X2 Distribution Research Articles

Limiting distributions of randomly accelerated motions

In this paper, the process {X(t); t>0}, representing the position of a uniformly accelerated particle (with Poisson-paced) changes of its acceleration, is studied. It is shown that the distribution ofX(t) (suitably normalized), conditionally on the numbern of changes of acceleration, tends in distribution to a normal variate asn goes to infinity. The asymptotic normality of the unconditional distribution ofX(t) for large values oft is also shown. The study of these limiting distributions is motivated by the difficulty of evaluating exactly the conditional and unconditional probability laws ofX(t). In fact, the results obtained in this paper permit us to give useful approximations of the probability distributions of the position of the particle.

TESTING FOR HOMOGENEITY OF GAMETIC DISEQUILIBRIUM AMONG POPULATIONS.

Testing for homogeneity of gametic disequilibria among populations can be informative in discriminating among the evolutionary agents generating them in natural populations. The standard statistical procedure that tests for homogeneity of disequilibrium for the same locus pair in different populations is based on the Fisher's test of homogeneity among correlation coefficients (Fisher 1925; Steel and Torrie 1980; Sokal and Rohlf 1995). This statistical approach makes it necessary to measure the magnitude of disequilibrium between loci by the coefficient of correlation of gene frequencies. This coefficient for the two-allele, two-locus case is defined as r = D/l[p(l p)q(l q)]?12 (Hill and Robertson 1968). In this formula, D is the covariance of allelic frequencies defined as D = f(A1B1) pq, where f(A1B1) is the frequency of the haplotype AIB1, andp and q denote the allele frequencies at the loci (Lewontin and Kojima 1960). When samples of haplotypes for a given pair of loci are available from k populations, each correlation coefficient of allele frequencies is transformed to a variable normal z, and a weighted sum of squares of the corresponding z-values has a x2 distribution with k 1 degrees of freedom, which is used for testing the null hypothesis of no heterogeneity. This statistical approach is recommended by Weir (1996) in his influential book Genetic Data Analysis (pp. 136-138), and it has been used in testing for homogeneity of disequilibrium for either allozyme loci or DNA polymorphisms (see Laurie-Ahlberg and Weir 1979; Miyashita et al. 1993). The test for homogeneity of r-values is a valid test for r-values but is not suitable to detect disequilibrium heterogeneity among populations. This is because this statistical approach uses the strength of disequilibrium as the correlation coefficient of allele frequencies whose range is strongly frequency dependent (Sved 1971; Hedrick et al. 1978; Hedrick 1988; Lewontin 1988). Thus, the r coefficient ranges from 1.0 to 1.0 only if allele frequencies at the two loci are 0.5. The range of r will be smaller to the extent that allele frequencies at loci are more different. This property makes it inappropriate to compare disequilibrium values among populations that have different allelic frequencies. In general, any test of homogeneity based on a disequilibrium measure whose range is frequency dependent, such as r or D coefficients, will suffer from the same problem. Alternatively, tests for homogeneity of disequilibrium based on a measure that have the same range for all allelic frequencies are desirable. Although no measure of disequilibrium is completely independent of allele frequencies (Lewontin 1988), the D' disequilibrium coefficient suggested by Lewontin (1964) has a range that is frequency independent, and, therefore, D' is very useful for comparisons of disequilibrium for pairs of loci with different allelic frequencies (Hedrick 1987, 1988; Lewontin 1988). The D' coefficient is the ratio of the D coefficient to its theoretical maximum value, Dmax, given the gene frequencies and the sign of D. The Dmax value is min[p(l q), (1 p)q], when D > 0, or min [pq, (1 p)(l q)], when D < 0. Because D' is a measure of disequilibrium standardized by the Dmax value, its range varies from 1.0 to 1.0 for all combinations of allele frequencies at two loci. Of course, the r disequilibrium coefficient can

Infinite-Order Cointegrated Vector Autoregressive Processes

Estimation of cointegrated systems via autoregressive approximation is considered in the framework developed by Saikkonen (1992, Econometric Theory 8, 1-27). The asymptotic properties of the estimated coefficients of the autoregressive error correction model (ECM) and the pure vector autoregressive (VAR) representations are derived under the assumption that the autoregressive order goes to infinity with the sample size. These coefficients are often used for analyzing the relationships between the variables; therefore, they are important for applied work. Tests for linear restrictions on the coefficients of both the ECM and the pure VAR representation are considered under the present assumptions. It is found that they have limiting x2 distributions. Tests are also derived under the assumption that the number of restrictions goes to infinity with the sample size.

