Showing papers by "Ralph B. D'Agostino published in 1971"

An omnibus test of normality for moderate and large size samples

Abstract: SUMMARY We present a test of normality based on a statistic D which is up to a constant the ratio of Downton's linear unbiased estimator of the population standard deviation to the sample standard deviation. For the usual levels of significance Monte Carlo simulations indicate that Cornish-Fisher expansions adequately approximate the null distribution of D if the sample size is 50 or more. The test is an omnibus test, being appropriate to detect deviations from normality due either to skewness or kurtosis. Simulation results of powers for various alternatives when the sample size is 50 indicate that the test compares favourably with the Shapiro-Wilk W test, Vbl, b2 and the ratio of range to standard deviation. Shapiro & Wilk (1965) presented a test of normality based on a statistic W consisting essentially of the ratio of the square of the best, or approximately best, linear unbiased estimator of the population standard deviation to the sample variance. They supplied weights for the ordered sample observations needed in computing the numerator of W and also percentile points of the null distribution of W for samples of size 3 to 50. Subsequent investigation (Shapiro, Wilk & Chen, 1968) revealed that this test has surprisingly good power properties. It is an omnibus test, that is, it is appropriate for detecting deviations from normality due either to skewness or kurtosis, which appears to be superior to 'distance' tests, e.g. the chi-squared and Kolmogorov-Smirnov tests. It also usually dominates such standard tests as 1bl, third standardized sample moment; b2, fourth standardized sample moment; and u, ratio of the sample range to the sample standard deviation. Shapiro and Wilk did not extend their test beyond samples of size 50. A number of reasons indicate that it is best not to make such an extension. First, there is the problem of the appropriate weights for the ordered observations for the numerator of W. Each sample size requires a new set. The proliferation of tables is obvious and undesirable. However, even if the appropriate weights were computed from the expected values of the ordered observations from the standardized normal distribution (Harter, 1961), there would still be the uninviting problem of finding the appropriate null distribution of W. Because W's moments beyond the first are unknown, Cornish-Fisher expansions or similar techniques are not applicable. Further, the extension of the normal approximation for W based on Johnson's bounded curves (Shapiro & Wilk, 1968) when the sample is greater than 50 would require an extrapolation and the procedure for implementing it is not available. Simulation runs seem to be the only available way to obtain the null distribution. We present a new test of normality applicable for samples of size 50 or larger which possesses the desirable omnibus property. It requires no tables of weights and for samples of

A second look at analysis of variance on dichotomous data

Abstract: The work of Lunney (1970) concerning the appropriateness of analysis of variance (ANOVA) techniques on dichotomous data is discussed and extended. Relations between standard statistical techniques for analyzing dichotomous data and ANOVA procedures are indicated. The need for usefulness of analyzing transformed data as opposed to direct analysis of dichotomous data are discussed. Required statistical procedures employing transformed data are outlined.

Simulation probability points of b2 for small samples

Abstract: SUMMARY A table of percentiles of b2, the fourth standardized sample moment, applicable for samples of size 50 or less is given. The table was constructed by means of simulations. It can be used for testing the hypothesis of normality against alternatives of nonnormality due to kurtosis.

Linear Estimation of the Weibull Parameters

Abstract: In this paper we consider the method of Johns and Lieberman (1966) for estimating the shape and the log of the scale parameters of the two parameter Weibull distribution. The estimates are linear, asymptotically jointly normal and efficient. The main thrust of the paper is that unlike most linear estimating procedures of the Weibull parameters, that require different sets (tables) of weights for the observations for each size and often for each different number of observations from the sample that are available, the present technique requires only the knowledge of the proportion of observations from the sample available for the estimates. Given this proportion and four numbers obtained from a table look-up, the weights of the observations are then expressible in simple closed forms. Further we include a simple scheme to update the estimates when more observations become available. Finally, small sample computations of mean squared errors indicate that these estimates compare very Pdvorably even here with ...

A normal approximation for testing the equality of two independent Chi-square variables

Abstract: A simple normal approximation is supplied to test the equality of two independent Chi-square variables. The approximation appears to be valid for degrees of freedom as small as two. Thus no new table of critical values is needed. The standard unit normal distribution tables are sufficient.

An Addendum to “Linear Estimation of the Weibull Parameters”

Abstract: Linear, asymptotically normal and efficient estimators are given for the shape parameter of the two parameter Weibull distribution when the scale parameter is known and for the log of the scale parameter when the shape parameter is known. The weights of the ordered observations and other constants needed for these estimators are readily obtainable from a previous article of the author.