Bounds for the variance of the graybill-deal estimator of the common mean of two normal distributions

In this note we derive sharp lower and upper bounds for the variance of the Graybill-Deal estimator of the common mean of two normal distributions with unknown variances when the sample sizes are not necessarily equal. We also derive similar bounds for the variance of the Brown-Cohen (1974) T _a(1) class of unbiased es-timators to which the Graybill-Deal estimator belongs. Further, we illustrate the sharpness of the bounds by numerical computations in the case of the Graybill-Deal estimator.     

Recurrence relations mid identities for moments of order statistics,i: arbitrary continuous distribution

In this paper, we review several recurrence relations and identities established for the single and product moments of order statistics from an arbitrary continuous distribution. We point out the interrelationships between many of these recurrence relations. We discuss the results giving the bounds for the number of single and double integrals needed to be evaluated in order to compute the first, second and product moments of order statistics in a sample of size n from an arbitrary continuous distribution, given these moments in samples of sizes n-1 and less. Improvements of these bounds for the case of symmetric continuous distributions are also discussed     

On testing the skew normal hypothesis

In this work two goodness-of-fit tests are proposed for the skew normal distribution, based on properties of this family of distributions and the sample correlation coefficient. The critical values for the tests are obtained by using Monte Carlo simulation for several sample sizes and levels of significance. The power of the proposed tests are compared with that of the tests studied by Mateu et al. (2007) and the one studied by Meintanis (2007) for several sample sizes and considering diverse alternatives. The results show that the proposed tests have greater power than those studied by Mateu et al. (2007) and Meintanis (2007) against some alternative distributions.     

A tool for systematically comparing the power of tests for normality

In this article, we describe a new approach to compare the power of different tests for normality. This approach provides the researcher with a practical tool for evaluating which test at their disposal is the most appropriate for their sampling problem. Using the Johnson systems of distribution, we estimate the power of a test for normality for any mean, variance, skewness, and kurtosis. Using this characterization and an innovative graphical representation, we validate our method by comparing three well-known tests for normality: the Pearson χ² test, the Kolmogorov–Smirnov test, and the D'Agostino–Pearson K ² test. We obtain such comparison for a broad range of skewness, kurtosis, and sample sizes. We demonstrate that the D'Agostino–Pearson test gives greater power than the others against most of the alternative distributions and at most sample sizes. We also find that the Pearson χ² test gives greater power than Kolmogorov–Smirnov against most of the alternative distributions for sample sizes between 18 and 330.     

Two-sample nonparametric stochastic order inference with an application in plant physiology

In this paper, a new nonparametric methodology is developed for testing whether the changing pattern of a response variable over multiple ordered sub-populations from one treatment group differs with the one from another treatment group. The question is formalized into a nonparametric two-sample comparison problem for the stochastic order among subsamples, through U-statistics with accommodations for zero-inflated distributions. A novel bootstrap procedure is proposed to obtain the critical values with given type I error. Following the procedure, bootstrapped p-values are obtained through simulated samples. It is proven that the distribution of the test statistics is independent from the underlying distributions of the subsamples, when certain sufficient statistics provided. Furthermore, this study also develops a feasible framework for power studies to determine sample sizes, which is necessary in real-world applications. Simulation results suggest that the test is consistent. The methodology is illustrated using a biological experiment with a split-plot design, and significant differences in changing patterns of seed weight between treatments are found with relative small subsample sizes.     

Lower bounds for expected sample size of sequential procedures for the multinomial selection problems

In this article, lower bounds for expected sample size of sequential selection procedures are constructed for the problem of selecting the most probable event of k-variate multinomial distribution. The study is based on Volodin’s universal lower bounds for expected sample size of statistical inference procedures. The obtained lower bounds are used to estimate the efficiency of some selection procedures in terms of their expected sample sizes.     

Bias in linear model power and sample size calculation due to estimating noncentrality

Data analysts frequently calculate power and sample size for a planned study using mean and variance estimates from an initial trial. Hence power,or the sample size needed to achieve a fixed power, varies randomly. Such claculations can be very inaccurate in the General Linear Univeriate Model (GLUM). Biased noncentrality estimators and censored power calculations create inaccuracy. Censoring occurs if only certain outcomes of an initial trial lead to a power calculation. For example, a confirmatory study may be planned (and a sample size estimated) only following a significant resulte in the initial trial.

Computing accurate point estimates or confidence bounds of GLUM noncentrality, power, or sample size in the presence of censoring involves truncated noncentral F distributions. We recommed confidence bounds, whether or not censoring occurs. A power analysis of data from humans exposed to carbon monoxide demonstrates the substantial impact on samle size that may occur. The results highlight potential; biases and should aid study planning and interpretation.     