Central limit theorem for stochastically continuous processes. Convergence to stable limit

LetX={X(t), t∈[0, 1]} be a stochastically continuous cadlag process. Assume that thek dimensional finite joint distributions ofX are in the domain of normal attraction of strictlyp-stable, 0 g(u−s) andΛp(|X(s−X(t|)∧|X(t)−X(u|)>f(u−s), 0 ≤s ≤t ≤u ≤ 1, conditions are found which imply that the distributions −(n−1/p(X1+···+Xn )),n≥1, converge weakly inD[0, 1] to the distribution of ap-stable process. HereX1,X2, ... are independent copies ofX andΛp(Z)=supt<0tpP{|Z|<t} denotes the weakpth moment of a random variable Z.

The occupation time of Brownian motion in a ball

LetW be a Wiener process of dimensiond≥3, starting from 0, and letX(t) be the total time spent byW in the ball centered at 0 with radiust. We give an affirmative answer to a conjecture of Taylor and Tricot(16) on the tail distribution ofX(t). Levy's lower functions ofX(t) are characterized by an integral test.

A Note on Bootstrapping Generalized Method of Moments Estimators

Recently, Arcones and Giné (1992, pp. 13–47, in R. LePage &amp; L. Billard [eds.], Exploring the Limits of Bootstrap, New York: Wiley) established that the bootstrap distribution of the M-estimator converges weakly to the limit distribution of the estimator in probability. In contrast, Brown and Newey (1992, Bootstrapping for GMM, Seminar note) discovered that the bootstrap distribution of the GMM overidentification test statistic does not converge weakly to the x2 distribution. In this paper, it is shown that the bootstrap distribution of the GMM estimator converges weakly to the limit distribution of the estimator in probability. Asymptotic coverage probabilities of the confidence intervals based on the bootstrap percentile method are thus equal to their nominal coverage probability.

Likelihood ratio tests for symmetry against one-sided alternatives

A random variableX is said to have a symmetric distribution (about 0) if and only ifX and −X are, identically distributed. By considering various types of partial orderings between the distributions ofX and −X, one obtains various notions of skewness or one-sided bias. In this paper we study likelihood ratio tests for testing the symmetry of a discrete distribution about zero against the alternatives, (i)X is stochastically greater than −X; and (ii) pr(X=j)≥pr(X=−j) for allj>0. In the process, we obtain maximum likelihood estimators of the distribution function under the above alternatives. The asymptotic null distributions of the test statistics have been obtained and are of the chi-bar square type. A simulation study was performed to compare the powers of these tests with other tests.

The Poisson Distribution and "Multiple Events" Cell Occupancy Problems

SUMMARY This article studies a variant of the birthday problem called the multiple events problem. It asks the following general cell occupancy question: if one distributes N items or units at random over c equally likely categories or cells, what is the number N of items required in order to ensure that the probability of at least one cell having k or more items is greater than or equal to P? The question is of particular interest to epidemiologists who investigate clusters of disease. Although the exact probability, with fairly sophisticated and numerically complicated solutions to this problem, has appeared in the literature, we prefer here to present an intuitive solution, readily explainable to non-mathematicians and easy to calculate using standard available statistical or spreadsheet software. The key to the result is to use two successive applications of the Poisson approximation and to take advantage of the underused relationship between the Poisson and x2 distributions. We illustrate the methods in two typical epidemiologic applications.

ON A SHARED ALLELE TEST OF RANDOM MATING

SummaryA simple random sample is observed from a population with a large number‘K’ of alleles, to test for random mating. Of n couples, nijkl have female genotype ij and male genotype kl (i, j, k, l{1,…, A‘}). The large contingency table is collapsed into three counts, n0, n1 and n2 where np is the number of couples with s alleles in common (s = 0,1, 2). The counts are estimated by np̂o where n0, is the estimated probability of a couple having s alleles in common under the hypothesis of random mating. The usual chi‐square goodness of fit statistic X2 compares observed (ns) with expected (np̂) over the three categories, s = 0,1,2. An empirical observation has suggested that X2 is close to having a chi‐square distribution with two degrees of freedom (X) despite a large number of parameters implicitly estimated in e. This paper gives two theorems which show that x is indeed the approximate distribution of X2 for large n and K1“, provided that no allele type over‐dominates the others.

The Markov branching random walk and systems of reaction-diffusion (Kolmogorov-Petrovskii-Piskunov) equations

A general model of a branching random walk inR 1 is considered, with several types of particles, where the branching occurs with probabilities determined by the type of a parent particle. Each new particle starts moving from the place where it was born, independently of other particles. The distribution of the displacement of a particle, before it splits, depends on its type. A necessary and sufficient condition is given for the random variable $$X^0 = \mathop {\sup max}\limits_{ n \geqq 0 1 \leqq k \leqq N_n } X_{n,k} $$ to be finite. Here,X n, k is the position of thek th particle in then th generation,N n is the number of particles in then th generation (regardless of their type). It turns out that the distribution ofX 0 gives a minimal solution to a natural system of stochastic equations which has a linearly ordered continuum of other solutions. The last fact is used for proving the existence of a monotone travelling-wave solution to systems of coupled non-linear parabolic PDE's.

