Asymptotic distributions in the projection pursuit based canonical correlation analysis

In this paper, associations between two sets of random variables based on the projection pursuit (PP) method are studied. The asymptotic normal distributions of estimators of the PP based canonical correlations and weighting vectors are derived.     

基于随机加权的空间聚集性检验及其在基础教育师资均衡性评价中的应用

使用空间统计检验方法研究北京基础教育资源分配的均衡性问题.对于空间分布均匀性的检验,常用的统计量是Moran's I统计量.但基于Moran's I统计量做推断的时候,人们往往用渐进正态分布或者用Bootstrap反复抽样得到经验分布来进行.提出使用随机加权法进行统计量的经验检验.Jin和Lee(2014)文中得出基于Bootstrap的Moran's I统计量满足一致逼近和渐进正态等性质.采用类似的统计工具证明了基于随机加权得到的统计量的渐进分布也满足这些良好性质.填补了用随机加权法在空间统计量的推断中理论保证的空白.通过模拟研究,证明了所提算法的有效性.方法应用于北京基础教育的师资-适龄儿童数比例,师资-在校生数比例的空间聚集性检验中得到了良好的应用,并与其它检验方法所得结论进行比较.结论显示在不同相邻概念(地理相邻、政策空间相邻)下,方法得到的结论符合常理.     

线性模型中M检验原假设分布的随机加权逼近

在线性模型中M-方法可以用于线性假设检验, 其中M检验、Wald检验和Rao的计分型检验是最常用的检验准则. 但是在计算这些检验的临界值时都涉及到未知参数的估计. 在本文中我们利用随机加权的方法来逼近这些检验的原假设分布. 结果表明在原假设和局部对立假设之下随机加权统计量的渐近分布与原检验统计量在原假设之下的渐近分布相同. 因此我们不需要对冗余参数进行估计,利用随机加权的方法就可以得到这些检验的临界值. 而且在局部对立假设之下可以实现对功效的计算. 当取不同的误差分布和不同的随机权时, 我们对本文的方法进行了蒙特卡洛模拟. 结果表明用随机加权方法来逼近原假设分布是非常精确的.     

协方差矩阵的齐性检验

基于ＰＰ技术、Ｂｏｏｔｓｔｒａｐ方法和数论方法,对于ｋ个总体协方差矩阵相等的检验,给出了ＰＰ型检验统计量,并讨论了它的渐近分布和Ｂｏｏｔｓｔｒａｐ逼近,最后给出了一些实际模拟结果。     

A note on runs of geometrically distributed random variables

Recently, Grabner et al. [Combinatorics of geometrically distributed random variables: run statistics, Theoret. Comput. Sci. 297 (2003) 261-270] and Louchard and Prodinger [Ascending runs of sequences of geometrically distributed random variables: a probabilistic analysis, Theoret. Comput. Sci. 304 (2003) 59-86] considered the run statistics of geometrically distributed independent random variables. They investigated the asymptotic properties of the number of runs and the longest run using the corresponding probability generating functions and a Markov chain approach. In this note, we reconsider the asymptotic properties of such statistics using another approach. Our approach of finding the asymptotic distributions is based on the construction of runs in a sequence of m-dependent random variables. This approach enables us to find the asymptotic distributions of many run statistics via the theorems established for m-dependent sequence of random variables. We also provide the asymptotic distribution of the total number of non-decreasing runs and the longest non-decreasing run.     

A modified two-factor multivariate analysis of variance: asymptotics and small sample approximations

In this paper, we present results for testing main, simple and interaction effects in heteroscedastic two factor MANOVA models. In particular, we suggest modifications to the MANOVA sum of squares and cross product matrices to account for heteroscedasticity. Based on these modified matrices, we define some multivariate test statistics and derive their asymptotic distributions under non-normality for the null as well as non-null cases. Derivation of these results relies on the perturbation method and limit theorems for independently distributed random matrices. Based on the asymptotic distributions, we devise small sample approximations for the quantiles of the null distributions. The numerical accuracy of the large sample as well as small sample approximations are favorable. A real data set from a Smoking Cessation Trial is analyzed to illustrate the application of the methods.     

