Optimum spacings for the joint estimation and tests of hypothesis of location and scale parameters of the cauchy distribution

A simultaneous test for the location and scale parameters of the Cauchy distribution is considered based on selected order statistics.. It is shown that optimum spacings that maximise the Pitman ARE of the test coincide with that of the optimum spacings for the estimation problem.     

Comparing diagnostic tests: test of hypothesis for likelihood ratios

Likelihood ratios (LRs) are used to characterize the efficiency of diagnostic tests. In this paper, we use the classical weighted least squares (CWLS) test procedure, which was originally used for testing the homogeneity of relative risks, for comparing the LRs of two or more binary diagnostic tests. We compare the performance of this method with the relative diagnostic likelihood ratio (rDLR) method and the diagnostic likelihood ratio regression (DLRReg) approach in terms of size and power, and we observe that the performances of CWLS and rDLR are the same when used to compare two diagnostic tests, while DLRReg method has higher type I error rates and powers. We also examine the performances of the CWLS and DLRReg methods for comparing three diagnostic tests in various sample size and prevalence combinations. On the basis of Monte Carlo simulations, we conclude that all of the tests are generally conservative and have low power, especially in settings of small sample size and low prevalence.     

SELECTION PROCEDURES FOR SCALE PARAMETERS USING TWO-SAMPLE U-STATISTICS

Let be k independent populations having the same known quantile of order p (0 p 1) and let F(x)=F(x/_i) be the absolutely continuous cumulative distribution function of the ith population indexed by the scale parameter ₁, i = 1,…, k. We propose subset selection procedures based on two-sample U-statistics for selecting a subset of k populations containing the one associated with the smallest scale parameter. These procedures are compared with the subset selection procedures based on two-sample linear rank statistics given by Gill & Mehta (1989) in the sense of Pitman asymptotic relative efficiency, with interesting results.     

Two sample nonparametric tests based on subsamples

The paper introduces a general class of nonparametric tests for the two-sample location problem based on subsamples. Includ- ed in this class is the Mann-Whitney (or the Wilcoxon rank sum) test. General formulas for the Pitman efficacy for different methods of subsampling are derived. A small sample power simu- lation compares the performance of members of this class     

Asymptotic optimality of a class of rank tests for replicated latin-square designs

This paper is concerned with the class of conditionally distribution-free rank tests introduced by Monga and Tardif (1994) for replicated Latin-square designs. It is possible to proceed with an enlargement of this class by making use of the method of ranking after substitution. The unconditional asymptotic behaviour of any member of the enlarged class is derived under the null hypothesis of no treatment effects as well as under a sequence of contiguous alternatives. This enables the establishment of the asymptotic Pitman efficiency of any member relative to the asymptotically minimax test and to conclude that at least one member of the class is asymptotically as efficient as the latter.     

On a class of nonparametric tests for the treatment vs control problem

The classical problem of testing treatment versus control is revisited by considering a class of test statistics based on a kernel that depends on a constant ‘a’. The proposed class includes the celebrated Wilcoxon-Mann-Whitnet statistics as a special case when ‘a’=1. It is shown that, with optimal choice of ‘a’ depending on the underlying distribution, the optimal member performs better (in terms of Pitman efficiency) than the Wilcoxon-Mann-Whitney and the Median tests for a wide range of underlying distributions. An extended Hodges-Lehmann type point estimator of the shift prameter corresponding to the proposed ‘optimal’ test statistic is also derived.     

Comparing diagnostic tests with missing data

When missing data occur in studies designed to compare the accuracy of diagnostic tests, a common, though naive, practice is to base the comparison of sensitivity, specificity, as well as of positive and negative predictive values on some subset of the data that fits into methods implemented in standard statistical packages. Such methods are usually valid only under the strong missing completely at random (MCAR) assumption and may generate biased and less precise estimates. We review some models that use the dependence structure of the completely observed cases to incorporate the information of the partially categorized observations into the analysis and show how they may be fitted via a two-stage hybrid process involving maximum likelihood in the first stage and weighted least squares in the second. We indicate how computational subroutines written in R may be used to fit the proposed models and illustrate the different analysis strategies with observational data collected to compare the accuracy of three distinct non-invasive diagnostic methods for endometriosis. The results indicate that even when the MCAR assumption is plausible, the naive partial analyses should be avoided.     

Comparison of some common ratio estimators using pitman measure of nearness

In this article we compare some common ratio estimators for estimating the population total of a given characteristic. The sampling schemes considered are simple random sampling (S.R.S.) and S.R.S.under stratification. The comparisons are made using the Pitman Nearness criterion under the model-based approach. The error term is assumed normal with mean zero and variance σg(x). The function g(x) is a known function of the auxiliary variable x. Special interest is on the cases of g(x) =l and x. The result is found the same as that using MSE criterion, although the PN is very different from the MSE intrinsically.     