A note on chi-squared approximations for tests of rank interaction with ordered categorical data

The idea of rank interaction in factorial experiments when the response is only measured on an ordinal scale was introduced by de Kroon and van der Laan (1981). In this paper, the exact null distributions of test statistics for main effects and rank interaction are compared with their independent asymptotic x2 distributions, in particular for experiments with an ordered categorical response using an arbitrary score. Expressions are derived for the exact null variances and covariance of the two statistics for balanced experiments. Simulation is used to make further comparisons between the exact and the asymptotic distributions, and to consider experiments which are not balanced. Some recommendations are made as to when the asymptotic distributions may safely be used

Efficient computation of the noncentral x2distribution

In this article we compare and contrast finite algorithms for computing the noncentral x2 distribution function for odd or even degrees of freedom with the algorithms proposed recently by Ashour & Abdel-Samad (1990). We also obtain an alternative error bound for Ruben's (1974a) algorithm for even degrees of freedom and analyze the rate of convergence of two common infinite series representations for computing the cdf

On testing for independence against right tail increasing in bivariate models

A random variableY is right tail increasing (RTI) inX if the failure rate of the conditional distribution ofX givenY>y is uniformly smaller than that of the marginal distribution ofX for everyy≥0. This concept of positive dependence is not symmetric inX andY and is stronger than the notion of positive quadrant dependence. In this paper we consider the problem of testing for independence against the alternative thatY is RTI inX. We propose two distribution-free tests and obtain their limiting null distributions. The proposed tests are compared to Kendall's and Spearman's tests in terms of Pitman asymptotic relative efficiency. We have also conducted a Monte Carlo study to compare the powers of these tests.

Better approximate confidence intervals for a binomial parameter

AbstractThis paper discusses five methods for constructing approximate confidence intervals for the binomial parameter Θ, based on Y successes in n Bernoulli trials. In a recent paper, Chen (1990) discusses various approximate methods and suggests a new method based on a Bayes argument, which we call method I here. Methods II and III are based on the normal approximation without and with continuity correction. Method IV uses the Poisson approximation of the binomial distribution and then exploits the fact that the exact confidence limits for the parameter of the Poisson distribution can be found through the x2 distribution. The confidence limits of method IV are then provided by the Wilson‐Hilferty approximation of the x2. Similarly, the exact confidence limits for the binomial parameter can be expressed through the F distribution. Method V approximates these limits through a suitable version of the Wilson‐Hilferty approximation. We undertake a comparison of the five methods in respect to coverage probability and expected length. The results indicate that method V has an advantage over Chen's Bayes method as well as over the other three methods.

Measurement, Design and Analysis: An Integrated Approach.

Contents: Preface. Overview. Part I: Measurement. Measurement and Scientific Inquiry. Criterion-Related Validation. Construct Validation. Reliability. Selected Approaches to Measurement in Sociobehavioral Research. Part II: Design. Science and Scientific Inquiry. Definitions and Variables. Theories, Problems, and Hypotheses. Research Design: Basic Principles and Concepts. Artifacts and Pitfalls in Research. Experimental Designs. Quasi-Experimental Designs. Nonexperimental Designs. Introduction to Sampling. Part III: Analysis. Computers and Computer Programs. Simple Regression Analysis. Multiple Regression Analysis. A Categorical Independent Variable. Multiple Categorical Independent Variables: Factorial Designs. Attribute--Treatments--Interactions Analysis of Covariance. Exploratory Factor Analysis. Confirmatory Factor Analysis. Structural Equation Modeling. Appendices: Critical Values for F. Percentile Points for X2 Distribution.

The nontruncated marginal of a truncated bivariate normal distribution

Inference is considered for the marginal distribution ofX, when (X, Y) has a truncated bivariate normal distribution. TheY variable is truncated, but only theX values are observed. The relationship of this distribution to Azzalini's “skew-normal” distribution is obtained. Method of moments and maximum likelihood estimation are compared for the three-parameter Azzalini distribution. Samples that are uniformative about the skewness of this distribution may occur, even for largen. Profile likelihood methods are employed to describe the uncertainty involved in parameter estimation. A sample of 87 Otis test scores is shown to be well-described by this model.

