Asymptotic expansions relating to discrimination based on two-step monotone missing samples

This paper proposes two asymptotic expansions relating to discrimination based on two-step monotone missing samples. These asymptotic expansions have been obtained by Okamoto (1963) and McLachlan (1973) for complete data under multivariate normality. This paper extends the results up to the terms of the first order in the case of two-step monotone missing samples, respectively. Especially, these asymptotic expansions play important roles in obtaining the asymptotic approximations for the probabilities of misclassification in discriminant analysis. The simulation studies have been also conducted in order to evaluate the accuracy of the approximation derived in this paper.     

Testing Normality Against the Logistic Distribution Using Saddlepoint Approximation

We consider the problem of testing normality against the logistic distribution, based on a random sample of observations. Since the two families are separate (non nested), the ratio of maximized likelihoods (RML) statistic does not have the usual asymptotic chi-square distribution. We derive the saddlepoint approximation to the distribution of the RML statistic and show that this approximation is more accurate than the normal and Edgeworth approximations, especially for tail probabilities that are the main values of interest in hypothesis testing. It is also shown that this test is almost identical to the most powerful invariant test.     

A modification of the Fisher-Cornish approximation for the student t percentiles

An asymptotic expansion of the Student t distribution is derived by expanding the standardized Student t distribution in terms of the normal distribution. This expansion is inverted to obtain corresponding asymptotic expansions for the Student t percentiles as functions of the standard normal percentiles0 Using the first two, three or four terms of these expansions, we get approximations of the Student t percentiles which are generally more accurate than the approximations given by Fisher and Cornish(1960) and Koehler (1983).An approximation of the distribution function obtained from this expansion is compared with the approximations discussed by Ling (1978) andfound to be more accurate for moderate degrees of freedom.     

Approximations for the doubly noncentral-F distribution

The approximate normality of the cube root of the noncentral chi-square observed by Aty (1954) and an Edgeworth-series expansion are used to derive an approximation for the doubly noncentral-F distribution. Another approximation in terms of a noncentral-F distribution is also proposed. Both these approximations are seen to compare favorably with some earlier approximations due to Das Gupta (1968) and Tiku (1972). The problem of approximating the cumulants of the doubly noncentral-F variable, which is pivotal in Tiku’s approximation, is examined and use of a noncentral-F distribution is seen to provide a good solution for it. A FORTRAN routine for the Edgeworth-series approximation is given.     

Asymptotic Expansions in Multi-Group Analysis of Moment Structures with an Application to Linearized Estimators

Asymptotic expansions of the joint distributions of functions of sample means and central moments up to an arbitrary order in multiple populations are given by Edgeworth expansions. The asymptotic distributions of the parameter estimators in moment structures under null/fixed alternative hypotheses and the chi-square statistics based on asymptotically distribution-free theory under fixed alternatives are given as applications of the above results. Asymptotic expansions of the null distributions of the chi-square statistics are also derived. For parameter estimators with the chi-square statistic, the linearized estimators are dealt with as well as fully iterated estimators.     

Box‐Cox transformations in linear models: Large sample theory and tests of normality

The authors provide a rigorous large sample theory for linear models whose response variable has been subjected to the Box‐Cox transformation. They provide a continuous asymptotic approximation to the distribution of estimators of natural parameters of the model. They show, in particular, that the maximum likelihood estimator of the ratio of slope to residual standard deviation is consistent and relatively stable. The authors further show the importance for inference of normality of the errors and give tests for normality based on the estimated residuals. For non‐normal errors, they give adjustments to the log‐likelihood and to asymptotic standard errors.     

Edgeworth Corrections for Realized Volatility

The quality of the asymptotic normality of realized volatility can be poor if sampling does not occur at very high frequencies. In this article we consider an alternative approximation to the finite sample distribution of realized volatility based on Edgeworth expansions. In particular, we show how confidence intervals for integrated volatility can be constructed using these Edgeworth expansions. The Monte Carlo study we conduct shows that the intervals based on the Edgeworth corrections have improved properties relatively to the conventional intervals based on the normal approximation. Contrary to the bootstrap, the Edgeworth approach is an analytical approach that is easily implemented, without requiring any resampling of one's data. A comparison between the bootstrap and the Edgeworth expansion shows that the bootstrap outperforms the Edgeworth corrected intervals. Thus, if we are willing to incur in the additional computational cost involved in computing bootstrap intervals, these are preferred over the Edgeworth intervals. Nevertheless, if we are not willing to incur in this additional cost, our results suggest that Edgeworth corrected intervals should replace the conventional intervals based on the first order normal approximation.     

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).     

