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列联表独立性的混合 Dirichlet 贝叶斯检验：稳健性与敏感性AI 解读

A mixed-Dirichlet prior was previously used to model the hypotheses of "independence" and "dependence" in contingency tables, thus leading to a Bayesian test for independence. Each Dirichlet has a main hyperparameter $\kappa$ and the mixing is attained by assuming a hyperprior for $\kappa$. This hyperparameter can be regarded as a flattening or shrinking constant. We here review the method, generalize it and check the robustness and sensitivity with respect to variations in…

离散数据的 M 估计：渐近分布理论及其含义AI 解读

The asymptotic distribution of an $M$-estimator is studied when the underlying distribution is discrete. Asymptotic normality is shown to hold quite generally within the assumed parametric family. When the specification of the model is inexact, however, it is demonstrated that an $M$-estimator whose score function is not everywhere differentiable, e.g., a Huber estimator, has a nonnormal limiting distribution at certain distributions, resulting in unstable inference in the…

线性模型的一步 L 估计量AI 解读

We propose and investigate the asymptotic properties of a class of estimators of the regression parameter in the general linear model. The estimators depend on a preliminary estimate of the regression parameter and the residuals based on it. For the location model, the estimators are linear combinations of the order statistics and the robustness and efficiency properties of this class of estimators carry over to the general linear model. The estimators settle the doubts…

回归分析中的异方差估计AI 解读

Consider the regression model $Y_i = g(t_i) + \varepsilon_i, 1 \leq i \leq n$, with nonrandom design variables $(t_i)$ and measurements $(Y_i)$ for the unknown regression function $g(\cdot)$. We assume that the data are heteroscedastic, i.e., $E(\varepsilon^2_i) = \sigma^2_i \not\equiv \operatorname{const.}$ and investigate how to estimate $\sigma^2_i$. If $\sigma^2_i = \sigma^2(t_i)$ with a smooth function $\sigma^2(\cdot)$, initial estimators $\tilde{\sigma}^2_i$ can be…

通过非参数估计构造的一类线性回归参数估计量AI 解读

Given a $(p + 1)$-dimensional random vector $(X, Y)$ where $f$ is the unknown density of $X$, the parameters of the multiple linear regression function $\alpha(x) = E(Y/X = x) = x\beta$ may be estimated from a sample $\{(X_1, Y_1), \cdots, (X_n, Y_n)\}$ by minimizing the functional $\hat{\psi}(\beta) = \int(\hat{\alpha}_n(x) - x\beta)^2\hat{f}_n(x) dx$, where $\hat{\alpha}_n$ and $\hat{f}_n$ may be any of a large class of nonparametric estimators of $\alpha$ and $f$. The…

最小二乘估计的新方法及其应用AI 解读

The regression model $\mathbf{y} = g(\mathbf{x}) + \mathbf{\varepsilon}$ and least-squares estimation are studied in a general context. By making use of empirical process theory, it is shown that entropy conditions on the class $\mathscr{G}$ of possible regression functions imply $L^2$-consistency of the least-squares estimator $\hat{\mathbf{g}}_n$ of $g$. This result is applied in parametric and nonparametric regression.

随机解释变量下单调回归最小二乘估计的强一致性AI 解读

In this paper it is shown that in the monotone regression model the unknown regression function can be consistently estimated by the least-squares method.

回归自适应最大似然估计的 Monte Carlo 证据AI 解读

This paper reports Monte Carlo evidence on the fixed sample size properties of adaptive maximum likelihood estimates of a linear regression. The focus is on the problem of selecting the smoothing and trimming parameters used in estimating the score function. We examine the performance of adaptive maximum likelihood estimators when these parameters are preselected or, alternatively, are determined by a data-based bootstrap method.

变量测量误差模型中的有效估计AI 解读

We consider efficient estimation of the slope in the errors in variables model with normal error when either the ratio of error variances is known and the distribution of the independent is arbitrary and unknown or the distribution of the independent variable is not Gaussian or degenerate. We calculate information bounds and exhibit estimates achieving these bounds using an initial minimum distance estimate and suitable estimates of the efficient score function.

使用估计得分函数的线性秩统计量的局部渐近分析AI 解读

Modified versions of linear rank statistics $S_N(\varphi)$ with score function $\varphi$ are studied for the two-sample testing problem of randomness. Depending on the unknown underlying alternatives, some score function $\varphi = b$ is known to be approximately optimal. Behnen, Neuhaus and Ruymgaart (1983) proposed estimating $b$ by some estimator $\hat b_N$ obtained by the kernel method based on ranks and used the quadratic rank statistic $S_N(\hat{b}_N)$ for testing the…

乘法强度模型中的筛法极大似然估计AI 解读

For point processes comprising i.i.d. copies of a multiplicative intensity process, it is shown that even though log-likelihood functions are unbounded, consistent maximum likelihood estimators of the unknown function in the stochastic intensity can be constructed using the method of sieves. Conditions are given for existence and strong and weak consistency, in the $L^1$-norm, of suitably defined maximum likelihood estimators. A theorem on local asymptotic normality of…

U 过程的收敛速率AI 解读

This paper introduces a new stochastic process, a collection of $U$-statistics indexed by a family of symmetric kernels. Conditions are found for the uniform almost-sure convergence of a sequence of such processes. Rates of convergence are obtained. An application to cross-validation in density estimation is given. The proofs adapt methods from the theory of empirical processes.

