A simple approximation for the doubly noncentral t-distribution

A simple approximation for the doubly noncentral t-distribution, based upon the Fieller-Geary Theorem (1930) and approximate normality of the square root of the noncentral chi-square variable observed by Patnaik (1949), is developed, This approximation and an Edgeworth series expansion associated with it are evaluated. The simple approximation is seen to be reasonably accurate for most practical purposes.     

A comparison of some approximations to the Wilcoxon-Mann-Whitney distribution

The following approximations to the exact distribution of the Wilcoxon rank sum test (Mann-Whitney U-test) are compared at or near significance levels .05, .025, .01 and .005: the normal approximation with continuity correction, the simple and complete version of the Edgeworth series approximation proposed by Fix and Hodges (1955), Buckle, Kraft and van Eeden’s (1969a) uniform approximation and Iman’s (1976) recently proposed approximation. The comparison takes into account simplicity of application as well as closeness, and some preference rules are suggested for different user objectives.     

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.     

An approximation for the hotelling-lawley trace in the noncentral case

A procedure for estimating power in conjunction with the Hotelling-Lawley trace is developed. By approximating a non-central Wishart distribution with a central Wishart, and using McKeon's (1974) F-type approximation, a relatively simple procedure for obtaining power estimates is obtained. The accuracy of the approximation is investigated by comparing the approximate results with those for a wide range of conditions given in Olson's (1973) extensive Monte Carlo study. Siotani's (1971) asymptotic expansion is used to provide further comparative assessments. It is demonstrated that the approximation is of sufficient accuracy to be used in practical applications.     

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.     

The kullback - leibler approximation of the marginal posterior density: An application to the linear functional model

The main purpose of this paper is to introduce the Kullback - Leibler approximation to investigate the estimation of the unobserved x values in a simple linear regression model when the variables are measured vith error. The mean and the variance of the Kullback- Leibler approximatin are shown to coincide with the estimators considered by Fuller, (1987, p.38). The Kulback-Leibler approximation in the modal approximation (Kass et al., 1990) of the exact marginal posterior of the real value x.     

Likelihood inference for dynamic linear models with Markov switching parameters: on the efficiency of the Kim filter

The Kim filter (KF) approximation is widely used for the likelihood calculation of dynamic linear models with Markov regime-switching parameters. However, despite its popularity, its approximation error has not yet been examined rigorously. Therefore, this study investigates the reliability of the KF approximation for maximum likelihood (ML) and Bayesian estimations. To measure the approximation error, we compare the outcomes of the KF method with those of the auxiliary particle filter (APF). The APF is a numerical method that requires a longer computing time, but its numerical error can be sufficiently minimized by increasing simulation size. According to our extensive simulation and empirical studies, the likelihood values obtained from the KF approximation are practically identical to those of the APF. Furthermore, we show that the KF method is reliable, particularly when regimes are persistent and sample size is small. From the Bayesian perspective, we show that the KF method improves the efficiency of posterior simulation. This study contributes to the literature by providing evidence to justify the use of the KF method in both ML and Bayesian estimations.     

An approximation to the exact distribution of the wilcoxon-mann-whitney rank sum test statistic

An approximation to the exact distribution of the Wilcoxon rank sum test (Mann-Whitney U-test) and the Siegel-Tukey test based on a linear combination of the two-sample t-test applied to ranks and the normal approximation is compared with the usual normal approximation. The normal approximation results in a conservative test in the tails while the linear combination of the test statistics provides a test that has a very high percentage of agreement with tables of the exact distribution. Sample sizes 3≤m, n≤50 were considered.     

An approximation to the distribution of the ratio of two general quadratic forms with application

Based on mixed cumulants up to order six, this paper provides a four moment approximation to the distribution of a ratio of two general quadratic forms in normal variables. The approximation is applied to calculate the percentile points of modified F-test statistics for testing treatment effects when standard F-ratio test is misleading because of dependence among observations. For the special case, when data is generated by an AR(1) process, the approximation is evaluated by a simulation study. For the general SARMA (p,q)(P,Q)_s process, a modified F-test statistic Is given, and its distribution for the (0,1)(0,l)₁₂ process, is approximated by the moment approximation technique.     

A new approximation to the incomplete beta function

A new approximation to the incomplete beta function is proposed. This approximation compares favourably well with the Mudholkar-Chaubey (1976) approximation and it is infact superior for equal degrees of freedom. Moreover’ it is simpler to evaluate.     

