EXACT TWO-SAMPLE MEDIAN TESTS WHEN ONE "SAMPLE" VALUE POSSIBLY FROM A DIFFERENT POPULATION

Random samples are assumed for the univariate two-sample problem. Sometimes this assumption may be violated in that an observation in one “sample”, of size m, is from a population different from that yielding the remaining m—1 observations (which are a random sample). Then, the interest is in whether this random sample of size m—1 is from the same population as the other random sample. If such a violation occurs and can be recognized, and also the non-conforming observation can be identified (without imposing conditional effects), then that observation could be removed and a two-sample test applied to the remaining samples. Unfortunately, satisfactory procedures for such a removal do not seem to exist. An alternative approach is to use two-sample tests whose significance levels remain the same when a non-conforming observation occurs, and is removed, as for the case where the samples were both truly random. The equal-tail median test is shown to have this property when the two “samples” are of the same size (and ties do not occur).     

A two- sample partially sequential probability ratio test

A two-sample partially sequential probability ratio test (PSPRT) is considered for the two-sample location problem with one sample fixed and the other sequential. Observations are assumed to come from two normal poptilatlons with equal and known variances. Asymptotically in the fixed-sample size the PSPRT is a truncated Wald one sample sequential probability test. Brownian motion approximations for boundary-crossing probabilities and expected sequential sample size are obtained. These calculations are compared to values obtained by Monte Carlo simulation.     

Evaluating the Risk in Sample Size Determination

This article considers experimental costs, besides power evaluation, in order to determine the sample size of an experiment. We focus on the use of standard tools of decision theory in the context of sample size determination. The loss function is defined, from the perspective of an experimenter which adopts the classical frequentist approach, and the risk function is computed. Then, we show the behavior of the risk function in the two-sample t-test, for a small sample experimental setting, with a medium-sized sample, and with large samples. Moreover, an objective criterion for a convenient sample size choice is introduced. Finally, a practical example of sample size determination, which also considers risk computation, is shown.     

Sample size determination for the parallel model in a survey with sensitive questions

Recently, a new non-randomized parallel design is proposed by Tian (2013) for surveys with sensitive topics. However, the sample size formulae associated with testing hypotheses for the parallel model are not yet available. As a crucial component in surveys, the sample size formulae with the parallel design are developed in this paper by using the power analysis method for both the one- and two-sample problems. We consider both the one- and two-sample problems. The asymptotic power functions and the corresponding sample size formulae for both the one- and two-sided tests based on the large-sample normal approximation are derived. The performance is assessed through comparing the asymptotic power with the exact power and reporting the ratio of the sample sizes with the parallel model and the design of direct questioning. We numerically compare the sample sizes needed for the parallel design with those required for the crosswise and triangular models. Two theoretical justifications are also provided. An example from a survey on ‘sexual practices’ in San Francisco, Las Vegas and Portland is used to illustrate the proposed methods.     

Two-sample tests and tests of fit

There is a close analogy between the problems of testing the hypothesis that two samples come from the same continuous population (the two-sample problem) and testing the hypothesis that a single sample comes from a completely specified continuous distribution (a test of fit problem). In an earlier paper, asymptotic distribution theory was developed for the test of fit problem, under both the hypothesis being tested and interesting alternatives. In this paper, essentially the same asymptotic theory is shown to hold for the two-sample problem. Applications are given.     

A TEST OF HOMOGENEITY AGAINST SCALE ALTERNATIVES USING SUBSAMPLE EXTREMA1

In this paper a new class of non-parametric tests for testing homogeneity of several populations against scale alternatives is proposed. For this, independent samples of fixed sizes are drawn from each population and from these samples, all possible sub-samples of the same size are drawn and their maxima and minima are computed. Using these extreme the class of tests is obtained. Tests of this type have been offered for the two-sample slippage problem by Kochar (1978). Under certain conditions, this class of tests is shown to be consistent against ‘difference in scale’ alternatives. The test has been compared with Bhapkar's V-test (1961), Deshpande's D-test (1965), Sugiura's D_rs-test (1965) and with a classical test given by Lehmann (1959, pp. 273–275). It is shown that some members of this proposed class of tests are more efficient than the first three tests in the case of uniform, Laplace and normal distributions, when the number of populations compared is small.     