Approximation by random weighting method for M-test in linear models

The M-test has been in common use and widely studied in testing the linear hypotheses in linear models. However, the critical value for the test is usually related to the quantities of the unknown error distribution and the estimate of the nuisance parameters may be rather involved, not only for the M-test method but also for the existing bootstrap methods. In this paper we suggest a random weighting resampling method for approximating the null distribution of the M-test statistic. It is shown that, under both the null and the local alternatives, the random weighting statistic has the same asymptotic distribution as the null distribution of the M-test. The critical values of the M-test can therefore be obtained by the random weighting method without estimating the nuisance parameters. A distinguished feature of the proposed method is that the approximation is valid even the null hypothesis is not true and the power evaluation is possible under the local alternatives.     

Near-exact distributions for the independence and sphericity likelihood ratio test statistics

In this paper we show how, based on a decomposition of the likelihood ratio test for sphericity into two independent tests and a suitably developed decomposition of the characteristic function of the logarithm of the likelihood ratio test statistic to test independence in a set of variates, we may obtain extremely well-fitting near-exact distributions for both test statistics. Since both test statistics have the distribution of the product of independent Beta random variables, it is possible to obtain near-exact distributions for both statistics in the form of Generalized Near-Integer Gamma distributions or mixtures of these distributions. For the independence test statistic, numerical studies and comparisons with asymptotic distributions proposed by other authors show the extremely high accuracy of the near-exact distributions developed as approximations to the exact distribution. Concerning the sphericity test statistic, comparisons with formerly developed near-exact distributions show the advantages of these new near-exact distributions.     

Limit distributions and comparative asymptotic efficiency of the Smirnov — Kolmogorov statistics with a random index

Limit distributions for certain statistics of Smirnov — Kolmogorov type are obtained which consider the weak convergence of the corresponding empirical process. Approximate and precise asymptotic efficiencies of these statistics are computed. It is shown that they are worse in a certain sense than the classical Kolmogorov — Smirnov statistics.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 55, pp. 185–194, 1976.     

The cumulative quantile regression function with censored and truncated covariate

In this paper, we consider the empirical estimator of the cumulative quantile regression (CQR) functionwhen the covariate is subjected to random truncation and censorship. Strong Gaussian approximations for the associated CQR process are established under appropriate assumptions. A functional law of the iterated logarithm for the CQR process is also derived. These results provide a foundation for the asymptotic theory of functional statistics based on these processes.     

基于投影偏度和投影峰度的投影寻踪自助法的正态性检验

本文研究了多元分布的正态性检验问题,用投影寻踪自助法,获得了投影偏度和投影峰度正态性检验统计量,证明了在零假设成立时,所提出的偏度和峰度检验统计量的极限分布为一高斯过程的上界.为计算机模拟计算提供了有力的手段和依据.     

Composing the cumulative quantile regression function and the Goldie concentration curve

The model we discuss in this paper deals with inequality in distribution in the presence of a covariate. To elucidate that dependence, we propose to consider the composition of the cumulative quantile regression (CQR) function and the Goldie concentration curve, the standardized counterpart of which gives a fraction to fraction plot of the response and the covariate. It has the merit of enhancing the visibility of inequality in distribution when the latter is present. We shall examine the asymptotic properties of the corresponding empirical estimator. The associated empirical process involves a randomly stopped partial sum process of induced order statistics. Strong Gaussian approximations of the processes are constructed. The result forms the basis for the asymptotic theory of functional statistics based on these processes.     

Random weighting estimation for survival function under right censorship

The random weighting method is an emerging computing method in statistics. In this paper, we propose a novel estimation of the survival function for right censored data based on the random weighting method. Under some regularity conditions, we prove the strong consistency of this estimation.     

The Asymptotic and Numerical Inversion of the Marcum Q‐Function

The generalized Marcum functions appear in problems of technical and scientific areas such as, for example, radar detection and communications. In mathematical statistics and probability theory these functions are called the noncentral gamma or the noncentral chi‐squared cumulative distribution functions. In this paper, we describe a new asymptotic method for inverting the generalized Marcum Q‐function and for the complementary Marcum P‐function. Also, we show how monotonicity and convexity properties of these functions can be used to find initial values for reliable Newton or secant methods to invert the function. We present details of numerical computations that show the reliability of the asymptotic approximations.     

