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关于经验贝叶斯估计的若干思考AI 解读

Examples are given to illustrate the "linear" and "general" empirical Bayes approaches to estimation. A final example concerns testing the null hypothesis that a treatment has had no effect.

We study the asymptotic accuracy of the bootstrap approximation to the distribution of a $k$-sample studentized mean.

利用拉格朗日乘数检验估计ARMA模型阶数AI 解读

A stepwise testing procedure using Lagrangian multiplier tests is developed to determine the order of an ARMA-process. The strong consistency of this procedure under slightly weaker assumptions than in Hannan (1980) (proof of the consistency of the order estimators obtained via BIC) is proved.

对称统计量、泊松点过程与多重维纳积分AI 解读

The asymptotic behaviour of symmetric statistics of arbitrary order is studied. As an application we describe all limit distributions of square integrable $U$-statistics. We use as a tool a randomization of the sample size. A sample of Poisson size $N_\lambda$ with $EN_\lambda = \lambda$ can be interpreted as a Poisson point process with intensity $\lambda$, and randomized symmetric statistics are its functionals. As $\lambda \rightarrow \infty$, the probability distribution…

大规模有限总体中的逐次抽样AI 解读

The permutation distribution induced upon a finite population by the order of selection under successive sampling is closely related to the order statistics of independent exponentially distributed waiting times. This characterization is applied to obtain necessary and sufficient conditions for asymptotic normality of the sum of characteristics observed in a successive sample from a finite population. The necessary and sufficient conditions generalize previous results for…

截断回归斜率的 K–S 型检验之理论依据AI 解读

A $K - S$ type statistic computed from sequential ranks has been proposed in the astrophysics literature for testing the slope of a truncated regression. There is an easy heuristic justification for the test in the nontruncated case, but it fails under truncation. This paper extends the heuristic justification to the truncated case and outlines a more complete proof of the asymptotic property.

通过线性组合两个给定统计量进行估计AI 解读

We consider the problems of (i) covariance adjustment of an unbiased estimator, (ii) combining two unbiased estimators, and (iii) improving upon an unbiased estimator. All these problems consist in determining a minimum dispersion linear unbiased combination of given two statistics, one of which is an unbiased estimator of a vector parameter $\mathbf{\theta} \in \mathscr{H}$, and the expectation of the other is a zero vector in the problem of covariance adjustment, is equal…

位置参数 L 估计量与 M 估计量的渐近关系及效率AI 解读

This paper gives conditions under which $L$- and $M$-estimators of a location parameter have asymptotically the same distribution, conditions under which they are asymptotically equivalent and conditions under which $L$-estimators are asymptotically efficient relative to the Cramer-Rao lower bound. Our results differ from analogous results of Jung (1955), Bickel (1965), Chernoff, Gastwirth, Johns (1967), Jaeckel (1971) and Rivest (1978, 1982) in that we do not require that…

一般多元方差分析模型中的同时区间估计AI 解读

Let $M \in M(m, p)$ be the matrix of means of interest in the GMANOVA problem. Our main results characterize all confidence sets for $M$ in a given class (invariant plus a weak additional restriction) that are exact for the families of parametric functions $a'Mb$ for all $a \in \mathbb{R}^m, b \in \mathbb{R}^p$ and $\operatorname{tr} N'M$ for all $N \in M(m, p)$. The corresponding families of smallest exact simultaneous confidence intervals are also given. Similar results…

Ferguson 分布函数贝叶斯估计量的稳健性AI 解读

We derive an explicit expression for the Bayes risk (using weighted squared error loss) of Dalal's Bayes estimator of a symmetric distribution under a $\mathscr{G}$-invariant Dirichlet process prior. We compare this risk to the risk of Ferguson's estimator of an arbitrary distribution under the $\mathscr{G}$-invariant prior. This enables us to (i) assess the savings in risk attained by incorporating known symmetry structure in the model and (ii) provide information about the…

置换检验功效的 Edgeworth 展开AI 解读

A randomization model for a two-sample situation with additive treatment effects is considered. Edgeworth expansions for the power of the usual permutation test are derived, under some conditions on the unit errors, from previously obtained expansions under the null hypothesis of no treatment effect. A general error structure is considered and conditions for the validity of the expansions for both conditional and unconditional power are examined. The results are shown to…

原假设下简单线性秩统计量的Edgeworth展开AI 解读

An Edgeworth expansion with remainder $o(N^{-1})$ is established for simple linear rank statistics under the null-hypothesis. The theorem is proved for a wide class of scores generating functions which includes the normal quantile function.

用 Sheffer 多项式计算 Takacs 拟合优度分布AI 解读

How often does the empirical distribution pass over an acceptance band around the hypothetical distribution function? Sheffer polynomials are applied to derive the exact distribution of this goodness-of-fit distribution and its two-sample analogue. This method leads to new proofs and extensions of Takacs's (1971) results.

