带多余位置一刻度参数的非参数两样本问题

<正> 设x_1,…,x_m,y_1,…,y_n为m+n个独立随机变量,x_i有连续分布函数F,y_i有连续分布函数G.要依据这些观测值x_i,i=1,…,m,y_i,j=1,…,n来检验假设(注意F和G都是未知的) H_o:存在常数A和B,B>0,致     

半参数回归模型的几乎无偏岭估计

提出了半参数回归模型的几乎无偏岭估计,并与岭估计进行了比较,在均方误差意义下,几乎无偏岭估计优于岭估计. 然后讨论了有偏参数的选取问题. 最后,用模拟算例和实际应用说明了几乎无偏岭估计的有效性和可行性.     

Asymptotically distribution free test for parameter change in a diffusion process model

A test procedure to detect a change in the value of the parameter in the drift of a diffusion process is proposed. The test statistic is asymptotically distribution free under the null hypothesis that the true parameter does not change. Also, the test is shown to be consistent under the alternative that there exists a change point.     

The new test criterion for the homogeneity of parameters of several populations

Summary The new test criterion for testing the homogeneity of parameters of several populations is proposed and the test properties of it is discussed. The asymptotic expansions of the distributions of test criterion are discussed under (i) null hypothesis, (ii) fixed alternative hypothesis and (iii) local alternative hypothesis converging to the null hypothesis with appropriate rate of convergence as the sample size increases. As a particular case the asymptotic theory of a statistic for a homogeneity of variances of normal populations is also discussed and the exact moments of it under a null hypothesis can be used to obtain a percentage point by a Pearsonian curve fitting. This Institute of statistical Mathematics     

假设检验中的原假设选取问题

针对假设检验中"对一个具体问题,在相同的显著性水平α下进行假设检验,如果交换原假设和备择假设,得到的结论可能完全相反"这一问题进行分析,并给出根据实际问题的要求如何正确建立原假设和备择假设的方法.     

The impact of sampling methods on bias and variance in stochastic linear programs

Stochastic linear programs can be solved approximately by drawing a subset of all possible random scenarios and solving the problem based on this subset, an approach known as sample average approximation (SAA). The value of the objective function at the optimal solution obtained via SAA provides an estimate of the true optimal objective function value. This estimator is known to be optimistically biased; the expected optimal objective function value for the sampled problem is lower (for minimization problems) than the optimal objective function value for the true problem. We investigate how two alternative sampling methods, antithetic variates (AV) and Latin Hypercube (LH) sampling, affect both the bias and variance, and thus the mean squared error (MSE), of this estimator. For a simple example, we analytically express the reductions in bias and variance obtained by these two alternative sampling methods. For eight test problems from the literature, we computationally investigate the impact of these sampling methods on bias and variance. We find that both sampling methods are effective at reducing mean squared error, with Latin Hypercube sampling outperforming antithetic variates. For our analytic example and the eight test problems we derive or estimate the condition number as defined in Shapiro et al. (Math. Program. 94:1–19, 2002). We find that for ill-conditioned problems, bias plays a larger role in MSE, and AV and LH sampling methods are more likely to reduce bias.     

Testing for no effect via splines

A spline-based test statistic for a constant mean function is proposed based on the penalized residual sum-of-squares difference between the null model and a B-spline model in which the regression function is approximated with P-splines approach. When the number of knots is fixed, the limiting null distribution of the test statistic is shown to be the distribution of a linear combination of independent chi-squared random variables, each with one degree of freedom. A smoothing parameter is selected by setting a specified value equal to the expected value of the test statistic under the null hypothesis. Simulation experiments are conducted to study the proposed spline-based test statistic’s finite-sample properties.     

A bartlett-type adjustment for the likelihood ratio statistic with an ordered alternative

For testing the equality of normal variances with an increasing alternative, under the null hypothesis the likelihood ratio test statistic is asymptotically distributed as a mixture of chi-squared distributions. In this paper a Bartlett-type adjustment is proposed to improve the approximation of the null distribution of the likelihood ratio test statistic with an ordered alternative.     

