n)$ where $k$ is a bounded nonnegative function on $\mathbb{R}^d$ with compact support and $\{h_n\}$ is…

基于删失数据的分布函数经验贝叶斯估计札记AI 解读

Susarla and Van Ryzin exhibited an empirical Bayes estimator of a distribution function $F$ based on randomly right-censored observations. In a later paper they obtained a different estimator which alleviates the weaknesses of their earlier estimator and showed that it is asymptotically optimal with rate of convergence $n^{-1}$. The purpose of this note is to present a slightly different estimator which is simpler and is also asymptotically optimal with the same rate of…

两个正态均值之差的序贯点估计AI 解读

A sequential procedure for estimating the difference of two normal means when the variances are unknown and not necessarily equal is proposed, and an asymptotic expression for "regret" is given. This generalizes the corresponding one sample result of Woodroofe.

通过线性回归刻画指数分布及相关分布AI 解读

Let $X_1, \cdots, X_n (n \geqslant 2)$ be a random sample on a rv $X$, and $Y_1 < \cdots < Y_n$ be the corresponding order statistics. Define $Z_k = \frac{1}{n-k}\Sigma^n_{i=k+1}(Y_i - Y_k), 1 \leqslant k \leqslant n - 1, W_k = \frac{1}{k-1}\Sigma^{k-1}_{i=1}(Y_k - Y_i),$ $2 \leqslant k \leqslant n$. Using the properties $E(Z_k\mid Y_k = y) = \alpha y + \beta$ and $E(W_k\mid Y_k = y) = \alpha y + \beta$, a.e. $(dF)$, where $\alpha$ and $\beta$ are constants, we obtain…

两个正态总体共同均值的 Graybill–Deal 估计量方差与分布AI 解读

Let $x$ and $y$ be two independent normal variables with mean $\mu$ and variances $\sigma^2_1$ and $\sigma^2_2$ respectively. Also let $S^2_1$ and $S^2_2$ be two independent estimators of $\sigma^2_1$ and $\sigma^2_2$ such that $mS^2_1\sigma^{-2}_1$ and $nS^2_2\sigma^{-2}_2$ are chi-squares with $m$ and $n$ degrees of freedom respectively. The Graybill-Deal estimator of $\mu$ is $\hat{\mu} = (S^{-2}_1x + S^{-2}_2y)/(S^{-2}_1 + S^{-2}_2)$. In this paper an expression for the…

共同均值估计与区组间信息恢复AI 解读

Consider the problem of combining two unbiased estimators of a parameter when the estimators are known to be independent normal variables with unknown variances possibly unequal. The two one parameter families of estimators studied in Brown and Cohen, and Khatri and Shah, are accommodated in a single two parameter family studied in this paper and the results in the two papers are unified. For the type of estimators considered by Brown and Cohen, this paper not only offers a…

正相依诱导的随机序与渐近检验水平的单调性AI 解读

An ordering of distributions related to a concept of positive dependence is studied and stochastic monotonicity with respect to this ordering is established for a wide class of two-sample test statistics. Asymptotic conservativeness of test sizes under certain departures from independence between samples is discussed. For example, if the observations are paired and the joint density is positive semidefinite then tests such as Kolmogorov-Smirnov, $\chi^2$ and Cramer-von…

Dynkin恒等式在泊松过程速率的贝叶斯序贯估计中的应用AI 解读

Conditional on the value of $\theta, \theta > 0$, let $X(t), t \geqslant 0$, be a homogeneous Poisson process. Sequential estimation procedures of the form $(\sigma, \theta(\sigma))$ are considered. To measure loss due to estimation, a family of functions, indexed by $p$, is used: $L(\theta, \hat{\theta}) = \theta^{-p}(\theta - \hat{\theta})^2$, and the cost of sampling involves cost per arrival and cost per unit time. The notion of "monotone case" for total cost functions…

自回归模型中最小二乘估计量的渐近性质AI 解读

Consistency and asymptotic normality of least-squares estimators are discussed for the linear autoregressive model with explanatory variables. Few assumptions are made about the error sequence. The case of stochastic explanatory variables is also considered.

二次型检验的效率与最优性AI 解读

Let $Z_1, Z_2, \cdots$ be i.i.d. standard normal variables. Results are obtained which relate to the tail behavior as $x \rightarrow \infty$ of distributions of the form $F(x) = P\{\sum^\infty_{k=1}\lambda_k\lbrack(Z_k + a_k)^2 - 1\rbrack \leqslant x\}$. For test statistics which have such limiting distributions $F$, asymptotic relative efficiency measures are discussed. One of these is the limiting approximate Bahadur efficiency. Applications are to tests of fit and tests…

从稳健检验导出的估计量AI 解读

In this paper, an asymptotic minimax theory for robust estimation of a one-dimensional parameter is derived, which is an asymptotic counterpart, and generalization to an arbitrary parameter, of Huber's finite sample minimax theory for the location case. A particular variability measure and results from robust asymptotic testing are employed. The results show a relationship of this approach to Hampel's local theory of robustness.

