Cramer-von Mises Type Statistics Research Articles

Log-Rank-Type Tests for Equality of Distributions in High-Dimensional Spaces

Motivated by applications in high-dimensional settings, we propose a novel approach for testing the equality of two or more populations by constructing a class of intensity centered score processes. The resulting tests are analogous in spirit to the well-known class of weighted log-rank statistics that are widely used in survival analysis. The test statistics are nonparametric, computationally simple, and applicable to high-dimensional data. We establish the usual large sample properties by showing that the underlying log-rank score process converges weakly to a Gaussian random field with zero mean under the null hypothesis and with a drift under the contiguous alternatives. For the Kolmogorov–Smirnov-type and the Cramer-von Mises-type statistics, we also establish the consistency result for any fixed alternative. Cutoff points for the test statistics are obtained by permutations or a simulation-based resampling method. The new approach is applied to a study of brain activation measured by functional magnetic resonance imaging when performing two linguistic tasks and also to a prostate cancer DNA microarray dataset. Supplementary materials for this article are available online.

Change-point tests under local alternatives for long-range dependent processes

We consider the change-point problem for the marginal distribution of subordinated Gaussian processes that exhibit long-range dependence. The asymptotic distributions of Kolmogorov-Smirnov- and Cramér-von Mises type statistics are investigated under local alternatives. By doing so we are able to compute the asymptotic relative efficiency of the mentioned tests and the CUSUM test. In the special case of a mean-shift in Gaussian data it is always $1$. Moreover, our theory covers the scenario where the Hermite rank of the underlying process changes. In a small simulation study, we show that the theoretical findings carry over to the finite sample performance of the tests.

Asymptotic behavior of weighted multivariate Cramér-von Mises-type statistics under contiguous alternatives

In this paper, we study the behavior of the weighted quadratic functionals of the multivariate empirical copula processes under sequences of contiguous alternatives. The Karhunen-Loeve expansions of the corresponding limiting Gaussian processes are derived by using the results, in a series of papers, by Deheuvels, which are used to obtain the asymptotic distribution of the weighted multivariate Cramer-von Mises-type statistics. These results are applied to compute the relative local asymptotic efficiency of the considered statistics, in the spirit of Genest et al. (2006, 2007), and discuss briefly some aspects regarding the power of these statistical tests. Finally, we give some additional results concerning the integrated copula processes.

A mixed-type test for linearity in time series

Abstract We propose a new test for linearity of time-series model by introducing a mixed-type statistic. It consists of a Cramer–Von Mises-type statistic and a goodness-of-fit statistic concerned only with fitting a linear autoregressive model. The computation involved in the new test is considerably simple and the curse of dimensionality is partly avoided. It is shown that the test is consistent against all fixed alternatives to linearity in stationary autoregressive series. The simulation results show that the test has good performance of power.

Modification of statistics based on the empirical characteristic function to yield asymptotic normality

A modification method of the Cramer-von Mises type statistic to yield asymptotic normality proposed by Ahmad(1993) is applied to the characteristic func tion based statistic. Weak convergence theorems of the empirical characteristic process are given. Similar modification of Epps and Pulley's statistic in testing composite hypothesis for normality is also considered.

Power of goodness of fit tests for the two-parameter weibull distribution with estimated parameters

The power of several EDF goodness-of-fit statistics was evaluated for the two-parameter Weibull distribution following estimation by maximum likelihood estimators (MLEs), good linear unbiased estimators (GLUEs), and modified GLUEs (MGLUEs). Relative power of Kolmogorov-Smirnov and Cramer-von Mises-type statistics was a function of level of Type I1 censoring, sample size, and method of parameter estimation. In generalW 2 and A 2 displayed more power than D U 2, and VPower was a function of method of estimation, but no one method was superior in all situations.

The Relative Efficiency of Goodness-of-Fit Statistics in the Simple and Composite Hypothesis-Testing Problem

Abstract Let X 1, …., X n be a sequence of independent and identically distributed random variables with an unknown underlying continuous cumulative distribution function F. Relative to this unknown distribution function, suppose one would like to test a null hypothesis concerning the goodness of fit of F to some distribution function using a Cramer-von Mises-type statistic. In some applications the null hypothesis is simple, whereas in others it may be composite. The subject of this article is the problem of comparing the efficiency of tests for the simple and composite hypotheses. Dyer (1974) and Stephens (1974) noticed that for testing normality using the Cramer-von Mises statistic the test procedures for which the nuisance parameters needed to be estimated were more powerful than the procedures in which the parameters were specified. We shall analytically compare these tests of the simple and composite hypotheses using the concepts of Bahadur efficiency and Pitman efficiency and thereby give a theoret...

