On Rao score and PearsonX 2 statistics in generalized linear models

The identity of the Rao score and PearsonX ² statistics is well known in the areas where the latter was first introduced: goodness-of-fit in contingency tables and binary responses. We show in this paper that the same identity holds when the two statistics are used for testing goodness-of-fit of Generalized Linear Models. We also highlight the connections that exist between the two statistics when they are used for the comparison of nested models. Finally, we discuss some merits of these unifying results. Work financially supported by cofin. MIUR grants 2000 and 2002.     

On the asymptotic efficiency of certain correlation tests of fit

Tests of fit based on correlation-type statistics are investigated for the exponential, extreme-value, and logistic distributions. The statistics are shown to be asymptotically normal at the rate log^½n. The result is used to show that such tests have 0 asymptotic relative efficiency.     

Cramér-von Mises tests of fit for the Poisson distribution

Goodness-of-fit tests based on the Cramér-von Mises statistics are given for the Poisson distribution. Power comparisons show that these statistics, particularly A², give good overall tests of fit. The statistic A² will be particularly useful for detecting distributions where the variance is close to the mean, but which are not Poisson.     

Robust statistics for testing mean vectors of multivariate distributions

We develop a ‘robust’ statistic T² _R, based on Tiku's (1967, 1980) MML (modified maximum likelihood) estimators of location and scale parameters, for testing an assumed meam vector of a symmetric multivariate distribution. We show that T² _R is one the whole considerably more powerful than the prominenet Hotelling T² statistics. We also develop a robust statistic T² _D for testing that two multivariate distributions (skew or symmetric) are identical; T² _D seems to be usually more powerful than nonparametric statistics. The only assumption we make is that the marginal distributions are of the type (1/σ_k)f((x-μ_k)/σ_k) and the means and variances of these marginal distributions exist.     

Sharp upper bounds for the expected values of non-extreme order statistics from discrete distributions

We consider the problem of determining sharp upper bounds on the expected values of non-extreme order statistics based on i.i.d. random variables taking on N values at most. We show that the bound problem is equivalent to the problem of establishing the best approximation of the projection of the density function of the respective order statistic based on the standard uniform i.i.d. sample onto the family of non-decreasing functions by arbitrary N  -valued functions in the norm of L²(0,1)$L^2(0,1)$ space. We also present an algorithm converging to the local minima of the approximation problems.     

Empirical process of long-range dependent sequences when parameters are estimated

In this paper, we study the asymptotic behavior of empirical processes when parameters are estimated, assuming that the underlying sequence of random variables is long-range dependent. We show completely different phenomena compared to i.i.d. situation, as well as compared to ordinary empirical processes of long-range dependent sequences. Applications include Kolmogorov–Smirnov and Cramer–Smirnov–von Mises goodness-of-fit statistics.     

Robust test for means when population variances are unequal

We consider the problem of testing the equality of two population means when the population variances are not necessarily equal. We propose a Welch-type statistic, say T^* _c, based on Tiku!s ‘1967, 1980’ modified maximum likelihood estimators, and show that this statistic is robust to symmetric and moderately skew distributions. We investigate the power properties of the statistic T^* _c; T^* _c clearly seems to be more powerful than Yuen's ‘1974’ Welch-type robust statistic based on the trimmed sample means and the matching sample variances. We show that the analogous statistics based on the ‘adaptive’ robust estimators give misleading Type I errors. We generalize the results to testing linear contrasts among k population means     

Two-way ANOVA when the distribution of the error terms is skew t

In this article, we assume that the distribution of the error terms is skew t in two-way analysis of variance (ANOVA). Skew t distribution is very flexible for modeling the symmetric and the skew datasets, since it reduces to the well-known normal, skew normal, and Student's t distributions. We obtain the estimators of the model parameters by using the maximum likelihood (ML) and the modified maximum likelihood (MML) methodologies. We also propose new test statistics based on these estimators for testing the equality of the treatment and the block means and also the interaction effect. The efficiencies of the ML and the MML estimators and the power values of the test statistics based on them are compared with the corresponding normal theory results via Monte Carlo simulation study. Simulation results show that the proposed methodologies are more preferable. We also show that the test statistics based on the ML estimators are more powerful than the test statistics based on the MML estimators as expected. However, power values of the test statistics based on the MML estimators are very close to the corresponding test statistics based on the ML estimators. At the end of the study, a real life example is given to show the implementation of the proposed methodologies.     

