Point process diagnostics based on weighted second-order statistics and their asymptotic properties

A new approach for point process diagnostics is presented. The method is based on extending second-order statistics for point processes by weighting each point by the inverse of the conditional intensity function at the point’s location. The result is generalized versions of the spectral density, R/S statistic, correlation integral and K-function, which can be used to test the fit of a complex point process model with an arbitrary conditional intensity function, rather than a stationary Poisson model. Asymptotic properties of these generalized second-order statistics are derived, using an approach based on martingale theory.     

Weak convergence of non-stationary multivariate marked processes with applications to martingale testing

This paper establishes the weak convergence of a class of marked empirical processes of possibly non-stationary and/or non-ergodic multivariate time series sequences under martingale conditions. The assumptions involved are similar to those in Brown's martingale central limit theorem. In particular, no mixing conditions are imposed. As an application, we propose a test statistic for the martingale hypothesis and we derive its asymptotic null distribution. Finally, a Monte Carlo study shows that the asymptotic results provide good approximations for small and moderate sample sizes. An application to the S&P 500 is also considered.     

Asymptotic properties of some goodness-of-fit tests based on the L 1-norm

Some goodness-of-fit tests based on the L ₁-norm are considered. The asymptotic distribution of each statistic under the null hypothesis is the distribution of the L ₁-norm of the standard Wiener process on [0,1]. The distribution function, the density function and a table of some percentage points of the distribution are given. A result for the asymptotic tail probability of the L ₁-norm of a Gaussian process is also obtained. The result is useful for giving the approximate Bahadur efficiency of the test statistics whose asymptotic distributions are represented as the L ₁-norms of Gaussian processes.     

Consistency of general bootstrap methods for degenerate -type and -type statistics

We provide general results on the consistency of certain bootstrap methods applied to degree-2 degenerate statistics of U-type and V-type. While it follows from well known results that the original statistic converges in distribution to a weighted sum of centred chi-squared random variables, we use a coupling idea of Dehling and Mikosch to show that the bootstrap counterpart converges to the same distribution. The result is applied to a goodness-of-fit test based on the empirical characteristic function.     

Goodness-of-Fit Tests for the Cauchy Distribution Based on the Empirical Characteristic Function

Let X ₁,...,X _n be independent observations on a random variable X. This paper considers a class of omnibus procedures for testing the hypothesis that the unknown distribution of X belongs to the family of Cauchy laws. The test statistics are weighted integrals of the squared modulus of the difference between the empirical characteristic function of the suitably standardized data and the characteristic function of the standard Cauchy distribution. A large-scale simulation study shows that the new tests compare favorably with the classical goodness-of-fit tests for the Cauchy distribution, based on the empirical distribution function. For small sample sizes and short-tailed alternatives, the uniformly most powerful invariant test of Cauchy versus normal beats all other tests under discussion.     

Goodness-of-fit test for switching diffusion

We consider two Cramér–von Mises goodness-of-fit tests for hypotheses that the observed diffusion process has sign-type trend coefficient based on empirical distribution function and empirical density function. It is shown that the limit distributions of the proposed tests statistics are defined by the integral type functionals of continuous Gaussian processes. We study the behavior of these statistics under the alternative hypothesis and we prove that the tests are consistent. We provide the Karhunen-Loève expansion on \mathbbR{\mathbb{R}} of the corresponding limiting processes and we show that the eigenfunctions in these expansions have expressions in term of Bessel functions.     

Diagnostic checking of multivariate nonlinear time series models with martingale difference errors

In this article, we derive the asymptotic distribution of residual autocovariance and autocorrelation matrices for a general class of multivariate nonlinear time series models by assuming only that the error term is a martingale difference sequence. Two types of applications are developed: global test statistics of the portmanteau type and one-lag test statistics, which describe the residual correlation at individual lags. To illustrate the proposed methodology, simulation results are reported for diagnosing multivariate threshold time series models. The following test statistics are compared: the classical test statistics presuming independent errors and the proposed methodology which supposes only martingale difference errors.     