The Relative Efficiency of Goodness-of-Fit Statistics in the Simple and Composite Hypothesis-Testing Problem

Abstract Let X 1, …., X n be a sequence of independent and identically distributed random variables with an unknown underlying continuous cumulative distribution function F. Relative to this unknown distribution function, suppose one would like to test a null hypothesis concerning the goodness of fit of F to some distribution function using a Cramer-von Mises-type statistic. In some applications the null hypothesis is simple, whereas in others it may be composite. The subject of this article is the problem of comparing the efficiency of tests for the simple and composite hypotheses. Dyer (1974) and Stephens (1974) noticed that for testing normality using the Cramer-von Mises statistic the test procedures for which the nuisance parameters needed to be estimated were more powerful than the procedures in which the parameters were specified. We shall analytically compare these tests of the simple and composite hypotheses using the concepts of Bahadur efficiency and Pitman efficiency and thereby give a theoretical basis for the paradoxical simulation results of Dyer and Stephens.

A correlation type procedure for assessing multivariate normality

A correlation-type statistic for assessing multivariate normality is described. Its estimated finite sample distribution is tabulated, and its performance against certain alternatives is compared with that of a competing Cramer-von Mises type statistic in a Monte Carlo power study. A set of quadrivariate data is examined as illustration of the procedure.

Tests for symmetry about an unknown value based on the empirical distribution function

A class of Kolmogorov-Smirnov and Cramer-von Mises type statistics for testing symmetry about an unknown value is described. These statistics are not distribution-free, however, and, indeed, are not readily amenable to calculation. A linear rank statistic analog of the first component of the Cramer-von Mises type statistic is investigated. Asymptotic non-null properties of these procedures in the normal case are studied, and an efficiency comparison of the Cramer-vonMises statistic, the linear rank statistic analog, the modified Wil-coxon statistic, and the likelihood ratio test is reported.

A class of invariant procedures for assessing multivariate normality

SUMMARY Distribution theory pertaining to a class of invariant procedures for assessing multivariate normality is described. A Cramer-von Mises type statistic belonging to this class is investigated in greater detail: its asymptotic distribution is tabulated and compared with a Monte Carlo simulation of its finite sample distribution, and a set of quadrivariate data is examined with it.

Asymptotic distribution of a Cramér-von mises type statistic for testing symmetry when the center is estimated

In this paper we investigate the effect of estimating the center of symmetry on a Cramer-von Mises type statistic for testing the symmetry of a distribution function. The test statistic is defined by $$nT_0 \left[ {F_n } \right] = n\int_\infty ^\infty {\left\{ {F_n \left( x \right) + Fn\left( {2S\left[ {F_n } \right] - x} \right) - 1} \right\}^2 } dF_n \left( x \right),$$ whereF n is the empirical distribution function andS[F n] is an estimator of the center ofF which is consistent with the ordern 1/2 and has von Mises derivative. The asymptotic distribution ofnT 0[Fn] under the null hypothesis is obtained. The distribution depends on the distributionF and on the estimatorS[F n].

On a Cramér-von Mises-Type Statistic for Testing Symmetry

Abstract A Cramer-von Mises-type statistic for testing symmetry, proposed independently by Orlov (1972) and Rothman and Woodroofe (1972), is decomposed into components in the manner of Durbin and Knott (1972); the components are found to be related to Fourier series expansions of the underlying distribution function. Asymptotic power properties of the statistic and its components in testing symmetry about the origin against location shift alternatives when observations have double exponential or normal distributions are described. From these power considerations, it is suggested that the first component is a more useful statistic for the testing problem than the overall Cramer-von Mises-type statistic. The components are shown to be asymptotically equivalent to linear rank statistics. On the basis of computational convenience, a new linear rank statistic for testing symmetry, the analog of the first component, is proposed.