作为估计得分线性秩检验的 Galton 检验及其局部渐近效率AI 解读

In the general two-sample testing problem $F = G$ versus $F \leq G, F \neq G$ the shift-model score function $f' \circ F^{-1}/f \circ F^{-1}$ has to be replaced by the nonparametric score function $b = \bar{f} - \bar{g}$, where $\bar{f} = d(F \circ H^{-1})/dx, \bar{g} = d(G \circ H^{-1})/dx, H = (mF + nG)/(m + n)$, and adaption of linear rank tests should be based on rank estimators of $b$. We consider an easy but rough and inconsistent rank estimator of $b$. The resulting…

经验分布正则泛函的分布与分位数展开及非参数置信区间AI 解读

Let $T(\cdot)$ be a suitably regular functional on the space of distribution functions, $F$, on $R^s$. A method is given for obtaining the derivatives of $T$ at $F$. This is used to obtain asymptotic expansions for the distribution and quantiles of $T(F_n)$ where $F_n$ is the empirical distribution of a random sample of size $n$ from a distribution $F$ with an absolutely continuous component. One- and two-sided confidence intervals for $T(F)$ are given of level $1 - \alpha + O(n^{-j/2})$…

Edgeworth 展开的反演AI 解读

We provide a method for inverting a general Edgeworth expansion, so as to correct a statistic for the effects of non-normality. This technique is applied to the special case of the "Studentized" mean. Explicit formulae are given for the correction terms.

未来收益贴现下股票市场的最优停止AI 解读

In the random walk stock market model, a stock is purchased at price $x$ and is sold at time $t$ for the price $x + S_t$ where $S_t = \sum^t_0 X_i, X_i$ is the price change during the $i$th epoch, and $X_1, X_2, \cdots$ are i.i.d. random variables with $\mu = E(X_1) > 0$ and finite $\sigma^2 = E(X^2_1) - \mu^2 > 0$. Discounting the future by a factor of $\gamma$ per epoch, $0 < \gamma < 1$, a selling or stopping policy $t$ has expected payoff or utility $u(t) = E\{\gamma^t(x + S_t)\}$…

抽样调查的极小极大方法AI 解读

Suppose that there is a population of $N$ identifiable units each with two associated values $x_i$ and $y_i$. All $N$ values of $x$ are given but $y_i$ is determined only after the $i$th unit is selected and observed. The objective is to estimate the population total $\sum^N_{i=1} y_i$. It is assumed that $y_i = \theta x_i + \delta_ig(x_i), 1 \leq i \leq N$, where $(\delta_1, \cdots, \delta_N)$ is in some bounded neighborhood of $(0, \cdots 0)$. The Rao-Hartley-Cochran and…

M 型平滑样条的渐近理论AI 解读

Limit theorems giving rates of convergence of nonparametric regression estimates obtained from smoothing splines are proved. The main emphasis is on nonlinear, robust smoothing splines, but new results are obtained for the usual (linear) case. It is assumed that the knots become asymptotically uniform in a vague sense. Convergence of derivatives is also investigated. The main mathematical tools are a linearization of the robust smoothing spline, and an approximation of the…

截断回归斜率的非参数估计AI 解读

A nonparametric estimate $\beta^\ast$ is presented for the slope of a regression line $Y = \beta_0X + V$ subject to the truncation $Y \leq y_0$. This model is relevant to a cosmological controversy which concerns Hubble's Law in Astronomy. The estimate $\beta^\ast$ corresponds to the zero-crossing of a random function $S_n(\beta)$, which for each $\beta$ is a Mann-Whitney type of statistic designed to measure heterogeneity among the calculated residuals $Y - \beta X$. The…

估计中的中偏差理论AI 解读

The performance of a sequence of estimators $\{T_n\}$ of $\theta$ can be measured by the probability concentration of the estimator in an $\varepsilon_n$-neighborhood of $\theta$. Classical choices of $\varepsilon_n$ are $\varepsilon_n = cn^{-1/2}$ (contiguous case) and $\varepsilon_n = \varepsilon$ fixed for all $n$ (non-local case). In this article all sequences $\{\varepsilon_n\}$ with $\lim_{n\rightarrow\infty} \varepsilon_n = 0$ and $\lim_{n\rightarrow\infty} \varepsilon_nn^{1/2} = \infty$…

混合设定下条件得分的效率AI 解读

The conditional score function is found to be generally fully informative concerning a parameter of interest when the conditioning statistic $S$ is sufficient for the nuisance parameter and has an exponential family distribution. Information is here measured by assuming the nuisance parameter to have been generated by an unknown mixing distribution and then computing the minimal Fisher information. The solution depends upon a study of the geometry of centered likelihood…

Min-Max 算法与保序回归AI 解读

There is growing interest in statistical inference under order restrictions. A major demand in this subject is to have a fast, direct method to solve the least squares problem of partially ordered isotonic regression. The Min-Max algorithm is such a method in which the user searches for the global minimum and the local maximum successively. A comparison of algorithms for partially ordered isotonic regression is included. As an application, using this efficient algorithm, it…

利用核函数平滑计数过程强度AI 解读

The kernel function method developed during the last twenty-five years to estimate a probability density function essentially is a way of smoothing the empirical distribution function. This paper shows how one can generalize this method to estimate counting process intensities using kernel functions to smooth the nonparametric Nelson estimator for the cumulative intensity. The properties of the estimator for the intensity itself are investigated, and uniform consistency and…