A Simple Estimator of Cointegrating Vectors in Higher Order Integrated Systems

Efficient estimators of cointegrating vectors are presented for systems involving deterministic components and variables of differing, higher orders of integration. The estimators are computed using GLS or OLS, and Wald Statistics constructed from these estimators have asymptotic x2 distributions. These and previously proposed estimators of cointegrating vectors are used to study long-run U.S. money (Ml) demand. Ml demand is found to be stable over 1900-1989; the 95% confidence intervals for the income elasticity and interest rate semielasticity are (.88,1.06) and (-.13, -.08), respectively. Estimates based on the postwar data alone, however, are unstable, with variances which indicate substantial sampling uncertainty.

Pattern-Mixture Models for Multivariate Incomplete Data

Consider a random sample on variables X1, …, Xv with some values of Xv missing. Selection models specify the distribution of X1 , …, XV over respondents and nonrespondents to Xv , and the conditional distribution that Xv is missing given X1 , …, Xv . In contrast, pattern-mixture models specify the conditional distribution of X 1, …, Xv given that XV is observed or missing respectively and the marginal distribution of the binary indicator for whether or not Xv is missing. For multivariate data with a general pattern of missing values, the literature has tended to adopt the selection-modeling approach (see for example Little and Rubin); here, pattern-mixture models are proposed for this more general problem. Pattern-mixture models are chronically underidentified; in particular for the case of univariate nonresponse mentioned above, there are no data on the distribution of Xv given X1 , …, XV–1 , in the stratum with Xv missing. Thus the models require restrictions or prior information to identify the paramet...

On the choice of the number of residual autocovariances for the portmanteau test of multivariate autoregressive models

The multivariate portmanteau test proposed by Hosking (1980) for testing the adequacy of an autoregressive moving average model is based on the first s residual autocovariances of the fitted model.In practice a value for s is chosen in dependence on the sample size n, mostly s = 20 for n between 50 and 200. In this paper it will be shown by simulations that the usual choiceof s = 20 oftenleads to a significant deviation of the sample distribution of the test statistic Pm from the asymptotic X2 distribution. In the case of pure multivariate AR models the Kolmogorow-Smirnow test is used to find those values of s for which the sample distribution shows the best agreement with X2.In this manner s depends not only on the sample size n but also on the order of themodel p and the dimension m. A table for the best choice of s is given for n between 100 and 1000,p between 1 and 5 and m between 1 and 12.

Analysis of fish behaviour data

An examination of the experimental protocol reveals that there are two kinds of binary trials involved. If a fish is in the tank at the beginning of a time period, then it can only stay in (SIN) or transition out (TROUT). If the fish is not in the tank at the beginning of a time period, then it can only stay out (SouT) or transition in (TRIN). Both SIN and TRIN are sociable behaviours when other fish are present; both TRIN and TROUT indicate activity. A fish may do only one kind of trial for an entire string, or it may alternate the two types. The behaviour data for each fish consists of 15 strings, corresponding to the three different time periods and five days (the first five of the seven days described by Ruzzante and Doyle). The approach used here is to relate the probabilities of the basic events TRIN and SIN, or of related compound events, to the main variables of interest-growth (DL), environment (ENV), and time period (BAD = before, after, during)-taking into account variability due to the different days (DAY) and fish (FISH). To allow for possible autocorrelation among the binary trials in each string, an overdispersion parameter is included in the logistic regression model (McCullagh and Nelder 1989). It is assumed that a series of binary trials, X1,...,XX, made under the same conditions, have constant P(Xj = 1) = n, but variance-covariance matrix it(1 r)C. Then, for example, Y = EXj, the number of TRINS, has mean nnt and variance nt(1 it)(1 + c/n), where c is the sum of the off-diagonal entries in C, and the overdispersion parameter is 02 = (1 + c/n). The overdispersion parameter is estimated using the residual-deviance or goodness-of-fit statistic divided by the number of degrees of freedom, and significance tests are based on the relative change in deviance divided by d2 and compared with the F rather than the X2 distribution [see McCullagh and Nelder (1989, Section 6.3.1), for example]. The fish are nested within envionments and each fish also has a single value for DL, SO that if FISH is treated as a fixed effect, then ENV, FISH, and DL are aliased. In addition, repeated measurements on the same fish at different time periods and on different days could be correlated. In a linear model with normal errors these features could be accommodated using random effects (for FISH and DAY). Software for recently developed analogous procedures for generalized linear models (e.g. Zeger, Liang, and Albert 1988) is not generally available at this time. To circumvent these problems, a preliminary analysis treats FISH and DAY as fixed effects and ignores the aliased factor DL. Subsequent analyses involving DL and ENV are simplified by compressing the data for each fish over nonsignificant factors, and by using only one observation per fish. Models for marginal probabilities of TRINS and SINs are discussed in Section 2. In Section 3, the previous action in the string is explicitly included in models for the conditional probabilities of TRINS and SINs.