Use of the squared ranks test to test for the equality of the coefficients of variation

A distribution-free test for the equality of the coefficients of variation from k populations is obtained by using the squared ranks test for variances, as presented by Conover and Iman (1978) and Conover (1980), on the original observations divided by their respective expected values. Substitution of the sample mean in place of the expected value results in the test being only asymptotically distribution-free. Results of a simulation study evaluating the size of the test for various coefficient of variation values and probability distributions are presented.     

Letters to the Editor

Although several authors have indicated that the median test has low power in small samples, it continues to be presented in many statistical textbooks, included in a number of popular statistical software packages, and used in a variety of application areas. We present results of a power simulation study that shows that the median test has noticeably lower power, even for the double exponential distribution for which it is asymptotically most powerful, than other readily available rank tests. We suggest that the median test be “retired” from routine use and recommend alternative rank tests that have superior power over a relatively large family of symmetric distributions.     

Asymptotic test statistics for coefficients of variation

A one-sample asymptotically normal test statistic Is derived for testing the hypothesis that the coefficient of variation of a normal population is equal to a specified value. Based on this derivation, an asymptotically noraml two-sample test statistic and an asymptotically chi-square k-sample test statistic are derived for testing the hypothesis that the coefficients of variation of k ≥2 normal populations are equal. The two and k-sample test statistics allow for unequal sample sizes. Results of a simulation study which evaluate the size and power of the test statistics and compare the test statistics to earlier ones developed by McKay (1932) and Bennett (1976) are presented.     

Testing for multivariate heteroscedasticity

In this article, we propose a testing technique for multivariate heteroscedasticity, which is expressed as a test of linear restrictions in a multivariate regression model. Four test statistics with known asymptotical null distributions are suggested, namely the Wald, Lagrange multiplier (LM), likelihood ratio (LR) and the multivariate Rao F-test. The critical values for the statistics are determined by their asymptotic null distributions, but bootstrapped critical values are also used. The size, power and robustness of the tests are examined in a Monte Carlo experiment. Our main finding is that all the tests limit their nominal sizes asymptotically, but some of them have superior small sample properties. These are the F, LM and bootstrapped versions of Wald and LR tests.     

Performance of the shrinkage preliminary test ridge regression estimators based on the conflicting of W,LR and LM tests

The shrinkage preliminary test ridge regression estimators (SPTRRE) based on the Wald (W), the likelihood ratio (LR) and the Lagrangian multiplier (LM) tests are considered in this paper. The bias and the risk functions of the proposed estimators are derived. The regions of optimality of the estimators are determined under the quadratic risk function. Under the null hypothesis, the SPTRRE based on LM test has the smallest risk, followed by the estimators based on LR and W tests. However, the SPTRRE based on W test performs the best followed by the LR and LM based estimators when the parameter moves away from the subspace of the restrictions. The conditions of superiority of the proposed estimator for both ridge and departure parameters are discussed. The optimum choice of the level of significance becomes the traditional choice by using the W test for all non-negative ridge parameters.     

On Small Samples Testing for Frailty Through Homogeneity Test

We derive a test in order to examine the need of modeling survival data using frailty models based on the likelihood ratio (LR) test for homogeneity. Test is developed for both complete and censored samples from a family of baseline distributions that satisfy a closure property. Approach motivated by I-divergence distance is used in order to determine “credible” regions for all parameters of baseline distribution for which homogeneity hypothesis is not rejected. Proposed test outperforms the usual asymptotic LR test both in very small samples with known frailty and for all small sample sizes under misspecified frailty.     

A METHOD FOR DETERMINING THE ASYMPTOTIC EFFICIENCY OF SOME SEQUENTIAL PROBABILITY RATIO TESTS

This paper gives a method for decomposing many sequential probability ratio tests into smaller independent components called “modules”. A function of some characteristics of modules can be used to determine the asymptotically most efficient of a set of statistical tests in which a, the probability of type I error equals β, the probability of type II error. The same test is seen also to give the asymptotically most efficient of the corresponding set of tests in which a is not equal to β. The “module” method is used to give an explanation for the super-efficiency of the play-the-winner and play-the-loser rules in two-sample binomial sampling. An example showing how complex cases can be analysed numerically using this method is also given.     