Control percentile test procedures for censored data

In life testing and survival analyses which involve the use of expensive equipment the cost of continuing an experiment until all the items on test have failed can be quite high. In these situations it is reasonable to make a statistical test when a pre-specified percentile, e.g. median of the control group has been observed. This article adapts some existing procedures for complete samples to randomly censored data. The results of Lo and Singh (1985) who extended the Bahadur representation of quantiles to the censored case enable us to use the methods of Gastwirth (1968) and Hettmansperger (1973) which were based on Bahadur's result to extend the procedures of Mathisen (1943), Gart (1963) and Slivka (1970).The large sample efficiency of the control median test is the same as that of Brookmeyer and Crowley's (1982) extension of the usual median test. For the two-sample shift problem with observations following the double-exponential law, the median remains the optimum percentile to use until the censoring becomes quite heavy. On the other hand, in the two-sample scale parameter problem for data from an exponential distribution the percentile (80th in the uncensored case) yielding the asymptotically most powerful test in the family of control percentile tests no longer is optimum. The effect becomes noticeable when 25% or more of the data is censored.     

The chernoff efficiency of the wilcoxon rank test

Exact limiting Chernoff efficiencies of the Wilcoxon rank test are derived using Hoadley's results. Efficiency curves are derived for the two-sample Wilcoxon rank test relative to the two-sample t test for normal shift alternatives when the null hypothesis is that of common normality. The comparisons with Bahadur efficiency and small sample Hodges-Lehmann efficiency-are also made.     

One-and two-sample sign tests for ranked set sample selective designs

Statistical inference based on a ranked set sample depends very much on the location of the quantified observations. A selective design which determines the location of the quantified observations in a ranked set sample is introduced. The paper investigates the effects of selective designs on one and two sample sign test statistics. The Pitman efficiencies of one- and two sample sign tests are calculated for selective designs and compared with ranked set samples of the same size. If the design quantifies observations at the center points, then the proposed procedure is superior to a ranked set sample of the same size in the sense of Pitman efficiency. Some practical problems are addressed for the two-sample sign test.     

The robustness of the two—sample t—test over the Pearson system

The present paper has as its objective an accurate quantification of the robustness of the two–sample t-test over an extensive practical range of distributions. The method is that of a major Monte Carlo study over the Pearson system of distributions and the details indicate that the results are quite accurate. The study was conducted over the range β ₁ =0.0(0.4)2.0 (negative and positive skewness) and β ₂ =1.4 (0.4)7.8 with equal sample sizes and for both the one-and two-tail t-tests. The significance level and power levels (for nominal values of 0.05, 0.50, and 0.95, respectively) were evaluated for each underlying distribution and for each sample size, with each probability evaluated from 100,000 generated values of the test-statistic. The results precisely quantify the degree of robustness inherent in the two-sample t-test and indicate to a user the degree of confidence one can have in this procedure over various regions of the Pearson system. The results indicate that the equal-sample size two-sample t-test is quite robust with respect to departures from normality, perhaps even more so than most people realize.     

Bayesian prediction with a random sample size for the burr lifetime distribution

This paper is concerned with the problem of deriving Bayesian prediction bounds for the Burr distribution when the sample size is a random variable. Prediction bounds for both the future observations (the case of two-sample prediction) and the remaining observations in the same sample (the case of one-sample prediction) will be derived. The analysis will depend mainly on assuming that the size of the sample is a random variable having the Poisson distribution. Finally, numerical examples are given to illustrate the results.     

Two-sample high-dimensional empirical likelihood

In this paper, we apply empirical likelihood for two-sample problems with growing high dimensionality. Our results are demonstrated for constructing confidence regions for the difference of the means of two p-dimensional samples and the difference in value between coefficients of two p-dimensional sample linear model. We show that empirical likelihood based estimator has the efficient property. That is, as p → ∞ for high-dimensional data, the limit distribution of the EL ratio statistic for the difference of the means of two samples and the difference in value between coefficients of two-sample linear model is asymptotic normal distribution. Furthermore, empirical likelihood (EL) gives efficient estimator for regression coefficients in linear models, and can be as efficient as a parametric approach. The performance of the proposed method is illustrated via numerical simulations.     

Bayesian approach to robust procedures based on censored samples

By adopting; the Bayesian method, we develop in this paper some robust procedures for the one and two-sample location problems based on symmetric Type-II censored samples and by assuming normality for the censored samples. The posterior distributions and the Highest Posterior Density (H.P.D.) intervals are? derived. Finally, we illustrate these procedures by applying the results to Darwin's data and to Brown lee's data     

Šidák-type tests for the two-sample problem based on precedence and exceedance statistics

This paper deals with a class of nonparametric two-sample tests for ordered alternatives. The test statistics proposed are based on the number of observations from one sample that precede or exceed a threshold specified by the other sample, and they are extensions of ?idák's test. We derive their exact null distributions and also discuss a large-sample approximation. We then study their power properties exactly against the Lehmann alternative and make some comparative comments. Finally, we present an example to illustrate the proposed tests.     