Lower confidence bounds as precision measure for truncated processes

Process capability index C_p has been the most popular one used in the manufacturing industry to provide numerical measures on process precision. For normally distributed processes with automatic fully inspections, the inspected processes follow truncated normal distributions. In this article, we provide the formulae of moments used for the Edgeworth approximation on the precision measurement C_p for truncated normally distributed processes. Based on the developed moments, lower confidence bounds with various sample sizes and confidence levels are provided and tabulated. Consequently, practitioners can use lower confidence bounds to determine whether their manufacturing processes are capable of preset precision requirements.     

Two separate effects of variance heterogeneity on the validity and power of significance tests of location

Heterogeneity of variances of treatment groups influences the validity and power of significance tests of location in two distinct ways. First, if sample sizes are unequal, the Type I error rate and power are depressed if a larger variance is associated with a larger sample size, and elevated if a larger variance is associated with a smaller sample size. This well-established effect, which occurs in t and F tests, and to a lesser degree in nonparametric rank tests, results from unequal contributions of pooled estimates of error variance in the computation of test statistics. It is observed in samples from normal distributions, as well as non-normal distributions of various shapes. Second, transformation of scores from skewed distributions with unequal variances to ranks produces differences in the means of the ranks assigned to the respective groups, even if the means of the initial groups are equal, and a subsequent inflation of Type I error rates and power. This effect occurs for all sample sizes, equal and unequal. For the t test, the discrepancy diminishes, and for the Wilcoxon–Mann–Whitney test, it becomes larger, as sample size increases. The Welch separate-variance t test overcomes the first effect but not the second. Because of interaction of these separate effects, the validity and power of both parametric and nonparametric tests performed on samples of any size from unknown distributions with possibly unequal variances can be distorted in unpredictable ways.     

Goodness-of-fit test based on empirical characterisitc function

This paper discusses a goodness-of-fit test that uses the integral of the squared modulus of the difference between the empirical characteristic function of the sample data and the characteristic function of the hypothesized distribution. Monte Carlo procedures are employed to obtain the empirical percentage points for testing the fit of normal, logistic and exponential distributions with unknown location and scale parameters. Results of Monte Carlo power comparisons with other well-developed goodness-of-fit tests are summarized. Tne proposed test is shown to have superior power for testing the fit of the logistic distibotion (for moderate sample sizes) against a wide range of alternative distributions.     

On lachin'S formulae for sample sizes of survival tests

Lachin [1981] and Lachin and Foulkes [1986] consider two groups of identically independently exponentially distributed random variables and four models of data sampling. The test problem they treat is to decide whether the two distributions are identical (null-hypothesis H⁰) or not (alternative hypothesis H¹). Basing the test on maximum-likelihood estimators and their asymptotic normal densities they obtain formulae for the group sizes necessary to yield asymptotic tests with guaranteed power under a prescribed level for specified hypotheses. It is intuitively reasonable to expect the sizes decrease the more the hypotheses differ. It the distance betwen H⁰ and H¹ is measured by the difference of the exponential parameters this assumption time or the deviation of the exponential parameter ratio from unity is the measure larger distances between the hypotheses do not necessarily lead to smaller sample sizes.     

A Bayesian sequential design with adaptive randomization for 2‐sided hypothesis test

Bayesian sequential and adaptive randomization designs are gaining popularity in clinical trials thanks to their potentials to reduce the number of required participants and save resources. We propose a Bayesian sequential design with adaptive randomization rates so as to more efficiently attribute newly recruited patients to different treatment arms. In this paper, we consider 2‐arm clinical trials. Patients are allocated to the 2 arms with a randomization rate to achieve minimum variance for the test statistic. Algorithms are presented to calculate the optimal randomization rate, critical values, and power for the proposed design. Sensitivity analysis is implemented to check the influence on design by changing the prior distributions. Simulation studies are applied to compare the proposed method and traditional methods in terms of power and actual sample sizes. Simulations show that, when total sample size is fixed, the proposed design can obtain greater power and/or cost smaller actual sample size than the traditional Bayesian sequential design. Finally, we apply the proposed method to a real data set and compare the results with the Bayesian sequential design without adaptive randomization in terms of sample sizes. The proposed method can further reduce required sample size.     