On the power of the t-test and some rank tests when outliers may be present

The power of some rank tests, used for testing the hypothesis of shift, is found when the underlying distributions contain outliers. The outliers are assumed to occur as the result of mixing two normal distributions with common variance. A small sample case shows how the scores for the rank tests are found and the exact power is computed for each of these rank tests. A Monte Carlo study provides an estimate of the power of the usual two sample t-test.     

Comparison of regression estimators using Pitman measures of nearness

Received: August 5, 1999; revised version: June 14, 2000     

Comparing a new Gini test with other symmetry tests when median is known

ABSTRACT

Wang et al. (2013 Wang, J. C., H. Auda, and M. Niewiadomska-Bugaj. 2013. Comparing a new Gini test with other symmetry tests when median is unknown. Communications in Statistics- Simulation and Computation 42:1616–27.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) provided a comprehensive study of 14 tests, including two tests based on the Gini mean difference (GMD) introduced by Auda (2006), applicable for data from populations with an unknown median. This paper is a similar study of symmetry tests applicable for data from populations with a known median. We are showing that GMD tests compare favorably with several existing procedures in controlling the type I error as well as in power. Results of the study are shown graphically, which makes the tests’ power easy to assess.     

Comparing least-squares lines for testing level changes in interrupted time-series analysis

A likelihood ratio test of equality of coefficients of variation or relative errors for two samples of unequal size is presented. The asymptotic distribution is found and the distribution is tabled for small sizes via computer simulation. Kolmogrov-Smirnov confidence intervals are constructed on the percentiles included in the table. Power studies are also simulated and the results indicate that the test has good power for small coefficients of variation     

Ratio tests under limiting normality

We propose a class of ratio tests that is applicable whenever a cumulation (of transformed) data is asymptotically normal upon appropriate normalization. The Karhunen–Loève theorem is employed to compute weighted averages. The test statistics are ratios of quadratic forms of these averages and hence scale-invariant, also called self-normalizing: The scaling parameter cancels asymptotically. Limiting distributions are obtained. Critical values and asymptotic local power functions can be calculated by standard numerical means. The ratio tests are directed against local alternatives and turn out to be almost as powerful as optimal competitors, without being plagued by nuisance parameters at the same time. Also in finite samples they perform well relative to self-normalizing competitors.     

Pitman closeness,monotonicity and consistency of best linear unbiased and invariant estimators for exponential distribution under Type II censoring

Comparisons of best linear unbiased estimators with some other prominent estimators have been carried out over the last 50 years since the ground breaking work of Lloyd [E.H. Lloyd, Least squares estimation of location and scale parameters using order statistics, Biometrika 39 (1952), pp. 88–95]. These comparisons have been made under many different criteria across different parametric families of distributions. A noteworthy one is by Nagaraja [H.N. Nagaraja, Comparison of estimators and predictors from two-parameter exponential distribution, Sankhyā Ser. B 48 (1986), pp. 10–18], who made a comparison of best linear unbiased (BLUE) and best linear invariant (BLIE) estimators in the case of exponential distribution. In this paper, continuing along the same lines by assuming a Type II right censored sample from a scaled-exponential distribution, we first compare BLUE and BLIE of the exponential mean parameter in terms of Pitman closeness (nearness) criterion. We show that the BLUE is always Pitman closer than the BLIE. Next, we introduce the notions of Pitman monotonicity and Pitman consistency, and then establish that both BLUE and BLIE possess these two properties.     

Comparing two independent incidence rates using conditional and unconditional exact tests

Several unconditional exact tests, which are constructed to control the Type I error rate at the nominal level, for comparing two independent Poisson rates are proposed and compared to the conditional exact test using a binomial distribution. The unconditional exact test using binomial p-value, likelihood ratio, or efficient score as the test statistic improves the power in general, and are therefore recommended. Unconditional exact tests using Wald statistics, whether on the original or square-root scale, may be substantially less powerful than the conditional exact test, and is not recommended. An example is provided from a cardiovascular trial.     

Goodness of fit tests with misclassified data

The most popular goodness of fit test for a multinomial distribution is the chi-square test. But this test is generally biased if observations are subject to misclassification, In this paper we shall discuss how to define a new test procedure when we have double sample data obtained from the true and fallible devices. An adjusted chi-square test based on the imputation method and the likelihood ratio test are considered, Asymptotically, these two procedures are equivalent. However, an example and simulation results show that the former procedure is not only computationally simpler but also more powerful under finite sample situations.     

On some tests in manocova model with different dispersion matrices

In this article we have considered the problem of testing linear hypothesis in MANOCOVA model with different dispersion Matrices by two test procedures in the line of Anderson (1963) and fihargava CI971). Also efficiencies of the two tests have often compared.