SELECTING THE m POPULATIONS WITH LARGEST MEANS FROM k NORMAL POPULATIONS WITH UNKNOWN VARIANCES

This paper gives a two-sample procedure for selecting the m populations with the largest means from k normal populations with unknown variances. The method is a generalization of a recent work by Ofosu [1973] and hence should find wider practical applications. The experimenter takes an initial sample of preset size N₀ from each population and computes an unbiased estimate of its variance. From this estimate he determines the second sample size for the population according to a table presented for this purpose. The populations associated with the m largest overall sample means will be selected. The procedure is shown to satisfy a confidence requirement similar to that of Ofosu.     

Robust nonparametric tests for the two-sample location problem

We construct and investigate robust nonparametric tests for the two-sample location problem. A test based on a suitable scaling of the median of the set of differences between the two samples, which is the Hodges-Lehmann shift estimator corresponding to the Wilcoxon two-sample rank test, leads to higher robustness against outliers than the Wilcoxon test itself, while preserving its efficiency under a broad range of distributions. The good performance of the constructed test is investigated under different distributions and outlier configurations and compared to alternatives like the two-sample t-, the Wilcoxon and the median test, as well as to tests based on the difference of the sample medians or the one-sample Hodges-Lehmann estimators.     

Fisher information in censored samples from Downton's bivariate exponential distribution

We develop a simple approach to finding the Fisher information matrix (FIM) for a single pair of order statistic and its concomitant, and Type II right, left, and doubly censored samples from an arbitrary bivariate distribution. We use it to determine explicit expressions for the FIM for the three parameters of Downton's bivariate exponential distribution for single pairs and Type II censored samples. We evaluate the FIM in censored samples for finite sample sizes and determine its limiting form as the sample size increases. We discuss implications of our findings to inference and experimental design using small and large censored samples and for ranked-set samples from this distribution.     

Likelihood ratio comparisons among spacings related to both one or two samples

In this paper, we investigate some stochastic comparisons in terms of likelihood ratio ordering between spacings from independent random variables exponentially distributed with different scale parameters. We partially solve some open problems in [Wen S, Lu Q, Hu T. Likelihood ratio orderings of spacings of heterogeneous exponential random variables. J Multivariate Anal. 2007;98:743–756] for a one-sample problem and in [Hu T, Lu Q, Wen S. Stochastic comparisons and dependence of spacings from two samples of exponential random variables. Commun Stat – Theory Methods 2006;35:979–988] for a two-sample problem. Specifically, we prove that the second spacing is always smaller than the third spacing in terms of the likelihood ratio order and we provide the ordering among all spacings in the case n=4. In the two-sample case, we establish comparisons between the second spacings related to each sample under certain conditions.     

Combined sample scores versus two-sample scores under censoring

For testing the equality of K-populations against ordered alternatives a quadratic form based on two-sample scores of (K-l) pairs of adjacent samples has been suggested as an alternative to the statistic proposed by Beslow(1970). The Pitman asymptotic efficiency of the proposed statistic relative to the Breslow's statistic is found to be one or very close to one in all cases considered. The Bahadur asymptotic efficiency has also been studied and found to be at least one for the situations taken up in this paper.     

Robustness of the two-sample t-test umdeh violations of the homogeneity of farjance assumptioi

When the two-sample t-test has equal sample slies, it is widely considered to be a robust procedure (with respect to the significaoce level) under violatioa of the assuaptioo of equal variances. This paper is coa-earned with a quantification of the amount of robustness which this procedure has under such violations, The approach is through the concept of "religion of robustness" and the resluts show an extremely strong degree of robustness for the equal an extremely strong degree of robustness for the equal sample size t-test, probably more so than most statistyicians realise. This extremely high level of robustness, however, reduces quickly as the sample sizes begin to vary from equality. The regions of robustnes obtained show that while most users would likely be satisfied with the degree of robustness inherent when the two sample sizes each vary by 10% from equality, most would wish to be much more cautions when the variation is 20%. The study covers sample sizes n₁ -= n ² = 5(5)30(10)50 plus 10% and 20% variations thereof for the two-tailed test and nominal significance levels of 0.01 and 0.05.