Generalized portmanteau statistics and tests of randomness

This paper considers the problem of testing the randomness of Gaussian and non–Gaussian time series. A general class of parametric portmanteau statistics, which include the Box–Pierce and the Ljung–Box statistics, is introduced. Using the exact first and second moments of the sample autocorrelations when the observations are i.i.d. normal with unknown mean, the exact expected value of any portmanteau statistics is obtained for this case. Two new portmanteau statistics, which exploit the exact moments of the sample autocorrelations, are studied. For the nonparametric case, a rank portmanteau statistic is introduced. The latter has the same distribution for any series of exchangeable random variables and uses the exact moments of the rank autocorrelations. We show that its asymptotic distribution is chi–squate. Simulation results indicate that the new portmanteau statistics are better approximated by the chi–square asymptotic distribution than the Ljung–Box statistics. Several analytical results presented in the paper were derived by usig a symbolic manipulation program.     

Improved Score Tests in Symmetric Linear Regression Models

The class of symmetric linear regression models has the normal linear regression model as a special case and includes several models that assume that the errors follow a symmetric distribution with longer-than-normal tails. An important member of this class is the t linear regression model, which is commonly used as an alternative to the usual normal regression model when the data contain extreme or outlying observations. In this article, we develop second-order asymptotic theory for score tests in this class of models. We obtain Bartlett-corrected score statistics for testing hypotheses on the regression and the dispersion parameters. The corrected statistics have chi-squared distributions with errors of order O(n ^?3/2), n being the sample size. The corrections represent an improvement over the corresponding original Rao's score statistics, which are chi-squared distributed up to errors of order O(n ^?1). Simulation results show that the corrected score tests perform much better than their uncorrected counterparts in samples of small or moderate size.     

Testing overdispersion in data with censoring using the mixture of exponential families

This resean h extends the mixture of exponential family developed by Dean (1992) to accommodate oveidispeision in data with censoring. Score statistics testing the (Existence of overdispersion based on the proposed model are obtained Simulations show that the test statistics have suffcient power in detecting the existence of overdispersion when sample size are sufficiently large and the degree of censoring is mild (i. e, 40%) The test statistics are applied to real data sets and are able to detect overdispersion in those data sets.     

Combinatorial analysis of the adaptive last particle method

The adaptive last particle method is a simple and interesting alternative in the class of general splitting algorithms for estimating tail distributions. We consider this algorithm in the space of trajectories and for general reaction coordinates. Using a combinatorial approach in discrete state spaces, we demonstrate two new results. First, we are able to give the exact expression of the distribution of the number of iterations in an perfect version of the algorithm where trajectories are i.i.d. This result is an improvement of previous known results when the cumulative distribution function has discontinuities. Second, we show that an effective computational version of the algorithm where trajectories are no more i.i.d. follows the same statistics than the idealized version when the reaction coordinate is the committor function.     

Maximum Entropy Approximations for Asymptotic Distributions of Smooth Functions of Sample Means

Abstract. We propose an information‐theoretic approach to approximate asymptotic distributions of statistics using the maximum entropy (ME) densities. Conventional ME densities are typically defined on a bounded support. For distributions defined on unbounded supports, we use an asymptotically negligible dampening function for the ME approximation such that it is well defined on the real line. We establish order n^?1 asymptotic equivalence between the proposed method and the classical Edgeworth approximation for general statistics that are smooth functions of sample means. Numerical examples are provided to demonstrate the efficacy of the proposed method.     

