Tables for the Friedman rank test

This note presents tables for Friedman's test for two-way analysis of variance by ranks. These tables are more accurate than those that are presented in the literature. After intensive simulations, we have found for particular critical values some discrepancies with tables published earlier. The tables are also more extensive than those previously available.     

Extended critical vawes of the multivariate extreme deviate test for detecting a single spurious observation

The Institute of Mathematical Statistics has published a table of critical values for the multivariate extreme deviate test. However, the critical values, derived by a Monte Carlo simulation, are given for only the dimensions 2 through 5. We present new critical values for the dimensions 6 through 10, 12, 15, and 20. The results are presented in both table and graphical form. All critical values for the test statistic have been generated by a Monte Carlo simulation using 10,000 observations per case. An example is presented using the new critical values.     

A weighted symmetric cointegration test

Weighted symmetric estimation is employed to develop a new test for cointegration. Using Monte Carlo simulation, the resulting test is shown to possess greater power than alternative existing tests.     

Power analysis of independence testing for three-way contingency tables of small sizes

The first aim of this paper is to introduce a modular test for the three-way contingency table (TT). The second aim is to describe the procedure of generating TT using the bar method. The third aim is on the one hand to suggest the measure of untruthfulness of H₀ and on the other hand to compare the quality of independence tests by using their power. Critical values for analyzed statistics were determined by simulating the Monte Carlo method.     

On the bootstrap quantile-treatment-effect test

Let {X ₁, …, X _n } and {Y ₁, …, Y _m } be two samples of independent and identically distributed observations with common continuous cumulative distribution functions F(x)=P(X≤x) and G(y)=P(Y≤y), respectively. In this article, we would like to test the no quantile treatment effect hypothesis H ₀: F=G. We develop a bootstrap quantile-treatment-effect test procedure for testing H ₀ under the location-scale shift model. Our test procedure avoids the calculation of the check function (which is non-differentiable at the origin and makes solving the quantile effects difficult in typical quantile regression analysis). The limiting null distribution of the test procedure is derived and the procedure is shown to be consistent against a broad family of alternatives. Simulation studies show that our proposed test procedure attains its type I error rate close to the pre-chosen significance level even for small sample sizes. Our test procedure is illustrated with two real data sets on the lifetimes of guinea pigs from a treatment-control experiment.     

On a test of independence for contingency tables

Using the concept of distributional distance, a test statistic is proposed FOR the hypothesis of independence in multidimensional contingency tables. A Monte Carlo Study is done to empirically compare the power of the proposed test to the Pearson x² and the likelihood ratio test- Further, the nonnull distribution under various spike alternatives is tabulated     

A likelihood ratio test of quasi-independence for sparse two-way contingency tables

We consider a likelihood ratio test of independence for large two-way contingency tables having both structural (non-random) and sampling (random) zeros in many cells. The solution of this problem is not available using standard likelihood ratio tests. One way to bypass this problem is to remove the structural zeroes from the table and implement a test on the remaining cells which incorporate the randomness in the sampling zeros; the resulting test is a test of quasi-independence of the two categorical variables. This test is based only on the positive counts in the contingency table and is valid when there is at least one sampling (random) zero. The proposed (likelihood ratio) test is an alternative to the commonly used ad hoc procedures of converting the zero cells to positive ones by adding a small constant. One practical advantage of our procedure is that there is no need to know if a zero cell is structural zero or a sampling zero. We model the positive counts using a truncated multinomial distribution. In fact, we have two truncated multinomial distributions; one for the null hypothesis of independence and the other for the unrestricted parameter space. We use Monte Carlo methods to obtain the maximum likelihood estimators of the parameters and also the p-value of our proposed test. To obtain the sampling distribution of the likelihood ratio test statistic, we use bootstrap methods. We discuss many examples, and also empirically compare the power function of the likelihood ratio test relative to those of some well-known test statistics.     

Minimally informative distributions with given rank correlation for use in uncertainty analysis

Minimum information bivariate distributions with uniform marginals and a specified rank correlation are studied in this paper. These distributions play an important role in a particular way of modeling dependent random variables which has been used in the computer code UNICORN for carrying out uncertainty analyses. It is shown that these minimum information distributions have a particular form which makes simulation of conditional distributions very simple. Approximations to the continuous distributions are discussed and explicit formulae are determined. Finally a relation is discussed to DAD theorems, and a numerical algorithm is given (which has geometric rate of covergence) for determining the minimum information distributions.     

The application of the durbin-watson test to the dynamic regression model under normal and non-normal errors

Until recently, a difficulty with applying the Durbin-Watson (DW) test to the dynamic linear regression model has been the lack of appropriate critical values. Inder (1986) used a modified small-disturbance distribution (SDD) to find approximate critical values. King and Wu (1991) showed that the exact SDD of the DW statistic is equivalent to the distribution of the DW statistic from the regression with the lagged dependent variables replaced by their means. Unfortunately, these means are unknown although they could be estimated by the actual variable values. This provides a justification for using the exact critical values of the DW statistic from the regression with the lagged dependent variables treated as non-stochastic regressors. Extensive Monte Carlo experiments are reported in this paper. They show that this approach leads to reasonably accurate critical values, particularly when two lags of the dependent variable are present. Robustness to non-normality is also investigated.     