On two asymptotic normal distributions for the generalized wilks lambda statistic

In this paper we.present a Normal asymptotic distribution for the logarithm of the generalized Wilks Lambda statistic based on an asymptotic distribution for the determinant of a Wishart matrix. This distribution is obtained through the combined use of Taylor expansions of random variables whose exponentials have chi-square distributions and the Lindeberg-Feller version of the Central Limit Theorem, Another asymptotic Normal distribution for the logarithm of the generalized Wilks Lambda statistic for the case when at most one of the sets has an odd number of variables is derived directly from the exact distribution. Both distributions are non-degenerate and non-singular. The first Normal distribution compares favorably with other known approximations and asymptotic distributions namely for large numbers of variables and small sample sizes, while the second Normal distribution, which has a more restricted application, compares in most cases highly favorably with other known asymptotic distributions and approximations. Finally, a method to compute approximate quantiles which lay very close and converge steadily to the exact ones is presented.     

Asymptotic normality for chi-bar-square distributions

Chi-bar-square distributions, which are mixtures of chi-square distributions, mixed over their degrees of freedom, often occur when testing hypotheses that involve inequality constraints. Here, necessary and sufficient conditions on the mixing or weighting distribution are found to ensure asymptotic normality of the corresponding chi-bar-square distribution. Essentially, asymptotic normality occurs for the chi-bar-square distribution if either the ratio of the mean to the variance of the mixing distribution goes to infinity, or the weighting distribution itself is asymptotically normal. Other than a combination of these two phenomena, this is also the only way for asymptotic normality to hold. Several examples of pertinent chi-bar-square distributions are shown to be asymptotically normal by the results in this paper.     

Large sample results for permutation tests of association

This paper deals with the asymptotics of permutation tests based on a certain rather general class of measures of association for R by C contingency tables, given marginal totals. This class includes the classical chi-square test, the T _b and γ indices of Goodman and Kruskall (1954) and the popular Rand (1971) index. The asymptotic distribution of this class of permutation tests for association is a weighted sum of non-central (gen-erally speaking) chi-squares. The formulae for the asymptotic moments of such tests are also given. If non-centrality holds under the null hypothe-sis of independence, the distribution in question converges to the normal distribution. The efficacies for such measures of association are obtained. Several applications are analysed in detail, including the above mentioned indices. Approximations to the permutation distribution are also discussed.     

Edgeworth Corrections for Realized Volatility

The quality of the asymptotic normality of realized volatility can be poor if sampling does not occur at very high frequencies. In this article we consider an alternative approximation to the finite sample distribution of realized volatility based on Edgeworth expansions. In particular, we show how confidence intervals for integrated volatility can be constructed using these Edgeworth expansions. The Monte Carlo study we conduct shows that the intervals based on the Edgeworth corrections have improved properties relatively to the conventional intervals based on the normal approximation. Contrary to the bootstrap, the Edgeworth approach is an analytical approach that is easily implemented, without requiring any resampling of one's data. A comparison between the bootstrap and the Edgeworth expansion shows that the bootstrap outperforms the Edgeworth corrected intervals. Thus, if we are willing to incur in the additional computational cost involved in computing bootstrap intervals, these are preferred over the Edgeworth intervals. Nevertheless, if we are not willing to incur in this additional cost, our results suggest that Edgeworth corrected intervals should replace the conventional intervals based on the first order normal approximation.     

Second-order and resampling approximation of finite population U-statistics based on stratified samples

We construct one-term Edgeworth expansions to distributions of U statistics and Studentized U-statistics, based on stratified samples drawn without replacement. Replacing the cumulants defining the expansions by consistent jackknife estimators, we obtain empirical Edgeworth expansions. The expansions provide second-order approximations that improve upon the normal approximation. Theoretical results are illustrated by a simulation study where we compare various approximations to the distribution of the commonly used Gini's mean difference estimator.     

Improvement of the quality of the chi-square approximation for the ADF test on a covariance matrix with a linear structure

The asymptotically distribution-free (ADF) test statistic was proposed by Browne (1984). It is known that the null distribution of the ADF test statistic is asymptotically distributed according to the chi-square distribution. This asymptotic property is always satisfied, even under nonnormality, although the null distributions of other famous test statistics, e.g., the maximum likelihood test statistic and the generalized least square test statistic, do not converge to the chi-square distribution under nonnormality. However, many authors have reported numerical results which indicate that the quality of the chi-square approximation for the ADF test is very poor, even when the sample size is large and the population distribution is normal. In this paper, we try to improve the quality of the chi-square approximation to the ADF test for a covariance matrix with a linear structure by using the Bartlett correction applicable under the assumption of normality. By conducting numerical studies, we verify that the obtained Bartlett correction can perform well even when the assumption of normality is violated.     