某些非参数模型中的分位数估计AI 解读

The deficiency of sample quantiles with respect to quasiquantiles is investigated under the assumption that the true density function has bounded derivatives. Then the sample quantile is still an efficient estimator of the true quantile but the relative deficiency of sample quantiles with respect to suitably defined quasiquantiles quickly tends to infinity for increasing sample sizes. If the second derivative of the true density function is bounded, then adaptive estimators…

非参数多重比较程序的前 k−1 组显著性水平AI 解读

We consider the nonparametric pairwise comparisons procedures derived from the Kruskal-Wallis $k$-sample test and from Friedman's test. For large samples the $(k - 1)$-mean significance level is determined, i.e., the probability of concluding incorrectly that some of the first $k - 1$ samples are unequal. We show that in general this probability may be larger than the simultaneous significance level $\alpha$. Even when the $k$th sample is a shift of the other $k - 1$ samples,…

自适应多元岭回归AI 解读

A multivariate version of the Hoerl-Kennard ridge regression rule is introduced. The choice from among a large class of possible generalizations is guided by Bayesian considerations; the result is implicitly in the work of Lindley and Smith although not actually derived there. The proposed rule, in a variety of equivalent forms is discussed and the choice of its ridge matrix considered. As well, adaptive multivariate ridge rules and closely related empirical Bayes procedures…

利用协变量的均值不变检验AI 解读

We consider the problem of testing a hypothesis about the means of a subset of the components of a multivariate normal distribution with unknown covariance matrix, when the means of a second subset (the covariates) are known. Because of the possible correlation between the two subsets, information provided by the second subset can be useful for inferences about the means of the first subset. In this paper attention is restricted to the class of procedures invariant under the…

对称统计量的渐近分布AI 解读

Sequences of $m$th order symmetric statistics are examined for convergence in law. Under appropriate conditions, a limiting distribution exists and is equivalent to that of a linear combination of products of Hermite polynomials of independent $N(0, 1)$ random variables. Connections with the work of von Mises, Hoeffding, and Filippova are noted.

线性过程参数估计中模型阶数的渐近有效选择AI 解读

Let $\{x_t\}$ be a linear stationary process of the form $x_t + \Sigma_{1\leqslant i<\infty}a_ix_{t-i} = e_t$, where $\{e_t\}$ is a sequence of i.i.d. normal random variables with mean 0 and variance $\sigma^2$. Given observations $x_1, \cdots, x_n$, least squares estimates $\hat{a}(k)$ of $a' = (a_1, a_2, \cdots)$, and $\hat{\sigma}^2_k$ of $\sigma^2$ are obtained if the $k$th order autoregressive model is assumed. By using $\hat{a}(k)$, we can also estimate coefficients of…

位置参数的最节省样本稳健选择程序AI 解读

Consider samples of size $n$ from each of $k$ symmetric populations, differing only in their location parameters. The decision problem is to select the best population--the one with the largest location parameter--with control on the probability of correct selection (PCS) whenever the largest parameter is at least $\Delta$ units larger than all others, and whenever the common error distribution belongs to a specified neighborhood of the standard normal. It is shown that, if…

非平稳时间序列中的信号提取误差AI 解读

It is supposed that an observable time series $\{x_t\}$ is representable as the sum of a "signal" $s_t$ and a "noise" $n_t$, and that it is desired to extract the signal $s_t$, i.e., to obtain an estimate $\hat{s}_t$ of $s_t$. Corresponding to any such estimate is a signal extraction error, $\delta_t = s_t - \hat{s}_t$, which for nondeterministic stochastic processes possesses a nonzero mean square. For the class of homogeneously nonstationary processes, characterizations of…

逆协方差矩阵估计：逆 Wishart 矩阵与单位矩阵的随机混合AI 解读

Let $S_{p \times p}$ have a nonsingular Wishart distribution with unknown matrix $\Sigma$ and $k$ degrees of freedom. For two different loss functions, estimators of $\Sigma^{-1}$ are given which dominate the obvious estimators $aS^{-1}, 0 < a \leqslant k - p - 1$. Our class of estimators $\mathscr{C}$ includes random mixtures of $S^{-1}$ and $I$. A subclass $\mathscr{C}_0 \subset \mathscr{C}$ was given by Haff. Here, we show that any member of $\mathscr{C}_0$ is dominated…