Tests of fit for the Rayleigh distribution based on the empirical Laplace transform

In this paper a class of goodness-of-fit tests for the Rayleigh distribution is proposed. The tests are based on a weighted integral involving the empirical Laplace transform. The consistency of the tests as well as their asymptotic distribution under the null hypothesis are investigated. As the decay of the weight function tends to infinity the test statistics approach limit values. In a particular case the resulting limit statistic is related to the first nonzero component of Neyman’s smooth test for this distribution. The new tests are compared with other omnibus tests for the Rayleigh distribution.     

About the rate function in concentration inequalities for suprema of bounded empirical processes

We provide new deviation inequalities in the large deviations bandwidth for suprema of empirical processes indexed by classes of uniformly bounded functions associated with independent and identically distributed random variables. The improvements we get concern the rate function which is, as expected, the Legendre transform of the suprema of the log-Laplace transform of the pushforward measure by the functions of the considered class (up to an additional corrective term). Our approach is based on a decomposition in martingale together with some comparison inequalities.     

Modified cramer-von mises goodness-of-fit tests for spectral distribution functions

Modifications to the Cramer-von Mises goodness-of-fit test statistic for spectral distributions are discussed. The modifications consist of inserting weight functions into the usual sto¬chastic integral for the test statistic. Conditions on the weight function are given under which the integral of the weighted square of the difference between the empirical and theoretical spectral distribution functions converges in distribution to the corresponding integral of a process related to Brownian Motion. The distributions of the test statistic under certain alternatives to the null hypothesis are also discussed. A discussion is given of the large sample distributions for weight function of the form ψ(t) = at ^k ,k < –2.     

A Partial Empirical Likelihood Based Score Test Under a Semiparametric Finite Mixture Model

We propose a score statistic to test the null hypothesis that the two-component density functions are equal under a semiparametric finite mixture model. The proposed score test is based on a partial empirical likelihood function under an I-sample semiparametric model. The proposed score statistic has an asymptotic chi-squared distribution under the null hypothesis and an asymptotic noncentral chi-squared distribution under local alternatives to the null hypothesis. Moreover, we show that the proposed score test is asymptotically equivalent to a partial empirical likelihood ratio test and a Wald test. We present some results on a simulation study.     

A test for the presence of stochastic ordering under censoring: the k-sample case

In this paper, we develop an empirical likelihood-based test for the presence of stochastic ordering under censoring in the k-sample case. The proposed test statistic is formed by taking the supremum of localized empirical likelihood ratio test statistics. Its asymptotic null distribution has a simple representation in terms of a standard Brownian motion process. Through simulations, we show that it outperforms in terms of power existing methods for the same problem at all the distributions that we consider. A real-life example is used to illustrate the applicability of this new test.

     

A new method for proving weak convergence results applied to nonparametric estimators in survival analysis

Using the limit theorem for stochastic integral obtained by Jakubowski et al. (Probab. Theory Related Fields 81 (1989) 111–137), we introduce in this paper a new method for proving weak convergence results of empirical processes by a martingale method which allows discontinuities for the underlying distribution. This is applied to Nelson–Aalen and Kaplan–Meier processes. We also prove that the same conclusion can be drawn for Hjort's nonparametric Bayes estimators of the cumulative distribution function and cumulative hazard rate.     

The Law of the Iterated Logarithm and Central Limit Theorem for L-Statistics

The Chung–Smirnov law of the iterated logarithm and the Finkelstein functional law of the iterated logarithm for empirical processes are used to establish new results on the central limit theorem, the law of the iterated logarithm, and the strong law of large numbers for L-statistics with certain bounded and smooth weight functions. These results are used to obtain necessary and sufficient conditions for almost sure convergence and for convergence in distribution of some well-known L-statistics and U-statistics, including Gini's mean difference statistic. A law of the logarithm for weighted sums of order statistics is also presented.     