A comparison of several adaptive tests for paired data

In the last few years, two adaptive tests for paired data have been proposed. One test proposed by Freidlin et al. [On the use of the Shapiro–Wilk test in two-stage adaptive inference for paired data from moderate to very heavy tailed distributions, Biom. J. 45 (2003), pp. 887–900] is a two-stage procedure that uses a selection statistic to determine which of three rank scores to use in the computation of the test statistic. Another statistic, proposed by O'Gorman [Applied Adaptive Statistical Methods: Tests of Significance and Confidence Intervals, Society for Industrial and Applied Mathematics, Philadelphia, 2004], uses a weighted t-test with the weights determined by the data. These two methods, and an earlier rank-based adaptive test proposed by Randles and Hogg [Adaptive Distribution-free Tests, Commun. Stat. 2 (1973), pp. 337–356], are compared with the t-test and to Wilcoxon's signed-rank test. For sample sizes between 15 and 50, the results show that the adaptive test proposed by Freidlin et al. and the adaptive test proposed by O'Gorman have higher power than the other tests over a range of moderate to long-tailed symmetric distributions. The results also show that the test proposed by O'Gorman has greater power than the other tests for short-tailed distributions. For sample sizes greater than 50 and for small sample sizes the adaptive test proposed by O'Gorman has the highest power for most distributions.     

A MONTE CARLO COMPARISON OF ELLIPTICAL-THEORY AND OTHER TESTS FOR EQUALITY OF COVARIANCE MATRICES

A Monte Carlo study of the size and power of tests of equality of two covariance matrices is carried out. Tests based upon normality assumptions, elliptical distribution assumptions as well as distribution-free tests are compared. Samples are generated from normal, elliptical and non-elliptical populations. The elliptical-theory tests, in particular, have poor size properties for both elliptical distributions with moderate sample sizes and for non-elliptical distributions.     

A step-down heteroscedastic multiple comparison procedure

All-pairs power in a one-way ANOVA is the probability of detecting all true differences between pairs of means. Ramsey (1978) found that for normal distributions having equal variances, step-down multiple comparison procedures can have substantially more all-pairs power than single-step procedures, such as Tukey’s HSD, when equal sample sizes are randomly sampled from each group. This paper suggests a step-down procedure for the case of unequal variances and compares it to Dunnett's T3 technique. The new procedure is similar in spirit to one of the heteroscedastic procedures described by Hochberg and Tamhane (1987), but it has certain advantages that are discussed in the paper. Included are results on unequal sample sizes.     

Empirical behavior of some tests for normality

The behavior of a range of tests assessing the normality of a sequence of independent and identically distributed random variables is investigated.An examination of the empirical significance level of the tests is undertaken for different sample sizes. The empirical power associated with these tests is also calculated under some alternative distributions.     

Detection of outliers in familial data

Procedures for detection of outliers in familial data is given for mean-slippage and dispersion-slippage model of outliers for equal and unequal family sizes. The distributions of the test statistics are derived and Bonferroni's bounds for the values of significant probabilities are given.     

Some nonparametric precedence-type tests based on progressively censored samples and evaluation of power

In this paper, we consider the problem of testing the equality of two distributions when both samples are progressively Type-II censored. We discuss the following two statistics: one based on the Wilcoxon-type rank-sum precedence test, and the second based on the Kaplan–Meier estimator of the cumulative distribution function. The exact null distributions of these test statistics are derived and are then used to generate critical values and the corresponding exact levels of significance for different combinations of sample sizes and progressive censoring schemes. We also discuss their non-null distributions under Lehmann alternatives. A power study of the proposed tests is carried out under Lehmann alternatives as well as under location-shift alternatives through Monte Carlo simulations. Through this power study, it is shown that the Wilcoxon-type rank-sum precedence test performs the best.     

A Goodness-of-Fit Test for Location-Scale Max-Stable Distributions

In this article, a technique based on the sample correlation coefficient to construct goodness-of-fit tests for max-stable distributions with unknown location and scale parameters and finite second moment is proposed. Specific details to test for the Gumbel distribution are given, including critical values for small sample sizes as well as approximate critical values for larger sample sizes by using normal quantiles. A comparison by Monte Carlo simulation shows that the proposed test for the Gumbel hypothesis is substantially more powerful than some other known tests against some alternative distributions with positive skewness coefficient.     

Pitman Closeness of Order Statistics to Population Quantiles

In this article, Pitman closeness of sample order statistics to population quantiles of a location-scale family of distributions is discussed. Explicit expressions are derived for some specific families such as uniform, exponential, and power function. Numerical results are then presented for these families for sample sizes n = 10,15, and for the choices of p = 0.10, 0.25, 0.75, 0.90. The Pitman-closest order statistic is also determined in these cases and presented.