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As we 2 . 1 Bar plot of the observed frequencies of Merriman’s target data. The ith belt number corresponds to (i − 1, i]. The smooth curve represents the estimated frequencies under the normality assumption. 2 (Mothers’ heights data) Pearson and Lee (1903) gave data on mother’s heights that are typical of much of the data used in the large heredity studies of the period. Snedecor and Cochran (1989, p. 2) presented the frequency data. In a later example (Snedecor and Cochran, 1989, p. 85 based on 6 degrees of freedom.

Once θ˜ Raphson search for the MOM estimate θ. 2. 24. 03 obtained by the χ12 approximation and by the parametric bootstrap. The use of components is again important in this example. 1 Introduction One of the criticisms aimed at X2 -type tests in the past has been that if the null hypothesis is rejected then there is no alternative distribution indicated. See for example Durbin and Knott (1972, p. 291). Were this true, there would be no alternative model the user could adopt. In fact this is not true for the Pearson X2 test, nor for the tests we will consider in later chapters.

J. Best © 2009 John Wiley & Sons (Asia) Pte Ltd. ISBN: 978-0-470-82442-9 34 Smooth Tests of Goodness of Fit The asymptotically optimal tests given here are similar in that they have good asymptotic properties and asymptotic χ2 distributions. However, the likelihood ratio test requires estimation under both the null and alternative hypotheses, whereas the score test requires estimation under the null hypothesis only, and the Wald test requires estimation under the alternative hypothesis only. Clearly if one of these estimations is difficult, that immediately handicaps the likelihood ratio test, but one of the other tests is available.