The Sampling Distribution of the Third Moment Coefficient--an Experiment

Pepper, Joseph

doi:10.1093/biomet/24.1-2.55

Cited by 9 publications

(2 citation statements)

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“…Pearson, 1927, p. iii)—Church (1926), “Sophister” (1928), and E. S. Pearson (1929) drew from 500 to 1,000 samples of sizes 2 through 10 to investigate the error properties of Student's t test under nonnormality, and Pepper (1932) drew 1,000 samples of size 10 to investigate the sampling distribution of the third sample moment. Kendall and Babington Smith (1938) and Yule (1938) subsequently showed, using a variety of statistical tests, that Tippett's (1927) random digits seemed, indeed, to be suitably random.…”

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confidence: 99%

Testing for robustness in Monte Carlo studies.

Serlin¹

2000

Psychological Methods

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Monte Carlo studies provide the information needed to help researchers select appropriate analytical procedures under design conditions in which the underlying assumptions of the procedures are not met. In Monte Carlo studies, the 2 errors that one could commit involve (a) concluding that a statistical procedure is robust when it is not or (b) concluding that it is not robust when it is. In previous attempts to apply standard statistical design principles to Monte Carlo studies, the less severe of these errors has been wrongly designated the Type I error. In this article, a method is presented for controlling the appropriate Type I error rate; the determination of the number of iterations required in a Monte Carlo study to achieve desired power is described; and a confidence interval for a test's true Type I error rate is derived. A robustness criterion is also proposed that is a compromise between W. G. Cochran's (1952) and J. V. Bradley's (1978) criteria.

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confidence: 99%

Testing for robustness in Monte Carlo studies.

Serlin¹

2000

Psychological Methods

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“…The construction of a significance test for the comparison of two skewness estimates requires knowledge about the sampling distribution of the skewness. Early work by Pepper () indicates that the distribution of a skewness estimate may be approximated by a normal distribution (see also Shenton & Bowman, ). D'Agostino () suggested a transformation of the skewness estimate to approach normality more closely.…”

Section: Statistical Inference On γε and γBold-italicε′mentioning

confidence: 99%

Significance tests to determine the direction of effects in linear regression models

Wiedermann

Hagmann

Eye

2014

Brit J Math & Statis

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Previous studies have discussed asymmetric interpretations of the Pearson correlation coefficient and have shown that higher moments can be used to decide on the direction of dependence in the bivariate linear regression setting. The current study extends this approach by illustrating that the third moment of regression residuals may also be used to derive conclusions concerning the direction of effects. Assuming non-normally distributed variables, it is shown that the distribution of residuals of the correctly specified regression model (e.g., Y is regressed on X) is more symmetric than the distribution of residuals of the competing model (i.e., X is regressed on Y). Based on this result, 4 one-sample tests are discussed which can be used to decide which variable is more likely to be the response and which one is more likely to be the explanatory variable. A fifth significance test is proposed based on the differences of skewness estimates, which leads to a more direct test of a hypothesis that is compatible with direction of dependence. A Monte Carlo simulation study was performed to examine the behaviour of the procedures under various degrees of associations, sample sizes, and distributional properties of the underlying population. An empirical example is given which illustrates the application of the tests in practice.

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Tables for the Determination of the Significance of Skewness and of the Significance of the Difference in the Skewness of Two Independent Distributions

Taylor¹

1947

Psychometrika

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Several of the more popular statistics texts used in psychology are reviewed with respect to their treatment of the measurement and interpretation of skewness. Some areas in psychology where measures of skewness of distributions may yield significant information are indicated. Tables of 1%, 2% and 5% points of moment measures of skewness and tables of the 1% and 5% points of the difference in skewness of two uncorrelated distributions are presented. These tables are approximations based on the approximate normality of skewness in large samples from normal populations. The limited confidence with which these tables can be used in the absence of exact knowledge of the distribution function of the underlying statistics is indicated.

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The Sampling Distribution of the Third Moment Coefficient--an Experiment

Cited by 9 publications

References 0 publications

Testing for robustness in Monte Carlo studies.

Testing for robustness in Monte Carlo studies.

Significance tests to determine the direction of effects in linear regression models

Tables for the Determination of the Significance of Skewness and of the Significance of the Difference in the Skewness of Two Independent Distributions

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