Suppose one is able to observe sequentially a series of independent observations $X_1, X_2, \ldots$, such that $X_1, X_2, \ldots, X_{\nu - 1}$ are i.i.d. with known density $f_0$ and $X_\nu, X_{\nu + 1}, \ldots$ are i.i.d. with density $f_\theta$ where $\nu$ is unknown. Define $R(n, \theta) = \sum^n_{k = 1} \prod^n_{i = k} \frac{f_\theta(X_i)}{f_0(X_i)}.$ It is known that rules, which call for stopping and raising an alarm the first time $n$ that $R(n, \theta)$ or a mixture…

二乘二列联表的合并及其几何性质AI 解读

If a pair of two-by-two contingency tables are amalgamated by addition it can happen that a measure of association for the amalgamated table lies outside the interval between the association measures of the individual tables. We call this the amalgamation paradox and we show how it can be avoided by suitable designs of the sampling experiments. We also study the conditions for the "homogeneity" of two subpopulations with respect to various measures of association. Some of…

兼具高崩溃点与高效率的稳健回归估计AI 解读

A class of robust estimates for the linear model is introduced. These estimates, called MM-estimates, have simultaneously the following properties: (i) they are highly efficient when the errors have a normal distribution and (ii) their breakdown-point is 0.5. The MM-estimates are defined by a three-stage procedure. In the first stage an initial regression estimate is computed which is consistent robust and with high breakdown-point but not necessarily efficient. In the…

Fieller解的一种刻画AI 解读

We consider the problem of finding $\alpha$-level confidence intervals for the ratio of two normally estimated means. We show that there is no procedure that with probability 1 gives bounded $\alpha$-level confidence intervals for the ratio, and we show that within a large class of sensible procedures the Fieller solution is the only one with exact coverage probability.

扩展 Paulson 序贯选择方法AI 解读

Principles of multiple comparisons with the best treatment are used to construct a sequential procedure for selecting one of $k$ normal populations with equal, known variance. The procedure makes a stronger confidence than traditional Indifference Zone procedures, without (asymptotically) inflating sample sizes. It is shown to be substantially superior in asymptotic sample size behavior to existing procedures making the stronger statement.

将单调回归假定为线性的代价AI 解读

For jointly distributed random variables $(X, Y)$ having marginal distributions $F$ and $G$ with finite second moments and $F$ continuous, the proportion of $\operatorname{Var}(Y)$ explained by linear regression is $\lbrack\operatorname{Corr}(X, Y)\rbrack^2$ while the proportion explained by $E(Y \mid X)$ can be arbitrarily near 1. However, if the true regression, $E(Y\mid X)$, is monotone, then the proportion of $\operatorname{Var}(Y)$ it explains is at most $\operatorname{Corr}\lbrack Y, G^{-1}(F(X))\rbrack$…

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Hájek 表示定理的一种替代正则条件AI 解读

Hajek's representation theorem states that under certain regularity conditions the limiting distribution of an estimator can be written as the convolution of a certain normal distribution with some other distribution. This result, originally developed for finite dimensional problems, has been extended to a number of infinite dimensional settings where it has been used, for example, to establish the asymptotic efficiency of the Kaplan-Meier estimator. The purpose of this note…

利用随机删失数据检验寿命分布的“新优于旧”性质AI 解读

Under a model of random censorship, we consider the test $H_0$: a life distribution is exponential, versus $H_1$: it is new better than used, but not exponential. This paper introduces a class of tests by using the Kaplan-Meier estimator for the sample distribution in the uncensored model. Under some regularity conditions, the asymptotic normality of statistics is derived by an application of von Mises' method, and asymptotically valid tests are obtained by using estimators…

若干修正 Shapiro–Wilk 统计量的渐近等价性：完整与删失样本AI 解读

The Shapiro-Wilk statistic and its modifications are widely applied in tests for normality. We establish the asymptotic equivalence of a class of statistics based on different choices of normal scores. In particular, we conclude that the Shapiro-Francia, Filliben, Weisberg-Bingham and de Wet-Venter versions of the statistic are asymptotically equivalent. Our results also apply to the Type I and Type II censored data cases.

代入参数估计量对卡方极限 U 与 V 统计量的影响AI 解读

Consider statistics $T_n(\lambda)$ that take the form of limiting chi-square (degenerate) $U$ or $V$ statistics. Here the phrase "limiting chi-square" means they have the same asymptotic distribution as a weighted sum of (possibly infinitely many) independent $\chi^2_1$ random variables. This paper examines the limiting distribution of $T_n(\hat{\lambda})$ and compares it to that of $T_n(\lambda)$, where $\hat{\lambda}$ denotes a consistent estimator of $\lambda$ based on…

For testing the goodness-of-fit of a $\log$-linear model to a multi-way contingency table with cell proportions estimated from survey data, Rao and Scott (1984) derived a first-order correction, $\delta\ldot$, to Pearson chi-square statistic, $X^2$ (or the likelihood ratio statistic, $G^2$) that takes account of the survey design. It was also shown that $\delta\ldot$ requires the knowledge of only the cell design effects (deffs) and the marginal deffs provided the model…