Testing for bivariate normality using the empirical distribution function

The problem of testing for bivariate normality using the empirical distribution function is considered. A Cramer-von Mises type statistic is defined and asymptotic percentage points for this statistic given. This involves solving a two-dimensional homogeneous integral equation. Unfortunately the Cramer-von Mises statistic is not invariant under orthogonal transformations of the data so that an invariant statistic is developed. Approximations for the distribution of this statistic are found by Monte Carlo. Applications of the statistics are given. It is shown that the statistics are particularly sensitive to certain kinds of pattern in the data and they could be useful in data analysis apart from providing a formal test of bivariate normality

On the global asymptotic behavior of Brownian local time on the circle

The asymptotic behavior of the local time of Brownian motion on the circle is investigated. For fixed time point t this is a (random) continuous function on S 1 {S^1} . It is shown that after appropriate norming the distribution of this random element in C ( S 1 ) C({S^1}) converges weakly as t → ∞ t\, \to \,\infty . The limit is identified as 2 ( B ( x ) − ∫ B ( y ) d y ) 2(B(x)\, - \,\int {B(y)\,dy)} where B is the Brownian bridge. The result is applied to obtain the asymptotic distribution of a Cramer-von Mises type statistic for the global deviation of the local time from the constant t on S 1 {S^1} .

Tests for the exponential distribution with censored data using Cramér–von Mises statistics

SUMMARY This paper considers modifications of Cramer-von Mfises type statistics so that the statistics can be used to test the goodness of fit of censored data to the exponential distribution. Asymptotic percentage points are given for the statistics. In this paper we give the asymptotic distributions of two Cramer-von Mises type statistics used to test for the exponential distribution with censored data when the scale parameter must be estimated. The distributions given are those of the statistics

A Cramér-von Mises statistic for randomly censored data

SUMMARY The asymptotic distributions of Cramer-von Mises type statistics based on the productlimit estimate of the distribution function of a certain class of randomly censored observations are derived; the asymptotic significance points of the statistics for various degrees of censoring are given. The statistics are also partitioned into orthogonal components in the manner of Durbin & Knott (1972). The asymptotic powers of the statistics and their components against normal mean and variance shifts, exponential scale shifts, and Weibull alternatives to exponentiality are compared. Data arising in a competing risk situation are examined, using the Crame6r-von Mises statistic. An important problem in the analysis of survival data is whether the observed survival pattern for a cohort of individuals can be closely explained by a particular mathematical model with readily interpretable characteristics. Because the data arising from survival studies are often censored, because of deaths from competing causes or failure to observe individuals until time of death, statistical procedures formally relying upon the empirical distribution function can instead be defined in terms of the standard life table, or productlimit, estimate of the survival distribution. Our aim in this paper is to examine a Oram6r-von Mises type statistic for tests of goodness of fit, based on the product-limit empirical distribution function, when the data are subject to random censorship. In ? 2 we derive its asymptotic distribution under particular models of censorship; we investigate also the asymptotic properties of the components of the Cramer-von Mises statistic, which have been shown by Durbin & Knott (1972) to be of particular value in testing for specific aspects of the data. In ? 3 we study briefly the asymptotic power of this Cramer-von Mises statistic and its components against normal mean shift and variance shift alternatives, exponential scale shifts, and Weibull alternatives to exponentiality. In ? 4 we examine survival data with competing risks. We have derived the asymptotic distribution of a Cramer-von Mises type statistic only for a particular distributional form of random censorship; nevertheless, we believe that this pattern of censorship often does occur, and we are confident that our methods can be extended to other random censorship patterns.

Modified Cramér-von Mises statistics for censored data

SUMMARY Cramer-von Mises type statistics are modified so that they can be used to test the goodness of fit of a censored sample, when the distribution function is known completely. Asymptotic percentage points are given, small-sample applications are discussed and asymptotic power considered.

A Cramer Von-Mises Type Statistic for Testing Symmetry

A Cramer von-Mises type statistic is proposed for testing the symmetry of a continuous distribution function. Its asymptotic null distribution is found explicitly, and its asymptotic distribution under a sequence of local alternatives is described. A Monte Carlo study indicates that the asymptotic formulae are accurate for sample sizes as small as twenty.