Asymptotic Percentage Points Research Articles

Goodness of Fit Tests for Marshal-Olkin Extended Rayleigh Distribution

A class of goodness of fit tests for Marshal-Olkin Extended Rayleigh distribution with estimated parameters is proposed. The tests are based on the empirical distribution function. For determination of asymptotic percentage points, Kolomogorov-Sminrov, Cramer-von-Mises, Anderson-Darling,Watson, and Liao-Shimokawa test statistic are used. This article uses Monte Carlo simulations to obtain asymptotic percentage points for Marshal-Olkin extended Rayleigh distribution. Moreover, power of the goodness of fit test statistics is investigated for this lifetime model against several alternatives.

The Anderson–Darling test of fit for the power-law distribution from left-censored samples

Maximum likelihood estimation and a test of fit based on the Anderson–Darling statistic are presented for the case of the power-law distribution when the parameters are estimated from a left-censored sample. Expressions for the maximum likelihood estimators and tables of asymptotic percentage points for the A 2 statistic are given. The technique is illustrated for data from the Dow Jones Industrial Average index, an example of high theoretical and practical importance in Econophysics, Finance, Physics, Biology and, in general, in other related sciences such as Complexity Sciences.

TEST OF FIT FOR THE INVERSE GAUSSIAN AND GAMMA DISTRIBUTIONS UNDER CENSORING

The problem of testing the fit of the inverse Gaussian and the gamma distribution when the sample is censored and some of the parameters are unknown, is studied. Empirical Distribution Function (EDF) statistics, namely Cramér-von Mises' W 2 and the Anderson-Darling's A 2, are used. The limiting covariance functions of the corresponding empirical processes are derived. Asymptotic percentage points are given for some parameter values and censoring proportions. Moreover, a numerical routine is made available upon request, to obtain p-values for both test statistics, thus eliminating the need of tables and interpolation. Finally, a simple Monte Carlo study is presented to evaluate first, the approximation when using the asymptotic distributions in finite samples and second, to support the use of estimated parameter values instead of the unknown parameters needed in the limiting covariance function.

A cramer-von mises type goodness-of-fit test with asymmetric weight function. the gaussian and exponential cases

A goodness of fit test of the Cramer - von Mises type which gives more weight to the upper (lower) tail of the distribution was proposed and studied by the authors, Rodriguez and Viollaz (1995), for the case of a simple null hypothesis. In the present paper that goodness of fit test is studied for a composite null hypothesis in the case of a Gaussian distribution with one or two unknown parameters and the exponential distribution with unknown parameter. Asymptotic percentage points of the distribution function of the test statistic are obtained for the cases studied: Case 1:Normal distribution with known variance, Case 2: Normal distribution with known mean, Case 3: normal distribution with unknown mean and variance and Case 4: exponential distribution with unknown parameter. For all cases asymptotic means and variances of the goodness of fit statistic are given, as well as some weights of the decomposition of the test as a sum of weighted x2(1)variables.

Asymptotic percentage points for Siegel's test statistic for compound periodicities

Fisher's test for a periodicity at a single Fourier frequency is often inadequate in practice where it is necessary to test for compound periodicities, i.e. spectral lines at several frequencies. Siegel (1980) derived a test statistic for compound periodicity and provided percentage points for m not exceeding 50, where N=2m+1 is the length of the time series. Here the distribution of the test statistic is rewritten to enable computation of exact percentage points for m up to 2000, and accurate empirical interpolation formulae derived

Some empirical distribution function tests for multivariate normality

This paper presents finite sample percentage points of the Anderson-Darling (AD) statistic for testing the composite hypothesis of Gaussianity (normality) for dimensionality 1≦p≦5 when the parameters are estimated from data. This paper also presents asymptotic percentage points for both the Anderson-Darling and the Cramervon Mises(CM) statistics for testing the composite hypothesis of Gaussianity for dimensionality 1≦p≦25 when the parameters are estimated from data. The AD test is developed from the fact that the quadratic form of the Gaussian density is distributed as chi-squared on p degrees of freedom. A small power study contrasts the finite sample performance of the AD and CM statistics. Several examples are discussed in the context of chi-squared probability plotting.

Testing for Normality in Arbitrary Dimension

The univariate weak convergence theorem of Murota and Takeuchi (1981) is extended for the Mahalanobis transform of the $d$-variate empirical characteristic function, $d \geq 1$. Then a maximal deviation statistic is proposed for testing the composite hypothesis of $d$-variate normality. Fernique's inequality is used in conjunction with a combination of analytic, numerical analytic, and computer techniques to derive exact upper bounds for the asymptotic percentage points of the statistic. The resulting conservative large sample test is shown to be consistent against every alternative with components having a finite variance. (If $d = 1$ it is consistent against every alternative.) Monte Carlo experiments and the performance of the test on some well-known data sets are also discussed.

On a separation theorem for generalized eigenvalues and a problem in the analysis of sample surveys

We obtain usable bounds for the asymptotic percentage points of chi-squared tests of fit for log-linear models fitted to contingency tables estimated from survey data, by applying some new separation inequalities for the generalized eigenvalues of a matrix X′AX with respect to a matrix X′BX, when both the matrices A and B are nonnegative definite. We also present some historical remarks on the Poincaré separation theorem for eigenvalues from which our new inequalities are shown to follow.

Tests of fit for the logistic distribution based on the empirical distribution function

SUMMARY Goodness-of-fit tests are given for the logistic distribution, based on statistics calculated from the empirical distribution function. Emphasis is on the statistics W2, U2 and A2, for which asymptotic percentage points are given, for each of the three cases where one or both of the parameters of the distribution must be estimated from the data. Slight modifications of the calculated statistics are given to enable the points to be used with small samples. Monte Carlo results are included also for statistics D+, D-, D and V.

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

Goodness of Fit for the Extreme Value Distribution

In this paper we present goodness of fit tests for the extreme value distribution, based on the empirical distribution function statistics W2, U2and A2. Asymptotic percentage points are given for each of the three statistics, for the three cases where one or both of the parameters of the distribution must be estimated from the data. Slight modifications of the calculated statistics are given to enable the points to be used with small samples.

Goodness of fit for the extreme value distribution

SUMMARY In this paper we present goodness of fit tests for the extreme value distribution, based on the empirical distribution function statistics W2, U2 and A2. Asymptotic percentage points are given for each of the three statistics, for the three cases where one or both of the parameters of the distribution must be estimated from the data. Slight modifications of the calculated statistics are given to enable the points to be used with small samples.

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

Testing the Normality of Several Independent Samples Using the Anderson-Darling Statistic

In a recent paper, the Anderson‐Darling goodness‐of‐fit statistic was shown to be a powerful omnibus test of normality when the mean and variance are unknown. In this paper, tables for calculating small sample percentage points of the statistic are given. The percentage points are derived from smoothing empirical percentage points using the asymptotic percentage points. These percentage points are then used to make a joint assessment of normality based on several independent samples using Fisher's method. Empirical power comparisons are then made with the method based on the Shapiro‐Wilk statistic. There is seen to be little difference between the methods based on the two different statistics.

Cramér-von Mises statistics for testing normality with censored samples

Asymptotic theory is developed for the distribution of Cramér-von Mises statistics for testing the goodness of fit of censored samples. Asymptotic percentage points are given and also small-sample Monte Carlo percentage points for testing for normality.

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.