A Criterion for Stepwise Regression

In “stepwise” regression analysis, the usual procedure enters or removes variables at each “step” on the basis of testing whether certain partial correlation coefficients are zero. An alternative method suggested in this paper involves testing the hypothesis that the mean square error of prediction does not decrease from one step to the next. This is equivalent to testing that the partial correlation coefficient is equal to a certain nonzero constant. For sample sizes sufficiently large, Fisher's z transformation can be used to obtain an asymptotically UMP unbiased test. The two methods are contrasted with an example involving actual data.     

The Effect of Pseudo-Exogenous Instrumental Variables on Hausman Test

Hausman test is popularly used to examine the endogeneity of explanatory variables in a regression model. To derive a well-defined asymptotic distribution of Hausman test, the correlation between the instrumental variables and the error term needs to converge to zero. However, it is possible that there remains considerable correlation in finite samples between the instruments and the error, even though their correlation eventually converges to zero. This article investigates the potential problem that such “pseudo-exogenous” instruments may create. We show that the performance of Hausman test is deteriorated when the instruments are asymptotically exogenous but endogenous in finite samples, through Monte Carlo simulations.     

Inferences on the lognormal mean for complete samples

In many engineering problems it is necessary to draw statistical inferences on the mean of a lognormal distribution based on a complete sample of observations. Statistical demonstration of mean time to repair (MTTR) is one example. Although optimum confidence intervals and hypothesis tests for the lognormal mean have been developed, they are difficult to use, requiring extensive tables and/or a computer. In this paper, simplified conservative methods for calculating confidence intervals or hypothesis tests for the lognormal mean are presented. In this paper, “conservative” refers to confidence intervals (hypothesis tests) whose infimum coverage probability (supremum probability of rejecting the null hypothesis taken over parameter values under the null hypothesis) equals the nominal level. The term “conservative” has obvious implications to confidence intervals (they are “wider” in some sense than their optimum or exact counterparts). Applying the term “conservative” to hypothesis tests should not be confusing if it is remembered that this implies that their equivalent confidence intervals are conservative. No implication of optimality is intended for these conservative procedures. It is emphasized that these are direct statistical inference methods for the lognormal mean, as opposed to the already well-known methods for the parameters of the underlying normal distribution. The method currently employed in MIL-STD-471A for statistical demonstration of MTTR is analyzed and compared to the new method in terms of asymptotic relative efficiency. The new methods are also compared to the optimum methods derived by Land (1971, 1973).     

Inference of Seasonal Long‐memory Time Series with Measurement Error

We consider the Whittle likelihood estimation of seasonal autoregressive fractionally integrated moving‐average models in the presence of an additional measurement error and show that the spectral maximum Whittle likelihood estimator is asymptotically normal. We illustrate by simulation that ignoring measurement errors may result in incorrect inference. Hence, it is pertinent to test for the presence of measurement errors, which we do by developing a likelihood ratio (LR) test within the framework of Whittle likelihood. We derive the non‐standard asymptotic null distribution of this LR test and the limiting distribution of LR test under a sequence of local alternatives. Because in practice, we do not know the order of the seasonal autoregressive fractionally integrated moving‐average model, we consider three modifications of the LR test that takes model uncertainty into account. We study the finite sample properties of the size and the power of the LR test and its modifications. The efficacy of the proposed approach is illustrated by a real‐life example.     

Estimation of the population mean when the coefficient of variation is known

Theory has been developed to provide an optimum estimator of the population mean based on a “mean per unit” estimator and the estimated standard deviation, assuming that the form of the distribution as well as its coefficient of variation (c.v.) are known. Theory has been extended to the case when an estimate of c.v. is available from an independent sample drawn in the past; the case when the form of the distribution is not known is also discussed. It is shown that the relative efficiency of the estimator with respect to “mean per unit estimator” is generally high for normal or near normal populations. For log-normal populations, an increase in efficiency of about 17 percent can be achieved. The results have been illustrated with data from biological populations.     

A monte carlo study of the wald,lm, and lr tests in a heteroscedastic linear model

In this paper, we examine by Monte Carlo experiments the small sample properties of the W (Wald), LM (Lagrange Multiplier) and LR (Likelihood Ratio) tests for equality between sets of coefficients in two linear regressions under heteroscedasticity. The small sample properties of the size-corrected W, LM and LR tests proposed by Rothenberg (1984) are also examined and it is shown that the performances of the size-corrected W and LM tests are very good. Further, we examine the two-stage test which consists of a test for homoscedasticity followed by the Chow (1960) test if homoscedasticity is indicated or one of the W, LM or LR tests if heteroscedasticity should be assumed. It is shown that the pretest does not reduce much the bias in the size when the sizecorrected citical values are used in the W, LM and LR tests.     

