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.     

EMPIRICALLY ACCELERATED CONVERGENCE OF THE SPACINGS STATISTICS TO NORMALITY

If the asymptotic normality of a statistic is inadequate for approximating its distribution in practice, then the statistic may be transformed in order to improve the approximation by accelerating the convergence to normality. We treat a goodness-of-fit statistic, the sum of the logarithms of generalized uniform spacings introduced by Cressie (1976, 1978), in this spirit. Specifically, we apply the method of maximum likelihood to simulations of the statistic in order to estimate a power transformation, as in Box & Cox (1964), and hence develop a small sample normal approximation. This approximation provides a more versatile method of applying the statistic than currently available tables of percentiles.     

A generalized improvement procedure for variance bounds for minimum variance unbiased estimator in inverse sampling

New bounds are obtained for the variance of the minimum variance unbiased estimator of p i n inverse sampling. A generalized procedure for further improving the bounds is also discussed.     

R-Symmetry and the Power Functions of the Tests for Inverse Gaussian Means

The two-parameter Inverse Gaussian (IG) distribution is often appropriate for modeling non negative right-skewed data due to the striking similarities with the Gaussian distribution in its basic properties and inference methods. There are about 40 such G-IG analogies developed in literature and were most recently tabulated by Mudholkar and Wang. Of these, the earliest and most commonly noted similarities are the significance tests based on student's t and F distribution for the homogeneity of one, two or several means of the IG populations. However, unlike the corresponding tests in Gaussian theory, little is known about the power function of the basic tests. In this article, we employ the IG-related root-reciprocal IG distribution and a notion of Reciprocal Symmetry to establish the monotonicity of the power function of the test of significance for the IG mean.     

Modeling the Effectiveness of Respiratory Protective Devices in Reducing Influenza Outbreak

Outbreaks of influenza represent an important health concern worldwide. In many cases, vaccines are only partially successful in reducing the infection rate, and respiratory protective devices (RPDs) are used as a complementary countermeasure. In devising a protection strategy against influenza for a given population, estimates of the level of protection afforded by different RPDs is valuable. In this article, a risk assessment model previously developed in general form was used to estimate the effectiveness of different types of protective equipment in reducing the rate of infection in an influenza outbreak. It was found that a 50% compliance in donning the device resulted in a significant (at least 50% prevalence and 20% cumulative incidence) reduction in risk for fitted and unfitted N95 respirators, high‐filtration surgical masks, and both low‐filtration and high‐filtration pediatric masks. An 80% compliance rate essentially eliminated the influenza outbreak. The results of the present study, as well as the application of the model to related influenza scenarios, are potentially useful to public health officials in decisions involving resource allocation or education strategies.     

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.     

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).