Non-Normal Universes Research Articles

Approximate test for testing a null variance ratio in the Unbalanced One-Way Random Model

The approximate test for testing the signicance of the random effect is presented in the unbalanced one-way random model in which both random effects and errors are from non-normal universes. The test is based on the asymptotic distribution of the F-statistic. Under the condition that the number of groups tends to infinity while the average of powers of the group sizes is bounded, the asymptotic dis tribution of the F-statistic is obtained. Robustness of the proposed test is given.

Testing for random effect in the Fuller–Battese model

We consider the Fuller–Battese model where random effects are allowed to be from non-normal universes. The asymptotic distribution of the F-statistic in this model is derived as the number of groups tends to infinity (is large) and sample size from any group is either fixed or large. The result is used to establish an approximate test for the significance of the random effect variance component. Robustness of the established approximate test is given.

The limiting distribution of the F-statistic from nonnormal universes

We consider a linear regression model with an unbalanced 1-fold nested error structure, where group effect and error are from nonnormal universes. The limiting distribution of the F-statistic in this model is derived, as the sample size is large and group sizes take values from a finite set of distinct integers. The result is used to approximate the F-distribution quantile and to test the significance of the random effect variance component. Results are also applicable to the F-statistic in the one-way random-effects model. The effects of departure from normality on the F-statistic distribution are given.

Inference for Variance Components in a Mixed Model for Unbalanced Split Plot Design

We consider the unbalanced split-plot design with the whole plot and the subplot effect from nonnormal universes. The three estimators for the whole plot effect variance component are obtained. An approximate test for significance of the whole plot effect variance component is presented.

A Robust Test for Multi‐Sample Location Problems with Unequal Group Variances and Nonnormality

AbstractA robust test (to be referred to as M* test) is proposed for testing equality of several group means without assuming normality and equality of variances. This test statistic is obtained by combining Tiku's MML robust procedure with the James statistic. Monte Carlo simulation studies indicate that the M* test is more powerful than the Welch test, the James test, and the tests based on Huber's M‐estimators over a wide range of nonnormal universes. It is also more powerful than the Brown and Forsythe test under most of nonnormal distributions and has substantially the same power as the Brown and Forsythe test under normal distribution. Comparing with Tan‐Tabatabai test, M* is almost as powerful as Tan‐Tabatabai test.

On approximating probability distributions of a statistic of brown-forsythe from normal and non-normal universes

This article considers a one-way ANOVA model as given in (1.1) with heterogeneous variances. The distributions of the Brown-Forsythe F * statistic for this model are derived and approximated by Laguerre polynomial expansions. The Type 1 error and power function values are then evaluated for both normal and mixtures of two normals populations. For the example with k=3n 1=8,n 2=10 and n 3=12. it is shown that the Satterthwaite approximation provides a close approximation to the null and non-null distributions in most o the cases. It is shown that the F * is quite robust with respect to departure from normality for Type 1 error in the case of mixtures of two normals.

On comparing several straight lines under heteroscedasticity and robustness with respect to departure from normality

This article considers the problem of testing slopes in k straight lines with'heterogeneous variances. The statistic Fβ is proposed and the null and non-null distributions of Fβ derived under normality assumption. The power function values are then approximated by Laguerre polynomial expansion for normal and non-normal universes. For the example given in Graybill ‘1976, p. 295’, it is shown that the Satterthwaite approximation provides a close approximation to the null and non-null distributions in all the cases; it is also shown that the Fβ test is quite robust with respect to departure from normality in the case of mixtures of two normals.

Cumulants of bilinear forms and quadratic forms and their applications

In this paper, mixed cumulants of general bilinear forms, quadratic forms and linear forms are given up to order 4. These results are useful in deriving approximations to the distributions of sums of squares and F-ratios from unbalanced analysis of variance models from non-normal universes. As an illustration, the results are applied to derive mixed cumulants of sums of squares in a three stages nested unbalanced random effects models from non-normal universes.

On Approximating the Null and Nonnull Distributions of the F Ration i Unbalanced Random-Effect Models from Nonnormal Universes

On Approximating the Null and Nonnull Distributions of the F Ration i Unbalanced Random-Effect Models from Nonnormal Universes

On the Roy-Tiku Approximation to the Distribution of Sample Variances from Nonnormal Universes

Abstract Some justifications for the Roy-Tiku (1962) approximation to the distributions of sample variances by Laguerre polynomial series are provided. It is shown that if one approximates any nonnegative continuous random variable by a finite series of Laguerre polynomials up to the kth degree, then the first k moments of the approximated distribution are equal to the first k moments of the random variable. We also derive an alternative Laguerre series approximation, including the Box (1953) approximation as a special case.

On the Roy-Tiku Approximation to the Distribution of Sample Variances from Nonnormal Universes

On the Roy-Tiku Approximation to the Distribution of Sample Variances from Nonnormal Universes

Optimum Tolerance Regions and Power When Sampling from Some Non-Normal Universes.

Optimum Tolerance Regions and Power When Sampling from Some Non-Normal Universes.

Optimum Tolerance Regions and Power When Sampling From Some Non-Normal Universes

We assume familiarity with the concepts defined in [1] and [2], where optimum $\beta$-expectation tolerance regions and their power functions were found for $k$-variate normal distributions. The method used is to reduce this problem to that of solving an equivalent hypothesis testing problem. It is the purpose of this paper to find optimum $\beta$-expectation tolerance regions for the single and double exponential distributions, and to exhibit the corresponding power functions. Let $X = (X_1, \cdots, X_n)$ be a random sample point in $n$ dimensions, where each $X_i$ is an independent observation, distributed by some continuous probability distribution function. It is often desirable to estimate on the basis of such a sample point a region, say $S(X_1, \cdots, X_n)$, which contains a given fraction $\beta$ of the parent distribution. We usually seek to estimate the center 100 $\beta$% of the distribution and/or one of the 100 $\beta$% tails of the parent distribution.

Sampling from Bivariate Non-Normal Universes by Means of Compound Normal Distributions

Sampling from Bivariate Non-Normal Universes by Means of Compound Normal Distributions

The Frequency Distribution of the Product-Moment Correlation Coefficient in Random Samples of Any Size Drawn from Non-Normal Universes

The Frequency Distribution of the Product-Moment Correlation Coefficient in Random Samples of Any Size Drawn from Non-Normal Universes

The Distribution of the Variance Ratio in Random Samples of Any Size Drawn from Non-Normal Universes

The Distribution of the Variance Ratio in Random Samples of Any Size Drawn from Non-Normal Universes

The Distribution of `Student's t in Random Samples of any Size Drawn from Non-Normal Universes

The Distribution of `Student's t in Random Samples of any Size Drawn from Non-Normal Universes

The Distribution of "Student's" Ratio for Samples of Two Items Drawn from Non-Normal Universes

In this paper, the formal expression for the distribution of t will be derived for samples of two drawn from any population having a continuous frequency function. A geometrical method similar to that employed by Rider will be used. Let the universe sampled have the frequency function, f(x), with zero mean, and let f(x) be greater than zero from x = a to x b and equal to zero elsewhere. Suppose the two observations are x1 and x2.

On Small Samples from Certain Non-Normal Universes

On Small Samples from Certain Non-Normal Universes

On the Distribution of the Ratio of Mean to Standard Deviation in Small Samples from Non-Normal Universes

On the Distribution of the Ratio of Mean to Standard Deviation in Small Samples from Non-Normal Universes