Improved testing for the binomial distribution using chi-squared components with data-dependent cells

A power study suggests that a good test of fit analysis for the binomial distribution is provided by a data-dependent Chernoff–Lehmann X ² test with class expectations greater than unity, and its components. These data-dependent statistics involve arithmetically simple parameter estimation, convenient approximate distributions and provide a comprehensive assessment of how well the data agree with a binomial distribution. We suggest that a well-performed single test of fit statistic is the Anderson–Darling statistic.     

Goodness of fit for the inverse Gaussian distribution

For testing the fit of the inverse Gaussian distribution with unknown parameters, the empirical distribution-function statistic A² is studied. Two procedures are followed in constructing the test statistic; they yield the same asymptotic distribution. In the first procedure the parameters in the distribution function are directly estimated, and in the second the distribution function is estimated by its Rao-Blackwell distribution estimator. A table is given for the asymptotic critical points of A². These are shown to depend only on the ratio of the unknown parameters. An analysis is provided of the effect of estimating the ratio to enter the table for A². This analysis enables the proposal of the complete operating procedure, which is sustained by a Monte Carlo study.     

The X2-goodness-of-fit tests with moment type estimators

In the x²-goodness-of-fit test the underlying null hypothesis usually involves unknown parameters. In this article we study the asymptotic distribution of the Pearson statistic when the unknown parameters are estimated by a moment type estimator based on the ungrouped data. As is expected the usual Pearson statistic is no longer asymptotically x²-distributed in this situation. We propose a statistic [Qcirc] which under certain regularity conditions is asymptotically x²-distributed. We also propose a statistic Q? for the goodness-of-fit test when the class boundaries are random. The asymptotic powers of [Qcirc] and [Qcirc]? tests are discussed.     

Quantile inference based on partially rank-ordered set samples

This paper develops statistical inference for population quantiles based on a partially rank-ordered set (PROS) sample design. A PROS sample design is similar to a ranked set sample with some clear differences. This design first creates partially rank-ordered subsets by allowing ties whenever the units in a set cannot be ranked with high confidence. It then selects a unit for full measurement at random from one of these partially rank-ordered subsets. The paper develops a point estimator, confidence interval and hypothesis testing procedure for the population quantile of order p. Exact, as well as asymptotic, distribution of the test statistic is derived. It is shown that the null distribution of the test statistic is distribution-free, and statistical inference is reasonably robust against possible ranking errors in ranking process.     

On the distributions of multivariate sample skewness

In this paper, we consider the multivariate normality test based on measure of multivariate sample skewness defined by Srivastava (1984). Srivastava derived asymptotic expectation up to the order N⁻¹ for the multivariate sample skewness and approximate χ²${\chi }^2$ test statistic, where N   is sample size. Under normality, we derive another expectation and variance for Srivastava's multivariate sample skewness in order to obtain a better test statistic. From this result, improved approximate χ²${\chi }^2$ test statistic using the multivariate sample skewness is also given for assessing multivariate normality. Finally, the numerical result by Monte Carlo simulation is shown in order to evaluate accuracy of the obtained expectation, variance and improved approximate χ²${\chi }^2$ test statistic. Furthermore, upper and lower percentiles of χ²${\chi }^2$ test statistic derived in this paper are compared with those of χ²${\chi }^2$ test statistic derived by Mardia (1974) which is used multivariate sample skewness defined by Mardia (1970).     

Empirical likelihood inference for probability density functions under association

In this paper, we consider to apply the empirical likelihood method to a probability density function under an associated sample. It is shown that the empirical likelihood ratio statistic is asymptotically χ²-type distributed under some mild conditions. The result is used to construct empirical likelihood-based confidence intervals on the probability density function.     

A test statistic for weighted runs

A new test statistic based on runs of weighted deviations is introduced. Its use for observations sampled from independent normal distributions is worked out in detail. It supplements the classic χ² test which ignores the ordering of observations and provides additional sensitivity to local deviations from expectations. The exact distribution of the statistic in the non-parametric case is derived and an algorithm to compute p-values is presented. The computational complexity of the algorithm is derived employing a novel identity for integer partitions.     

A new ranked set sampling protocol for the signed rank test

In this paper, we construct a new ranked set sampling protocol that maximizes the Pitman asymptotic efficiency of the signed rank test. The new sampling design is a function of the set size and independent order statistics. If the set size is odd and the underlying distribution is symmetric and unimodal, then the new sampling protocol quantifies only the middle observation. On the other hand, if the set size is even, the new sampling design quantifies the two middle observations. This data collection procedure for use in the signed rank test outperforms the data collection procedure in the standard ranked set sample. We show that the exact null distribution of the signed rank statistic W_RSS⁺ based on a data set generated by the new ranked set sample design for odd set sizes is the same as the null distribution of the simple random sample signed rank statistic W_SRS⁺ based on the same number of measured observations. For even set sizes, the exact null distribution of W_RSS⁺ is simulated.     

Empirical likelihood for probability density functions under negatively associated samples

In this paper, we study the construction of confidence intervals for a probability density function under a negatively associated sample by using the blockwise technique. It is shown that the blockwise empirical likelihood (EL) ratio statistic is asymptotically χ²‐type distributed. The result is used to obtain EL based confidence interval on the probability density function.     

Approximation of distributions by using the Anderson Darling statistic

ABSTRACT

In practice, it is often not possible to find an appropriate family of distributions which can be used for fitting the sample distribution with high precision. In these cases, it seems to be opportune to search for the best approximation by a family of distributions instead of an exact fit. In this paper, we consider the Anderson–Darling statistic with plugged-in minimum distance estimator for the parameter vector. We prove asymptotic normality of the Anderson–Darling statistic which is used for a test of goodness of approximation. Moreover, we introduce a measure of discrepancy between the sample distribution and the model class.     

