Handbook of the Normal Distribution, Second Edition by Jagdish K. Patel

By Jagdish K. Patel

"Traces the old improvement of the traditional legislations. moment version deals a complete remedy of the bivariate general distribution--presenting totally new fabric on general integrals, asymptotic normality, the asymptotic homes of order information, and aspect estimation and statistical intervals."

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Extra info for Handbook of the Normal Distribution, Second Edition (Statistics: A Series of Textbooks and Monographs)

Example text

Efficiency considerations We begin by considering efficiency of hypothesis testing using S. 15) In order to get a quantitative impression of the efficiency of S and its associated point estimate, relative to other statistics, particular examples of F have tobe considered. In the examples that follow, S is ONE-SAMPLE LOCATION PROBLEMS 27 compared with the appropriate likelihood procedure. :__=0 d(x;, t) at ~ L... 3 If the distribution of X is N(e, a 2 ), we have e5 (B) = (2/n) 112 ja eML(B) = 1/a giving Pitman ARE= 2/n.

05 we have the following: n t. 2 we mentioned that the point estimate of () resulting from the estimating equation defined by A is the sample mean x. While it is possible to make exact conditional distribution-free inferences, unconditional behaviour must be studied when assessing the efficiency of a procedure based on A relative to other statistics. The efficacy of A is eA (()) = 1/a where a 2 = var (X) if it exists; and, var (X) = a 2 jn. 3 Rank transformations Comparing the statistics Sand A discussed in previous sections, it will be noted that S can be regarded as having been derived from A by applying a transformation T(u) = 1 to the values oflx;- 81 in A.

28) is r. f. : 0. f. of Y. The explanation of the similar effects of these two transformations is that E( Y(r)l = ,p; 1 {r/(n + 1)} + 0(1/n); see, for example, David (1970, p. 65). 67 Note that the scores give a reasonable reproduction of the original observations; also note the closeness of the two sets of scores. The agreement between scores and original observations suggests that, in the normal case, procedures based on the scores could have efficiencies close to that of the likelihood method.

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