Application of tests of chi-square type

Some recent results on the theory of chi-square tests are given. Remarks of practical character on the use of these tests are made, reflecting on a large number of applied papers.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 158, pp. 49–71, 1987.     

General chi-square goodness-of-fit tests with data-dependent cells

Summary The goodness-of-fit of a parametric model for non-categorical data can be tested using the x ² statistic calculated after grouping the data into a finite number of disjoint cells. Work of Watson, ebyev, Moore and others shows that the classical limit distributions still hold even for certain methods of grouping that depend on the data themselves. These results are generalised to cover a much wider class of methods of grouping; the parameters can be estimated from either the grouped or the ungrouped data. The proofs use a Central Limit Theorem for Empirical Measures due to Dudley. The grouping cells are allowed to come from any Donsker class for the underlying sampling distribution.Dedicated to Leopold Schmetterer, on his sixtieth birthdayMost of the work for this paper was carried out with the support of a fellowship of the Alexander von Humboldt Foundation at the Ruhr-Universität Bochum     

Asymptotic power comparison of the chi-square and likelihood ratio tests

Summary A Hoeffding-type power comparison is made between the likelihood ratio and chi-square tests for a simple hypothesis in a multinomial. The power comparison is based on the fact that under an alternative hypothesis the distribution of the test statistic can be approximated by a normal distribution. The theory of large deviations is used to match the significance levels. This research was partially supported by the National Science Foundation Grant No. GP-33697X2, U.S. Energy Research and Development Agency Grant 7064100, Environmental Protection Agency Grant R805379-01-0 and U.S. Public Health Service Grant USPHS ES01299-15, at the University of California-Berkeley.     

Expansions for the multivariate chi-square distribution

Three classes of expansions for the distribution function of the χ_k²(d, R)-distribution are given, where k denotes the dimension, d the degree of freedom, and R the “accompanying correlation matrix.” The first class generalizes the orthogonal series with generalized Laguerre polynomials, originally given by Krishnamoorthy and Parthasarathy [12]. The second class contains always absolutely convergent representations of the distribution function by univariate chi-square distributions and the third class provides also the probabilities for any unbounded rectangular regions. In particular, simple formulas are given for the three-variate case including singular correlation matrices R, which simplify the computation of third order Bonferroni inequalities, e.g., for the tail probabilities of max{χ_i²|1 ≤ i ≤ k} (k > 3).     

Extremes of independent chi-square random vectors

We prove that the componentwise maximum of an i.i.d. triangular array of chi-square random vectors converges in distribution, under appropriate assumptions on the dependence within the vectors and after normalization, to the max-stable Hüsler–Reiss distribution. As a by-product we derive a conditional limit result.     

Stepwise test procedures and approximate chi-square analysis

Let an overall null hypothesisH be factored in a certain stepwise manner intok subhypotheses as . Suppose the test statisticw forH be correspondingly expressed asw=w ₁ w₂…w_k wherew _i is the test statistic forH _i. We consider the case where the Box method [2] is applicable for the distributions ofw andw _i's. Ifw _i's are independent underH, we obtain a stepwise test procedure forH on the basis of an approximate chi-square analysis. To demonstrate the procedure of this sort, the testing hypotheses of equality of several convariance matrices and of the multiple independence are discussed. Finally the related asymptotic distributions are shortly noted.     

A note on the noncentral chi-square distribution

For the noncentral chi-square random variable, the density and cumulative distribution function may be computed using tables of modified Bessel functions. In the case of odd degrees of freedom, a simple exact expression is given which is computable on a hand calculator.     

Spherical level-crossing measures for chi-square random fields

Spherical measures of exceeding a low level are considered for strongly dependent chi-square fields. The limit distributions are sums of multiple stochastic integrals.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 68, pp. 118–128, 1989.     

A chi-square test for dimensionality with non-Gaussian data

The classical theory for testing the null hypothesis that a set of canonical correlation coefficients is zero leads to a chi-square test under the assumption of multi-normality. The test has been used in the context of dimension reduction. In this paper, we study the limiting distribution of the test statistic without the normality assumption, and obtain a necessary and sufficient condition for the chi-square limiting distribution to hold. Implications of the result are also discussed for the problem of dimension reduction.     

The bivariate noncentral chi-square distribution - A compound distribution approach

This paper proposes the bivariate noncentral chi-square (BNC) distribution by compounding the Poisson probabilities with the bivariate central chi-square distribution. The probability density and cumulative distribution functions of the joint distribution of the two noncentral chi-square variables are derived for arbitrary values of the correlation coefficient, degrees of freedom(s), and noncentrality parameters. Computational procedures to calculate the upper tail probabilities as well as the percentile points for selected values of the parameters, for both equal and unequal degrees of freedom, are discussed. The graphical representation of the distribution for different values of the parameters are provided. Some applications of the distribution are outlined.     

Economic and economic-statistical design of a chi-square chart for CBM

In this paper, the economic and economic-statistical design of a χ² chart for a maintenance application is considered. The machine deterioration process is described by a three-state continuous time Markov chain. The machine state is unobservable, except for the failure state. To avoid costly failures, the system is monitored by a χ² chart. The observation process stochastically related to the machine condition is assumed to be multivariate, normally distributed. When the chart signals, full inspection is performed to determine the actual machine condition. The system can be preventively replaced at a sampling epoch and must be replaced upon failure; preventive replacement costs less than failure replacement. The objective is to find the optimal control chart parameters that minimize the long-run average maintenance cost per unit time. For the economic-statistical design, an additional constraint guaranteeing the occurrence of the true alarm signal on the chart before failure with given probability is considered. For both designs, the objective function is derived using renewal theory.     

A chi-square type test for time-invariant fiber pathways of the brain

Statistical Inference for Stochastic Processes - A longitudinal diffusion tensor imaging (DTI) study on a single brain can be remarkably useful to probe white matter fiber connectivity that may or...     

An estimation method for the Neyman chi-square divergence with application to test of hypotheses

We propose a new definition of the Neyman chi-square divergence between distributions. Based on convexity properties and duality, this version of the χ² is well suited both for the classical applications of the χ² for the analysis of contingency tables and for the statistical tests in parametric models, for which it is advocated to be robust against outliers.We present two applications in testing. In the first one, we deal with goodness-of-fit tests for finite and infinite numbers of linear constraints; in the second one, we apply χ²-methodology to parametric testing against contamination.     

A chi-square method for obtaining a priority vector from multiplicative and fuzzy preference relations

Decision makers (DMs)’ preferences on decision alternatives are often characterized by multiplicative or fuzzy preference relations. This paper proposes a chi-square method (CSM) for obtaining a priority vector from multiplicative and fuzzy preference relations. The proposed CSM can be used to obtain a priority vector from either a multiplicative preference relation (i.e. a pairwise comparison matrix) or a fuzzy preference relation or a group of multiplicative preference relations or a group of fuzzy preference relations or their mixtures. Theorems and algorithm about the CSM are developed. Three numerical examples are examined to illustrate the applications of the CSM and its advantages.     

Approximations to the probabilities of binomial and multinomial random variables and chi-square type statistics

Summary New lower and upper bounds are given to the probabilities of binomial and multinomial random variables. Exact bounds are also presented for the sampling distributions of chi-square type statistics and the K-L information number from a multinomial distribution. The Institute of Statistical Mathematics