Distributional Studies and the Computer: An Analysis of Durbin's Rank Test

A study of the distribution of a statistic involves two major steps: (a) working out its asymptotic, large n, distribution, and (b) making the connection between the asymptotic results and the distribution of the statistic for the sample sizes used in practice. This crucial second step is not included in many studies. In this article, the second step is applied to Durbin's (1951) well-known rank test of treatment effects in balanced incomplete block designs (BIB's). We found that asymptotic, χ², distributions do not provide adequate approximations in most BIB's. Consequently, we feel that several of Durbin's recommendations should be altered.     

On Non-Normal Invariance Principles for Multi-Response Permutation Procedures

A non-normal invariance principle is established for a restricted class of univariate multi-response permutation procedures whose distance measure is the square of Euclidean distance. For observations from a distribution with finite second moment, the test statistic is found asymptotically to have a centered chi-squared distribution. Spectral expansions are used to determine the asymptotic distribution for more general distance measures d, and it is shown that if d(x, y) = |x — y|^u, u? 2, the asymptotic distribution is not invariant (i.e. it is dependent on the distribution of the observations).     

Generalized bartlett correction

This paper provides a general method of modifying a statistic of interest in such a way that the distribution of the modified statistic can be approximated by an arbitrary reference distribution to an order of accuracy of O(n ^-1/2) or even O(n ^-1). The reference distribution is usually the asymptotic distribution of the original statistic. We prove that the multiplication of the statistic by a suitable stochastic correction improves the asymptotic approximation to its distribution. This paper extends the results of the closely related paper by Cordeiro and Ferrari (1991) to cope with several other statistical tests. The resulting expression for the adjustment factor requires knowledge of the Edgeworth-type expansion to order O(n^-1) for the distribution of the unmodified statistic. In practice its functional form involves some derivatives of the reference distribution. Certain difference between the cumulants of appropriate order in n of the unmodified statistic and those of its first-order approximation, and the unmodified statistic itself. Some applications are discussed.     

Comparisons of approximate confidence intervals for the smallest extreme value distribution simple linear regression model under time censoring

Exact confidence interval estimation for accelerated life regression models with censored smallest extreme value (or Weibull) data is often impractical. This paper evaluates the accuracy of approximate confidence intervals based on the asymptotic normality of the maximum likelihood estimator, the asymptotic X²distribution of the likelihood ratio statistic, mean and variance correction to the likelihood ratio statistic, and the so-called Bartlett correction to the likelihood ratio statistic. The Monte Carlo evaluations under various degrees of time censoring show that uncorrected likelihood ratio intervals are very accurate in situations with heavy censoring. The benefits of mean and variance correction to the likelihood ratio statistic are only realized with light or no censoring. Bartlett correction tends to result in conservative intervals. Intervals based on the asymptotic normality of maximum likelihood estimators are anticonservative and should be used with much caution.     

A Non‐parametric ANOVA‐type Test for Regression Curves Based on Characteristic Functions

This article studies a new procedure to test for the equality of k regression curves in a fully non‐parametric context. The test is based on the comparison of empirical estimators of the characteristic functions of the regression residuals in each population. The asymptotic behaviour of the test statistic is studied in detail. It is shown that under the null hypothesis, the distribution of the test statistic converges to a finite combination of independent chi‐squared random variables with one degree of freedom. The coefficients in this linear combination can be consistently estimated. The proposed test is able to detect contiguous alternatives converging to the null at the rate n ^{? 1 ∕ 2}. The practical performance of the test based on the asymptotic null distribution is investigated by means of simulations.     

A likelihood ratio test of a homoscedastic normal mixture against a heteroscedastic normal mixture

It is generally assumed that the likelihood ratio statistic for testing the null hypothesis that data arise from a homoscedastic normal mixture distribution versus the alternative hypothesis that data arise from a heteroscedastic normal mixture distribution has an asymptotic χ ² reference distribution with degrees of freedom equal to the difference in the number of parameters being estimated under the alternative and null models under some regularity conditions. Simulations show that the χ ² reference distribution will give a reasonable approximation for the likelihood ratio test only when the sample size is 2000 or more and the mixture components are well separated when the restrictions suggested by Hathaway (Ann. Stat. 13:795–800, 1985) are imposed on the component variances to ensure that the likelihood is bounded under the alternative distribution. For small and medium sample sizes, parametric bootstrap tests appear to work well for determining whether data arise from a normal mixture with equal variances or a normal mixture with unequal variances.     

Asymptotic expansions of the distributions of the chi-square statistic based on the asymptotically distribution-free theory in covariance structures

An asymptotic expansion of the null distribution of the chi-square statistic based on the asymptotically distribution-free theory for general covariance structures is derived under non-normality. The added higher-order term in the approximate density is given by a weighted sum of those of the chi-square distributed variables with different degrees of freedom. A formula for the corresponding Bartlett correction is also shown without using the above asymptotic expansion. Under a fixed alternative hypothesis, the Edgeworth expansion of the distribution of the standardized chi-square statistic is given up to order O(1/n). From the intermediate results of the asymptotic expansions for the chi-square statistics, asymptotic expansions of the joint distributions of the parameter estimators both under the null and fixed alternative hypotheses are derived up to order O(1/n).     

An L-statistic approach to a test of exponentiality against IFR alternatives

In this note we consider the problem of testing exponentiality against IFR alternatives. A measure of deviation from exponentiality is developed and a test statistic constructed on the basis of this measure. It is shown that the test statistic is an L-statistic. The asymptotic as well as the exact distribution of the test statistic is obtained and the test is shown to be consistent.