On the relationship between the matrix operators,vech and vecd

We introduce a matrix operator, which we call “vecd” operator. This operator stacks up “diagonals” of a symmetric matrix. This operator is more convenient for some statistical analyses than the commonly used “vech” operator. We show an explicit relationship between the vecd and vech operators. Using this relationship, various properties of the vecd operator are derived. As applications of the vecd operator, we derive concise and explicit expressions of the Wald and score tests for equal variances of a multivariate normal distribution and for the diagonality of variance coefficient matrices in a multivariate generalized autoregressive conditional heteroscedastic (GARCH) model, respectively.     

A Robust Score Test for Testing Several Coefficients of Variation with Unknown Underlying Distributions

A parametric robust test is proposed for comparing several coefficients of variation. This test is derived by properly correcting the normal likelihood function according to the technique suggested by Royall and Tsou. The proposed test statistic is asymptotically valid for general random variables, as long as their underlying distributions have finite fourth moments.

Simulation studies and real data analyses are provided to demonstrate the effectiveness of the novel robust procedure.     

Preliminary Test Estimators Induced by Three Large Sample Tests for Stochastic Constraints in a Regression Model with Multivariate Student-t Error

In this article, we consider the preliminary test approach to the estimation of the regression parameter in a multiple regression model with multivariate Student-t distribution. The preliminary test estimators (PTE) based on the Wald (W), Likelihood Ratio (LR), and Lagrangian Multiplier (LM) tests are given under the suspicion of stochastic constraints occurring. The bias, mean square error matr ix (MSEM), and weighted mean square error (WMSE) of the proposed estimators are derived and compared. The conditions of superiority of the proposed estimators are obtained. Finally, we conclude that the optimum choice of the level of significance becomes the traditional choice by using the W test.     

The exact risks of some pre-test and stein-type regression estimators umder balanced loss

In regression analysis we are often interested in using an estimator which is “precise” and which simultaneously provides a model with “good fit”, In this paper we consider the risk properties of several estimators of the regression coefficient vector "trader “balanced” loss, This loss function (Zellner, 1994) reflects both of the described attributes. Under a particular form of balanced loss, we derive the predictive risk of the pre-test estimator which results after a test for exact linear restrictions on the coefficient vector. The corresponding risks of Stein-rule and positive-part Stein-rale estimators are also established. The risks based on loss functions which allow only for estimation precision, or only for goodness of fit, are special cases of our results, and we draw appropriate comparisons, In particular, we show that some of the well-known results under (quadratic) precision-only loss are not robust to our generalization of the loss function     

A CLASS OF NONPARAMETRIC TESTS FOR THE ONE-SAMPLE LOCATION PROBLEM

A class of “optimal”U-statistics type nonparametric test statistics is proposed for the one-sample location problem by considering a kernel depending on a constant a and all possible (distinct) subsamples of size two from a sample of n independent and identically distributed observations. The “optimal” choice of a is determined by the underlying distribution. The proposed class includes the Sign and the modified Wilcoxon signed-rank statistics as special cases. It is shown that any “optimal” member of the class performs better in terms of Pitman efficiency relative to the Sign and Wilcoxon-signed rank statistics. The effect of deviation of chosen a from the “optimal” a on Pitman efficiency is also examined. A Hodges-Lehmann type point estimator of the location parameter corresponding to the proposed “optimal” test-statistics is also defined and studied in this paper.     

Imposing and Testing for Shape Restrictions in Flexible Parametric Models

In many economic models, theory restricts the shape of functions, such as monotonicity or curvature conditions. This article reviews and presents a framework for constrained estimation and inference to test for shape conditions in parametric models. We show that “regional” shape-restricting estimators have important advantages in terms of model fit and flexibility (as opposed to standard “local” or “global” shape-restricting estimators). In our empirical illustration, this is the first article to impose and test for all shape restrictions required by economic theory simultaneously in the “Berndt and Wood” data. We find that this dataset is consistent with “duality theory,” whereas previous studies have found violations of economic theory. We discuss policy consequences for key parameters, such as whether energy and capital are complements or substitutes.     

Exact calculations for sequential tests based on Bernoulli trials

We consider methods of computing exactly the probability of “acceptance” and the “average sample size needed” for the sequential probability ratio test (SPRT) and likewise the newer “2-SPRT,” concerning the value of a Bernoulli parameter. The methods permit one to approximate, iteratively, the desired operating characteristics for the test.