A simple approximation relation between the symmetric generalized logistic and the students t distributions

In this paper, we obtain a new approximation of the Student's t distribution by using the symmetric generalized logistic (SGL) distribution function. The error of this approximation is shown to be 0(1/n² )where nis the degrees of freedom of thetdistribution. In comparison to similar approximations by George and Ojo and George et al. (1986), this new approximation is much simpler and more accurate. It is also shown that under some conditions, the tdistribution is a good approximation of the SGL distribution. Therefore, the complicated expressions for the cumulants and moments of the SGL can be approximated by those of the t, distribution. Finally, numerical results are given.     

On the distribution of Fisher's transformation of the correlation coefficient

As the sample size increases, the coefficient of skewness of the Fisher's transformation, z = (1/2) log ((l+r)/(l-r)), of the correlation coefficient decreases much more rapidly than the excess of its kurtosis. Hence, the usual normal approximation for its distribution can be improved by adjusting for the excess of its kurtosis. This is accomplished by mixing the approximating normal distribution with a logistic distribution. The resulting mixture approximation which can be used to estimate the probabilities, as well as the percentiles, compares favorably in both accuracy and simplicity, with the two best earlier approximations, namely, those due to Ruben (1966) and Kraemer (1973).     

A quadratic approximation for Jackknife estimators of the variance of sample mean functions

Jackknife estimators of the variance of estimators which are functions of the sample mean are considered. A quadratic approximation of them is proposed and compared with a linear approximation by Monte Carlo experiments carried out by statistical software Minitab.     

On Using Hellinger Distance in Checking the Validity of Dynamic Approximations

The purpose of this article is to provide validation for the approximate algebraic propagation algorithms to accommodate non-Gaussian dynamic processes. These algorithms have been developed to carry out Bayesian analysis based on conjugate forms and presented with detailed examples of response distributions such as Poisson and Lognormal. The validity of the approximation algorithms can be checked by introducing a metric (Hellinger divergence measure) over the distribution of the states (parameters) and use it to judge the approximation. Theoretical bounds for the efficacy of such procedure are discussed.     

An alternative approximation to the variance of transformation score

An alternative approximation to the variance of transformation score is given, based on an asymptotic expansion of the transformation estimator. It is then compared with the variance approximation given by Lawrance (1987) in terms of standardized scores. Simulations show that the two standardized scores behave very similarly when model error standard deviation is small. However,when error standard deviation is not small, the new standardized score outperforms that of Lawrance,especially in the structured models.     

FITTING SPATIAL CORRELATION MODELS: APPROXIMATING A LIKELIHOOD APPROXIMATION

The paper considers a class of spatial correlation models (stationary Gaussian processes) which includes (spatial) conditional autoregressive, simultaneous autoregressive, moving average and direct covariance models. Given observations on a finite rectangular lattice, a likelihood approximation for estimating the parameters in the spectral density of the model is discussed. The approximation consists of applying the trapezoidal rule, with a her grid of frequencies than the usual Fourier frequencies, to compute the integral in an appraximation due to Whittle (1954) and later modified by Guyon (1984). With this approximation, a Fisher scoring type algorithm has a simple form and in some casea reduces to iteratively reweighted least squares. Methods for computing the unbiased two-dimensional periodogram required by the method are presented and the accuracy of the approximation is discussed. The asymptotic distribution of the parameter estimates computed from the likelihood approximation is also given.     

A new sampling plan under the exponential distribution

This article considers a sampling plan when the quality characteristic follows the exponential distribution. We provide the exact approach and propose an approximated approach. In the proposed approximation, a new statistic combined with a Weibull transformation is used for the normal approximation. The plan parameters are obtained through the two-point approach at the acceptable quality level (AQL) and the limiting quality level (LQL). The tables for plan parameters are reported according to various values of the AQL and the LQL when the producer's and the consumer's risks are given. A real example is given to illustrate the proposed approximation approach.     

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.     

Approximation of the Distribution of the Location Parameter in the Growth Curve Model

Abstract.  In this paper an Edgeworth-type approximation of order O(n ⁻² ) to the density of the estimator of the location parameter in the growth curve model has been derived. The approximation is a mixture of a normal and a Kotz-type distribution, thus being an elliptical distribution. A condition for unimodality of the mixture was found and marginal distribution of a subvector of the mixture distribution was derived. Finally, a small example was given to demonstrate an application of the approximation.