Estimation for the generalized half-normal distribution based on progressive type-II censoring

Based on progressively type-II censored data, the problem of estimating unknown parameters and reliability function of a two-parameter generalized half-normal distribution is considered. Maximum likelihood estimates are obtained by applying expectation-maximization algorithm. Since they do not have closed forms, approximate maximum likelihood estimators are proposed. Several Bayesian estimates with respect to different symmetric and asymmetric loss functions such as squared error, LINEX and general entropy are calculated. The Lindley approximation method is applied to determine Bayesian estimates. Monte Carlo simulations are performed to compare the performances of the different methods. Finally, one real data set is analysed.     

A MATLAB package for multivariate normality test

A MATLAB package testing for multivariate normality (TMVN) is implemented as an interactive and graphical tool to examine multivariate normality (MVN). Monte Carlo simulation studies have failed to find a uniformly most powerful MVN test, which requires a rather extensive statistical inference procedure. TMVN contains several competitive MVN tests and provides a flexible and extensive testing environment for univariate or multivariate data analyses. Simulated results provide information of which test may possess more power for the selected non-MVN alternatives. Fisher's Iris data are used to show how TMVN can be used in practice.     

The robustness of the likelihood ratio test,the nonparametric sum rank test,and f-ratio tests when the populations are from the negative binomial family

In this study, the robustness of power and significance level of several statistical testing methods was evaluated under the assumption that the test populations were from Poisson, negative binomial, or geometric distributions. The F-ratio test, with or without appropriate transformations, was shown to be both safe and robust for all distributions examined.     

Rank-based test procedures for interaction in the two-way layout with one observation per cell

New aligned-rank test procedures for the composite null hypothesis of no interaction effects (without placing restrictions on the two main effects) against appropriate composite general alternatives are developed for the standard two-way layout with a single observation per cell. Relative power performances of the two new aligned-rank procedures and existing tests due to Tukey (1949) and to de Kroon & van der Laan (1981) are examined via Monte Carlo simulation. Extensive power studies conducted on the 5 × 6 and 5 × 9 two-way layouts with one observation per cell show superior performance of the new procedures for a variety of interaction effects. Simulated critical values for the new procedures are provided in settings where the number of levels for each of the factors is between 3 and 9, inclusive.     

Prediction Intervals for an Ordered Observation from Weibull Distribution Based on Censored Samples

In this article, we provide some suitable pivotal quantities for constructing prediction intervals for the jth future ordered observation from the two-parameter Weibull distribution based on censored samples. Our method is more general in the sense that it can be applied to any data scheme. We present a simulation of our method to analyze its performance. Two illustrative examples are also included. For further study, our method is easily applied to other location and scale family distributions.     

Data driven chi-square test for uniformity with unequal cells

The power of Pearson's chi-square test for uniformity depends heavily on the choice of the partition of the unit interval involved in the form of the test statistic. We propose a selection rule which chooses a proper partition based on the data. This selection rule leads usually to essentially unequal cells well suited to the observed distribution. We investigate the corresponding data driven chi-square test and present a Monte Carlo simulation study. The conclusion is that this test achieves a high and very stable power for a large class of alternatives, and is much more stable than any other test we compare to.     

Monte Carlo Approximations of the Quantiles of a Sample Statistic

We consider in this article the problem of numerically approximating the quantiles of a sample statistic for a given population, a problem of interest in many applications, such as bootstrap confidence intervals. The proposed Monte Carlo method can be routinely applied to handle complex problems that lack analytical results. Furthermore, the method yields estimates of the quantiles of a sample statistic of any sample size though Monte Carlo simulations for only two optimally selected sample sizes are needed. An analysis of the Monte Carlo design is performed to obtain the optimal choices of these two sample sizes and the number of simulated samples required for each sample size. Theoretical results are presented for the bias and variance of the numerical method proposed. The results developed are illustrated via simulation studies for the classical problem of estimating a bivariate linear structural relationship. It is seen that the size of the simulated samples used in the Monte Carlo method does not have to be very large and the method provides a better approximation to quantiles than those based on an asymptotic normal theory for skewed sampling distributions.     

A modified Kolmogorov-Smirnov test for the inverse gaussian density with unknown parameters

The Kolmogorov-Smirnov (KS) test is an empirical distribution function (EDF) based goodness-of-fit test that requires the underlying hypothesized density to be continuous and completely specified. When the parameters are unknown and must be estimated from the data, standard tables of the KS test statistic are not valid. Approximate upper tail percentage points of the KS statistic for the inverse Gaussian (IG) distribution with unknown parameters are tabled in this paper.

A study of the power of the KS test for the IG distribution indicates that the test is able todiscriminate between the IG distribution and distributions such as the uniform and exponentialdistributions that are very different in shape, but is relatively unable to discriminate between the IG distribution and distributions that are similar in shape such as the lognormal and Weibull distributions. In modeling settings the former distinction is typically more important to make than the latter distinction.