A constructive hypothesis test for the single-index models with two groups

Comparison of two-sample heteroscedastic single-index models, where both the scale and location functions are modeled as single-index models, is studied in this paper. We propose a test for checking the equality of single-index parameters when dimensions of covariates of the two samples are equal. Further, we propose two test statistics based on Kolmogorov–Smirnov and Cramér–von Mises type functionals. These statistics evaluate the difference of the empirical residual processes to test the equality of mean functions of two single-index models. Asymptotic distributions of estimators and test statistics are derived. The Kolmogorov–Smirnov and Cramér–von Mises test statistics can detect local alternatives that converge to the null hypothesis at a parametric convergence rate. To calculate the critical values of Kolmogorov–Smirnov and Cramér–von Mises test statistics, a bootstrap procedure is proposed. Simulation studies and an empirical study demonstrate the performance of the proposed procedures.     

Nonparametric evaluation of the first passage time of degradation processes

This paper discusses a nonparametric method to approximate the first passage time (FPT) distribution of the degradation processes incorporating random effects if the process type is unknown. The FPT of a degradation process is unnecessarily observed since its density function can be approximated by inverting the empirical Laplace transform using the empirical saddlepoint method. The empirical Laplace transform is composed of the measured increments of the degradation processes. To evaluate the performance of the proposed method, the approximated FPT is compared with the theoretical FPT assuming a true underlying process. The nonparametric method discussed in this paper is shown to possess the comparatively small relative errors in the simulation study and performs well to capture the heterogeneity in the practical data analysis. To justify the fitting results, the goodness‐of‐fit tests including Kolmogorov‐Smirnov test and Cramér‐von Mises test are conducted, and subsequently, a bootstrap confidence interval is constructed in terms of the 90th percentile of the FPT distribution.     

Numerical comparison of classical and permutation statistical hypothesis testing methods

The article is devoted to the classical problem of statistical hypothesis testing for the equality of two distributions. For normal distributions, Student’s test is optimal in many senses. However, in practice, distributions to be compared are often not normal and, generally speaking, unknown. When nothing is known about the distributions to be compared, one usually applies the nonparametric Kolmogorov–Smirnov test to solve this problem. In the present paper, methods are considered that are based on permutations and, in recent years, have attracted interest for their simplicity, universality, and relatively high efficiency. Methods of stochastic simulation are applied to the comparative analysis of the power of a few permutation tests and classical methods (such as the Kolmogorov–Smirnov test, Student’s test, and the Mann–Whitney test) for a wide class of distribution functions. Normal distributions, Cauchy distributions, and their mixtures, as well as exponential, Weibull, Fisher’s, and Student’s distributions are considered. It is established that, for many typical distributions, the permutation method based on the sum of the absolute values of differences is the most powerful one. The advantage of this method over other ones is especially large when one compares symmetric distributions with the same centers. Thus, this permutation method can be recommended for application in cases when the distributions to be compared are different from normal ones.     

Analogues of Classical Goodness-of-Fit Tests for Distribution Tails

Doklady Mathematics - We propose analogues of the classical Kolmogorov–Smirnov and omega-squared tests for goodness-of-fit testing of distribution tails. The consistency of the proposed tests...     

Discovering focal regions of slightly-aggregated sparse signals

The characteristic aspects of dynamic distortions on a lengthy time series of i.i.d. pure noise when embedded with slightly-aggregating sparse signals are summarized into a significantly shorter recurrence time process of a chosen extreme event. We first employ the Kolmogorov–Smirnov statistic to compare the empirical recurrence time distribution with the null geometry distribution when no signal being present in the original time series. The power of such a hypothesis testing depends on varying degrees of aggregation of sparse signals: from a completely random distribution of singletons to batches of various sizes on the entire temporal span. We demonstrate the Kolmogorov–Smirnov statistic capturing the dynamic distortions due to slightly-aggregating sparse signals better than does Tukey’s Higher Criticism statistic even when the batch size is as small as five. Secondly, after confirming the presence of signals in the pure noise time series, we apply the hierarchical factor segmentation (HFS) algorithm again based on the recurrence time process to compute focal segments that contain a significantly higher intensity of signals than do the rest of the temporal regions. In a computer experiment with a given fixed number of signals, the focal segments identified by the HFS algorithm afford many folds of signal intensity which also critically depend on the degree of aggregation of sparse signals. This ratio information can facilitate better sensitivity, equivalent to a smaller false discovery rate, if the signal-discovering protocol implemented within the computed focal regions is different from that used outside of the focal regions. We also numerically compute the specificity as the total number of signals contained in the computed collection of focal regions, which indicates the inherent difficulty in the task of sparse signal discovery.     