Continuity corrected approximations for and ‘exact’ inference with Pearson's X2

The classical adjustments for the inadequacy of the asymptotic distribution of Pearson's X² statistic, when some cells are sparse or the cell expectations are small, use continuity corrections and exact moments; the recent approach is to use computer based ‘exact inference’. In this paper we observe that the original exact test due to Freeman and Halton (Biometrika 38 (1951), 141–149) and its computer implementation are theoretically unsound. Furthermore, the corrected algorithmic version for the exact p-value in StatXact is practically useful in very few cases, and the results of its present version which includes Monte Carlo estimates can be highly variable. We then derive asymptotic expansions for the moments of the null distribution of Pearson's X², introduce a new method of correcting for discreteness and finite range of Pearson's X² as an alternative to the classical continuity correction, and use them to construct new and improved approximations for the null distribution. We also offer diagnostic criteria applicable to the tables for selecting an appropriate approximation. The exact methods and the competing approximations are studied and compared using thirteen test cases from the literature. It is concluded that the accuracy of the appropriate approximation is comparable with the truly exact method whenever it is available. The use of approximations is therefore preferable if the truly exact computer intensive solutions are unavailable or infeasible.     

Study of the quality of several asymptotic and near-exact approximations based on moments for the distribution of the Wilks Lambda statistic

In this paper a measure of proximity of distributions, when moments are known, is proposed. Based on cases where the exact distribution is known, evidence is given that the proposed measure is accurate to evaluate the proximity of quantiles (exact vs. approximated). The measure may be applied to compare asymptotic and near-exact approximations to distributions, in situations where although being known the exact moments, the exact distribution is not known or the expression for its probability density function is not known or too complicated to handle. In this paper the measure is applied to compare newly proposed asymptotic and near-exact approximations to the distribution of the Wilks Lambda statistic when both groups of variables have an odd number of variables. This measure is also applied to the study of several cases of telescopic near-exact approximations to the exact distribution of the Wilks Lambda statistic based on mixtures of generalized near-integer gamma distributions.     

Saddlepoint approach to inference for response probabilities under the logistic response model

This paper studies lower confidence limits of response probabilities based on sensitivity testing data set. The saddlepoint approximation to a conditional distribution is developed. Based on it we give a modified algorithm to find approximate confidence limits for the parameter of interest. A simulation study shows that the saddlepoint approximation with proper corrections gives better coverage probability than the direct saddlepoint approximation and the asymptotic normality approximation. Finally, we apply the proposed approximation to a real data set.     

A new approximation to the distribution function of the studentized maximum root

The Studentized maximum root (SMR) distribution is useful for constructing simultaneous confidence intervals around product interaction contrasts in replicated two-way ANOVA. A three-moment approximation to the SMR distribution is proposed. The approximation requires the first three moments of the maximum root of a central Wishart matrix. These values are obtained by means of numerical integration. The accuracy of the approximation is compared to the accuracy of a two-moment approximation for selected two-way table sizes. Both approximations are reasonably accurate. The three-moment approximation is generally superior.     

A General formula for the upper tail significance levels of empirical distribution function test statistics

In this paper ve obtain an asymptotic expression for the upper tail area of the distribution of an infinite weighted sum of chi-square random variables and show how this can be applied to distributions of various goodness of fit test statistics. Results obtained by this general approach are comparable with those reported previously in the literature. In the case of the Cramer-von Mises statistic an empirical adjustment is given vhich significantly improves on previous approximations. For the Kuiper statistic the corresponding empirical adjustment leads to an existing highly accurate approximation.     

Development and comparative study of two near-exact approximations to the distribution of the product of an odd number of independent Beta random variables

Using the concept of near-exact approximation to a distribution we developed two different near-exact approximations to the distribution of the product of an odd number of particular independent Beta random variables (r.v.'s). One of them is a particular generalized near-integer Gamma (GNIG) distribution and the other is a mixture of two GNIG distributions. These near-exact distributions are mostly adequate to be used as a basis for approximations of distributions of several statistics used in multivariate analysis. By factoring the characteristic function (c.f.) of the logarithm of the product of the Beta r.v.'s, and then replacing a suitably chosen factor of that c.f. by an adequate asymptotic result it is possible to obtain what we call a near-exact c.f., which gives rise to the near-exact approximation to the exact distribution. Depending on the asymptotic result used to replace the chosen parts of the c.f., one may obtain different near-exact approximations. Moments from the two near-exact approximations developed are compared with the exact ones. The two approximations are also compared with each other, namely in terms of moments and quantiles.