On Tests of Symmetry Against One-Sided Alternatives

To test that an experimental treatment is better than an existing one (or control), one can equivalently consider the difference in their response and test if the distribution of the difference is symmetric (about zero) versus it exhibits positive bias (skewness to the right). In this paper, we test the symmetry (about zero) of a discrete distribution against two particular classes of one sided alternatives. We obtain the maximum likelihood estimators under each alternative. The asymptotic null distributions of the likelihood ratio statistics are shown to have chi-bar square type distributions. A power study is performed to compare these one-sided alternatives with other one-sided tests. The theory developed is illustrated by an example.     

Nonparametric likelihood ratio goodness-of-fit tests for survival data

Berk and Jones (Z. Wahrsch. Verw. Gebiete 47 (1979) 47) described a nonparametric likelihood test of uniformity that is more efficient, in Bahadur's sense, than any weighted Kolmogorov-Smirnov test at any alternative. This article shows how to obtain a nonparametric likelihood test of a general parametric family for incomplete survival data. A nonparametric likelihood ratio test process is employed to measure the discrepancy between a parametric family and the observed data. Large sample properties of the likelihood ratio test process are studied under both the null and alternative hypotheses. A Monte Carlo simulation method is proposed to estimate its null distribution. We show how to produce a likelihood ratio graphical check as well as a formal test of a parametric family based on the developed theory. Our method is developed for the right-censorship model, but can be easily extended to some other survival models. Illustrations are given using both real and simulated data.     

A novel trace test for the mean parameters in a multivariate growth curve model

A trace test for the mean parameters of the growth curve model is proposed. It is constructed using the restricted maximum likelihood followed by an estimated likelihood ratio approach. The statistic reduces to the Lawley-Hotelling trace test for the Multivariate Analysis of Variance (MANOVA) models. Our test statistic is, therefore, a natural extension of the classical trace test to GMANOVA models. We show that the distribution of the test under the null hypothesis does not depend on the unknown covariance matrix Σ. We also show that the distributions under the null and alternative hypotheses can be represented as sums of weighted central and non-central chi-square random variables, respectively. Under the null hypothesis, the Satterthwaite approximation is used to get an approximate critical point. A novel Satterthwaite type approximation is proposed to obtain an approximate power. A simulation study is performed to evaluate the performance of our proposed test and numerical examples are provided as illustrations.     

Inference of change-point in single index models

Single index models are widely used in medicine, econometrics and some other fields. In this paper, we consider the inference of a change point problem in single index models. Based on density-weighted average derivative estimation (ADE) method, we propose a statistic to test whether a change point exists or not. The null distribution of the test statistic is obtained using a permutation technique. The permuted statistic is rigorously shown to have the same distribution in the limiting sense under both null and alternative hypotheses. After the null hypothesis of no change point is rejected, an ADE-based estimate of the change point is proposed under assumption that the change point is unique. A simulation study confirms the theoretical results.     

A nonparametric lack-of-fit test for heteroscedastic regression models

A simple test is proposed for examining the correctness of a given completely specified response function against unspecified general alternatives in the context of univariate regression. The usual diagnostic tools based on residual plots are useful but heuristic. We introduce a formal statistical test supplementing the graphical analysis. Technically, the test statistic is the maximum length of the sequences of ordered (with respect to the covariate) observations that are consecutively overestimated or underestimated by the candidate regression function. Note that the testing procedure can cope with heteroscedastic errors and no replicates. Recursive formulae allowing one to calculate the exact distribution of the test statistic under the null hypothesis and under a class of alternative hypotheses are given.     