The exact and approximate distributions of linear combinations of selected order statistics from a uniform distribution

Summary The exact probability density function is given for linear combinations ofk=k(n) order statistics selected from whole order statistics based on random sample of sizen drawn from a uniform distribution. Normal approximation to the linear combinations is made with the aid of Berry-Esseen's theorem. Necessary and sufficient conditions of the asymptotic normality for the statistic are obtained, too. An exact distribution and its normal approximation of linear combination of mutually independent gamma variables with integer valued parameters are also given as associated consequences. The Institute of Statistical Mathematics     

Bayesian approach to prediction and the spacings in the exponential distribution

Summary A series of independent samples are drawn from a general population with positive variationf(x,ϕ), x>0. Based on the Bayesian approach, a general predictive distribution is given, to predict a statistic in the future sample based on the statistics in the earlier samples (or stages). Few general classes of distributions of this type like Koopman-Pitman family, power function family and Burr's class of distributions are considered to show how this procedure works in predicting order statistics in the future sample. Also, the sum of the spacings in the future samples from an exponential population is predicted in terms of similar sum of spacings in the earlier samples. Discussion on the variance of this predictive distribution is dealt with. Finally, an illustrative example with simulated samples from an exponential population gives actual prediction of an order statistic as well as the sum of spacings in the future sample.     

On the empirical process of multivariate, dependent random variables

A convergence theorem of Billingsley for the empirical process of stationary, real valued radom variables under a mixing condition is generalized to the k-dimensional and nonstationary case. Further a more general empirical process is treated, including the upper summation boundary as argument. Some applications are given to a Kolmogorov-Smirnov and a Cramér-von Mises type statistic.     

Average projection type weighted Cramér- von Mises statistics for testing some distributions

This paper addresses the problem of testing goodness-of-fit for several important multivariate distributions: (I) Uniform distribution on p-dimensional unit sphere; (II) multivariate standard normal distribution; and (III) multivariate normal distribution with unknown mean vector and covariance matrix. The average projection type weighted Cramér-von Mises test statistic as well as estimated and weighted Cramér-von Mises statistics for testing distributions (I), (II) and (III) are constructed via integrating projection direction on the unit sphere, and the asymptotic distributions and the expansions of those test statistics under the null hypothesis are also obtained. Furthermore, the approach of this paper can be applied to testing goodness-of-fit for elliptically contoured distributions.     

A multivariate empirical characteristic function test of independence with normal marginals

This paper proposes a semi-parametric test of independence (or serial independence) between marginal vectors each of which is normally distributed but without assuming the joint normality of these marginal vectors. The test statistic is a Cramér–von Mises functional of a process defined from the empirical characteristic function. This process is defined similarly as the process of Ghoudi et al. [J. Multivariate Anal. 79 (2001) 191] built from the empirical distribution function and used to test for independence between univariate marginal variables. The test statistic can be represented as a V-statistic. It is consistent to detect any form of dependence. The weak convergence of the process is derived. The asymptotic distribution of the Cramér–von Mises functionals is approximated by the Cornish–Fisher expansion using a recursive formula for cumulants and inversion of the characteristic function with numerical evaluation of the eigenvalues. The test statistic is finally compared with Wilks statistic for testing the parametric hypothesis of independence in the one-way MANOVA model with random effects.     

Asymptotic expansion of the null distribution of test statistic for linear hypothesis in nonnormal linear model

This paper is concerned with the null distribution of test statistic T for testing a linear hypothesis in a linear model without assuming normal errors. The test statistic includes typical ANOVA test statistics. It is known that the null distribution of T converges to χ² when the sample size n is large under an adequate condition of the design matrix. We extend this result by obtaining an asymptotic expansion under general condition. Next, asymptotic expansions of one- and two-way test statistics are obtained by using this general one. Numerical accuracies are studied for some approximations of percent points and actual test sizes of T for two-way ANOVA test case based on the limiting distribution and an asymptotic expansion.