Statistical inferences for partially linear single-index models with error-prone linear covariates

We consider statistical inference for partially linear single-index models (PLSIM) when some linear covariates are not observed, but ancillary variables are available. Based on the profile least-squared estimators of the unknowns, we study the testing problems for parametric components in the proposed models. It is to see whether the generalized likelihood ratio (GLR) tests proposed by Fan et al. (2001) are applicable to testing for the parametric components. We show that under the null hypothesis the proposed GLR statistics follow asymptotically the χ²-distributions with the scale constants and the degrees of freedom being independent of the nuisance parameters or functions, which is called the Wilks phenomenon. Simulated experiments are conducted to illustrate our proposed methodology.     

A comparative study on detection of influential observations in linear regression

A large number of statistics are used in the literature to detect outliers and influential observations in the linear regression model. In this paper comparison studies have been made for determining a statistic which performs better than the other. This includes: (i) a detailed simulation study, and (ii) analyses of several data sets studied by different authors. Different choices of the design matrix of regression model are considered. Design A studies the performance of the various statistics for detecting the scale shift type outliers, and designs B and C provide information on the performance of the statistics for identifying the influential observations. We have used cutoff points using the exact distributions and Bonferroni's inequality for each statistic. The results show that the studentized residual which is used for detection of mean shift outliers is appropriate for detection of scale shift outliers also, and the Welsch's statistic and the Cook's distance are appropriate for detection of influential observations.     

Asymptotics for multisample statistics

Consider a family of square-integrable R^d-valued statistics S_k = S_k(X_1,k1; X_2,k2;…; X_m,km), where the independent samples X_i,kj respectively have k_i i.i.d. components valued in some separable metric space X_i. We prove a strong law of large numbers, a central limit theorem and a law of the iterated logarithm for the sequence {S_k}, including both the situations where the sample sizes tend to infinity while m is fixed and those where the sample sizes remain small while m tends to infinity. We also obtain two almost sure convergence results in both these contexts, under the additional assumption that S_k is symmetric in the coordinates of each sample X_i,kj. Some extensions to row-exchangeable and conditionally independent observations are provided. Applications to an estimator of the dimension of a data set and to the Henze-Schilling test statistic for equality of two densities are also presented.     

A note on Cochran test for homogeneity in one-way ANOVA and meta-analysis

In this paper we provide a formal yet simple and straightforward proof of the asymptotic χ² distribution for Cochran test statistic. Then, we show that the general form of this type of test statistics is invariant for the choice of weights. This fact is important since in practice many such test statistics are constructed with more complicated forms which usually require calculating generalized inverse matrices. Based on our results, we can simplify the construction of the test statistics. More importantly, properties such as anti-conservativeness of this type of test statistics can be drawn from Cochran test statistic. Furthermore, one can improve the performance of the tests by using some modified statistics with correction for small sample size situations.     

Modified kolmogorov-smirnov tests of goodness of fit

The Kolmogorov-Smirnov (K–S) one-sided and two-sided tests of goodness of fit based on the test statistics D⁺ _n D^? _n and D_n are equivalent to tests based on taking the cumulative probability of the i–th order statistic of a sample of size n to be (i–.5)/n. Modified test statistics C⁺ _n, C^? _n and C_n are obtained by taking the cumulative probability to be i/(n+l). More generally, the cumula-tive probability may be taken to be (i?δ)/(n+l?2δ), as suggested by Blom (1958), where 0 less than or equal δ less than or equal .5. Critical values of the test statis-tics can be found by interpolating inversely in tables of the proba-bility integrals obtained by setting a=l/(n+l?2δ) in an expression given by Pyke (1959). Critical values for the D's (corresponding to δ=.5) have been tabulated to 5DP by Miller (1956) for n=1(1)100. The authors have made analogous tabulations for the C's (corresponding to δ=0) [previously tabulated by Durbin (1969) for n=1(1)60(2)100] and for the test statistics E⁺ _n, E^? _n and E_n corresponding to δ f.3. They have also made a Monte Carlo comparison of the power of the modified tests with that of the K–S test for several hypothetical distributions. In a number of cases, the power of the modified tests is greater than that of the K–S test, especially when the standard deviation is greater under the alternative than under the null hypo-thesis.