Two-sample Kolmogorov-Smirnov test using a Bayesian nonparametric approach

In this paper, a Bayesian nonparametric approach to the two-sample problem is proposed. Given two samples \(\text{X} = {X_1}, \ldots ,{X_{m1}}\;\mathop {\text~}\limits^{i.i.d.} F\) and \(Y = {Y_1}, \ldots ,{Y_{{m_2}}}\mathop {\text~}\limits^{i.i.d.} G\), with F and G being unknown continuous cumulative distribution functions, we wish to test the null hypothesis H ₀: F = G. The method is based on computing the Kolmogorov distance between two posterior Dirichlet processes and comparing the results with a reference distance. The parameters of the Dirichlet processes are selected so that any discrepancy between the posterior distance and the reference distance is related to the difference between the two samples. Relevant theoretical properties of the procedure are also developed. Through simulated examples, the approach is compared to the frequentist Kolmogorov–Smirnov test and a Bayesian nonparametric test in which it demonstrates excellent performance.     

Tests of symmetry for bivariate copulas

Tests are proposed for the hypothesis that the underlying copula of a continuous random pair is symmetric. The procedures are based on Cramér–von Mises and Kolmogorov–Smirnov functionals of a rank-based empirical process whose large-sample behaviour is obtained. The asymptotic validity of a re-sampling method to compute P values is also established. The technical arguments supporting the use of a Chi-squared test due to Jasson are also presented. A power study suggests that the proposed tests are more powerful than Jasson’s procedure under many scenarios of copula asymmetry. The methods are illustrated on a nutrient data set.     

金融资产收益分布的混合高斯分析

本文研究了金融资产收益的混合高斯分布模型 ,给出了混合高斯分布的 Kolmogorov-Smirnov检验的方法 ,分析了金融资产收益的非高斯性及市场价格运动的有效性 .此外 ,用成分数目 K*、拟合误差 DK*n和主成分系数 p*k 描述金融资产收益的性质 ,对外汇银行同业拆借市场和中国股票市场实证分析     

PP multivariate random weighting method

According to the Projection Pursuit (PP) method and the random weighting method, we propose a PP random weighting method, and set up the asymptotic distribution theory and strong limit theorem of PP random weighting empirical process. Applying this method, we obtain two kinds of goodness-of-fit test for a multivariate distribution function, i.e., we get the random weighting approximations of PP Kolmogorov Smirnov statistics (PPKS) and PP Smirnov Cramér Von Mises statistics (PPSC), we prove that the asymptotic distribution of PPKS and PPSC are the same as those of their respective random weighting approximations.Supported by the National Natural Science Foundation of China.     

Empirical processes with estimated parameters under auxiliary information

Empirical processes with estimated parameters are a well established subject in nonparametric statistics. In the classical theory they are based on the empirical distribution function which is the nonparametric maximum likelihood estimator for a completely unknown distribution function. In the presence of some “nonparametric” auxiliary information about the distribution, like a known mean or a known median, for example, the nonparametric maximum likelihood estimator is a modified empirical distribution function which puts random masses on the observations in order to take the available information into account [see Owen, Biometrika 75 (1988) 237–249, Ann. Statist. 18 (1990) 90–120, Empirical Likelihood, Chapman & Hall/CRC, London/Boca Raton, FL; Qin and Lawless, Ann. Statist. 22 (1994) 300–325]. Zhang [Metrika 46 (1997) 221–244] has proved a functional central limit theorem for the empirical process pertaining to this modified empirical distribution function. We will consider the corresponding empirical process with estimated parameters here and derive its asymptotic distribution. The limiting process is a centered Gaussian process with a complicated covariance function depending on the unknown parameter. The result becomes useful in practice through the bootstrap, which is shown to be consistent in case of a known mean. The performance of the resulting bootstrap goodness-of-fit test based on the Kolmogorov–Smirnov statistic is studied through simulations.     

Kolmogorov–Smirnov‐type testing for the partial homogeneity of Markov processes—with application to credit risk

In banking, the default behaviour of the counterpart is not only of interest for the pricing of transactions under credit risk but also for the assessment of a portfolio credit risk. We develop a test against the hypothesis that default intensities are chronologically constant within a group of similar counterparts, e.g. a rating class. The Kolmogorov–Smirnov‐type test builds up on the asymptotic normality of counting processes in event history analysis. The right censoring accommodates for Markov processes with more than one no‐absorbing state. A simulation study and two examples of rating systems demonstrate that partial homogeneity can be assumed, however occasionally, certain migrations must be modelled and estimated inhomogeneously. Copyright © 2007 John Wiley & Sons, Ltd.     