A Smirnov–Bickel–Rosenblatt Theorem for Compactly-Supported Wavelets

In nonparametric statistical problems, we wish to find an estimator of an unknown function f. We can split its error into bias and variance terms; Smirnov, Bickel and Rosenblatt have shown that, for a histogram or kernel estimate, the supremum norm of the variance term is asymptotically distributed as a Gumbel random variable. In the following, we prove a version of this result for estimators using compactly-supported wavelets, a popular tool in nonparametric statistics. Our result relies on an assumption on the nature of the wavelet, which must be verified by provably-good numerical approximations. We verify our assumption for Daubechies wavelets and symlets, with N=6,…,20 vanishing moments; larger values of N, and other wavelet bases, are easily checked, and we conjecture that our assumption holds also in those cases.     

Asymptotics for testing hypothesis in some multivariate variance components model under non-normality

We consider the problem of deriving the asymptotic distribution of the three commonly used multivariate test statistics, namely likelihood ratio, Lawley-Hotelling and Bartlett-Nanda-Pillai statistics, for testing hypotheses on the various effects (main, nested or interaction) in multivariate mixed models. We derive the distributions of these statistics, both in the null as well as non-null cases, as the number of levels of one of the main effects (random or fixed) goes to infinity. The robustness of these statistics against departure from normality will be assessed.Essentially, in the asymptotic spirit of this paper, both the hypothesis and error degrees of freedom tend to infinity at a fixed rate. It is intuitively appealing to consider asymptotics of this type because, for example, in random or mixed effects models, the levels of the main random factors are assumed to be a random sample from a large population of levels.For the asymptotic results of this paper to hold, we do not require any distributional assumption on the errors. That means the results can be used in real-life applications where normality assumption is not tenable.As it happens, the asymptotic distributions of the three statistics are normal. The statistics have been found to be asymptotically null robust against the departure from normality in the balanced designs. The expressions for the asymptotic means and variances are fairly simple. That makes the results an attractive alternative to the standard asymptotic results. These statements are favorably supported by the numerical results.     

Testing for cointegration using induced-order statistics

In this paper we explore the usefulness of induced-order statistics in the characterization of integrated series and of cointegration relationships. We propose a non-parametric test statistic for testing the null hypothesis of two independent random walks against wide cointegrating alternatives including monotonic nonlinearities and certain types of level shifts in the cointegration relationship. We call our testing device the induced-order Kolmogorov–Smirnov cointegration test (KS), since it is constructed from the induced-order statistics of the series, and we derive its limiting distribution. This non-parametric statistic endows the test with a number of desirable properties: invariance to monotonic transformations of the series, and robustness for the presence of important parameter shifts. By Monte Carlo simulations we analyze the small sample properties of this test. Our simulation results show the robustness of the induced order cointegration test against departures from linear and constant parameter models. This paper is an extension of the work of Aparicio and Granger (1995) and Aparicio and Escribano (1998).     

泛函型统计量的随机加权逼近

§ 1 引言及一般结果设X_,X_2,…X_n是来自分布为F(未知)的总体的n个iid.样本。它们的经验分布为 F_n(x)=1/n sum from i=1 to n I{X_ix}。 (1)常用的许多统计量都是通过经验分布而依赖于样本的,即能写成经验分布的泛函形式     

Approximation by randomly weighting method in censored regression model

Censored regression (“Tobit”) models have been in common use, and their linear hypothesis testings have been widely studied. However, the critical values of these tests are usually related to quantities of an unknown error distribution and estimators of nuisance parameters. In this paper, we propose a randomly weighting test statistic and take its conditional distribution as an approximation to null distribution of the test statistic. It is shown that, under both the null and local alternative hypotheses, conditionally asymptotic distribution of the randomly weighting test statistic is the same as the null distribution of the test statistic. Therefore, the critical values of the test statistic can be obtained by randomly weighting method without estimating the nuisance parameters. At the same time, we also achieve the weak consistency and asymptotic normality of the randomly weighting least absolute deviation estimate in censored regression model. Simulation studies illustrate that the performance of our proposed resampling test method is better than that of central chi-square distribution under the null hypothesis. This work was supported by National Natural Science Foundation of China (Grant No. 10471136), PhD Program Foundation of the Ministry of Education of China, and Special Foundations of the Chinese Academy of Sciences and University of Science and Technology of China     

高维随机截尾数据的PP拟合优度检验

关于一维删截数据的拟合优度检验，已有相当多的文献，但高维截尾数据的拟合优度检验尚不多见．本文用PP技巧讨论了高维截尾数据的拟合优度检验，得到了检验统计量的渐近分布，并讨论了其Bootstrap逼近及逼近的相容性和检验的渐近功效．