Two advanced methods for adjusting the main coefficient in logistic regression

A binary disease outcome is commonly modeled with continuous covariates (e.g., biochemical concentration) in medical research, and the corresponding exploration may employ a normal discrimination approach. The covariate relationship affects the estimated association between binary outcome and the interesting covariate. The method of value deviated from a fitted value (fractional polynomial), which is abbreviated as VDFV, may reduce the estimation bias especially when the relationship between the covariates is nonlinear. However, when the extraneous variable relates to the outcome, the pooled data (cases and controls) are replaced by the control data only for the purpose of fitting values. Based on two association patterns, the extraneous variable unrelated to the outcome (I) and that related to the outcome (II), the simulation study reveals that VDFV-p (using pooled data) is reliable, with less bias and a smaller mean square error (MSE) in pattern (I) and that VDFV-c (using control data) shows less bias in pattern (II). The conventional covariate adjustment performs worse in (I) but fairly well in (II). Note that a huge MSE is never observed in VDFV-p or VDFV-c, while this is a common issue related to small sample size or sparse data in logistic regression. Two fetal studies are illustrated—one for pattern (I) and one for pattern (II).     

Testing for additivity with B-splines

Regression splines are often used for fitting nonparametric functions, and they work especially well for additivity models. In this paper, we consider two simple tests of additivity: an adaptation of Tukey's one degree of freedom test and a nonparametric version of Rao's score test. While the Tukey-type test can detect most forms of the local non-additivity at the parametric rate of O(n-1/2), the score test is consistent for all alternative at a nonparametric rate. The asymptotic distribution of these test statistics is derived under both the null and local alternative hypotheses. A simulation study is conducted to compare their finite-sample performances with some existing kernel-based tests. The score test is found to have a good overall performance.     

Adaptive minimax testing in the discrete regression scheme

We consider the problem of testing hypotheses on the regression function from n observations on the regular grid on [0,1]. We wish to test the null hypothesis that the regression function belongs to a given functional class (parametric or even nonparametric) against a composite nonparametric alternative. The functions under the alternative are separated in the L₂-norm from any function in the null hypothesis. We assume that the regression function belongs to a wide range of Hölder classes but as the smoothness parameter of the regression function is unknown, an adaptive approach is considered. It leads to an optimal and unavoidable loss of order Open image in new window in the minimax rate of testing compared with the non-adaptive setting. We propose a smoothness-free test that achieves the optimal rate, and finally we prove the lower bound showing that no test can be consistent if in the distance between the functions under the null hypothesis and those in the alternative, the loss is of order smaller than the optimal loss.     

奇异超线性(k,n-k)多点边值问题的正解

应用锥理论和不动点指数方法,在与相应的线性算子第一特征值有关的条件下,获得奇异超线性(k,n-k)多点边值问题正解的存在性结果,推广和改进了已有的主要结论.     

Nonparametric tests for a change in the coefficient of variation

We consider the problem of testing the null hypothesis of no change against the alternative of exactly one change point when the change is expressed in terms of the value of the coefficient of variation. We propose a number of nonparametric test statistics for this problem. The asymptotic theory of the proposed tests is developed.      

Goodness-of-Fit Tests for Continuous Regression

A family of transformations of the processes of accumulated residues of linear models is used to construct tests of fit of the models, consistent for any alternative, and focused on alternatives in the direction selected by the user. The resulting tests are asymptotically distribution free, both under the null hypothesis of fit, and under the selected alternatives. An interesting feature is that this distributions do not depend on (possible) parameter estimations.     

Objective Bayesian model selection approach to the two way analysis of variance

An objective Bayesian procedure for testing in the two way analysis of variance is proposed. In the classical methodology the main effects of the two factors and the interaction effect are formulated as linear contrasts between means of normal populations, and hypotheses of the existence of such effects are tested. In this paper, for the first time these hypotheses have been formulated as objective Bayesian model selection problems. Our development is under homoscedasticity and heteroscedasticity, providing exact solutions in both cases. Bayes factors are the key tool to choose between the models under comparison but for the usual default prior distributions they are not well defined. To avoid this difficulty Bayes factors for intrinsic priors are proposed and they are applied in this setting to test the existence of the main effects and the interaction effect. The method has been illustrated with an example and compared with the classical method. For this example, both approaches went in the same direction although the large P value for interaction (0.79) only prevents us against to reject the null, and the posterior probability of the null (0.95) was conclusive.