Hodges-Lehmann asymptotic efficiency of the Kolmogorov and Smirnov goodness-of-fit tests

One considers the Hodges-Lehmann asymptotic efficiency of the Kolmogorov and Smirnov goodness-of-fit tests, which measures the rate of the exponential decrease of the errors of the second kind, under a fixed significance level. It is shown that the Kolmogorov test is always asymptotically optimal in this sense, while the one-sided Smirnov test is asymptotically optimal under additional conditions imposed on the parametric family of distributions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 142, pp. 119–123, 1985.     

The Smirnov distribution

Summary Adequacy of the Smirnov approximation to P (D _mn ≧c/mn), of the exact distribution of the two sample Kolmogorov-Smirnov criterion,D _mn ,m<=n is examined. The main finding is that accuracy depends as much on the sample sizes,m andn, as on the ratio, ρ=m/n. The Smirnov approximation is good whenn is an integer multiple ofm, especiallyn=m, 2m; poor otherwise. This contrasts with the Smirnov approximation to P (D _mn >c/mn), where this ordering is reversed, the cases forn=m, 2m being poor. The merit of continuity correction, of order ofn ⁻¹ in theD _mn scale, is demonstrated for the Smirnov approximation, .1≦ρ, as well as for the Kolmogorov approximation, ρ<.1. As an argument for the optimum choice of ρ, the casen=m+1 is shown to have much to recommend it over the casen=m. Finally the usefulness of theχ ² distribution with 2 degress of freedom is illustrated as an approximation to the Smirnov distribution. This study was in part supported by USPHS-NIH Grant number 2 DO4 AH 0167 07 PHT Development Project.     

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 Nonparametric Test for the Equivalence of Populations Based on a Measure of Proximity of Samples

We propose a new measure of proximity of samples based on confidence limits for the bulk of a population constructed using order statistics. For this measure of proximity, we compute approximate confidence limits corresponding to a given significance level in the cases where the null hypothesis on the equality of hypothetical distribution functions may or may not be true. We compare this measure of proximity with the Kolmogorov–Smirnov and Wilcoxon statistics for samples from various populations. On the basis of the proposed measure of proximity, we construct a statistical test for testing the hypothesis on the equality of hypothetical distribution functions.     

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.     

Testing long memory based on a discretely observed process

In this paper we consider the problem of testing long memory for a continuous time process based on high frequency data. We provide two test statistics to distinguish between a semimartingale and a fractional integral process with jumps, where the integral is driven by a fractional Brownian motion with long memory. The small–sample performances of the statistics are evidenced by means of simulation studies. The real data analysis shows that the fractional integral process with jumps can capture the long memory of some financial data.     

Improved inference in dispersion models

We derive a general matrix Bartlett–type correction factor to the gradient statistic in the class of dispersion models. The correction improves the large–sample χ² approximation to the null distribution of the gradient statistic when the sample size is finite. We conduct Monte Carlo simulation experiments to evaluate and compare the performance of various different tests, namely the usual Wald, likelihood ratio, score, and gradient tests, the Bartlett–corrected versions of the likelihood ratio, score, and gradient tests, and bootstrap–based tests. The simulation results suggest that the analytical and computational corrections are effective in removing size distortions of the type I error probability with no power loss. The impact of the corrections in two real data applications is considered for illustrative purposes.     

Probability distribution fitting of schedule overruns in construction projects

The probability of schedule overruns for construction and engineering projects can be ascertained using a ‘best fit’ probability distribution from an empirical distribution. The statistical characteristics of schedule overruns occurring in 276 Australian construction and engineering projects were analysed. Skewness and kurtosis values revealed that schedule overruns are non-Gaussian. Theoretical probability distributions were then fitted to the schedule overrun data; including the Kolmogorov–Smirnov, Anderson–Darling and Chi-Squared non-parametric tests to determine the ‘Goodness of Fit’. A Four Parameter Burr probability function best described the behaviour of schedule overruns, provided the best overall distribution fit and was used to calculate the probability of a schedule overrun being experienced. The statistical characteristics of contract size and schedule overruns were also analysed, and the Wakeby (<AU$1?m and AU$11–50?m), Three Parameter Log-logistic (AU$1–A$10?m) and Beta (AU$51–A$100?m and >AU$101?m) models provided the best distribution fits and were used to calculate schedule overrun probabilities by contract size.     

二阶随机控制变点的Kolmogrov型检验和估计

基于Kolmogrov型统计量和Kiefer过程,对一样本情形,我们讨论了二阶随机控制变点的检验和估计到了检验统计量的渐近分布且用模拟方法给出了其有限样本的分位数,并证明了变点的估计为强相合的。