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By S. Kotz

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Applications of empirical characteristic functions in some multivariate problems. Can. J. , 20, 429–440. ) 13. Hall, P. and Welsh, A. H. (1983). A test for normality based on the empirical characteristic function. Biometrika, 70, 485–489. ) 14. Mardia, K. V. (1980). Tests of univariate and multivariate normality. In Handbook of Statistics, vol. 1, P. R. Krishnaiah, ed. NorthHolland, Amsterdam, pp. 279–320. ) 15. Murota, K. and Takeuchi, K. (1981). The Studentized empirical characteristic function and its application to tests for the shape of the distribution.

Theorem 3. If φ and ψ are two procedures in Cn and f is strictly TPn+1 , then [φ(x) − ψ(x)]f (x, θ )dµ(x) has less than n zeroes counting multiplicities. Assume f (x, θ ) is strictly TP2 . For a one-sided testing problem, existence of a uniformly most powerful level α test can be easily established by Theorems 2 and 3. For further discussion and other applications see Karlin [10–12] and Karlin and Rubin [20]. APPLICATIONS IN PROBABILITY AND STOCHASTIC PROCESSES∗ Let P(t, x, E) be the transition probability function of a homogeneous strong Markov process∗ whose state space is an interval on the real line and that possesses a realization in which almost all sample paths are continuous.

1981). Math. Oper. , 6, 113–121. ´ 15. , Cs¨org¨o, S. and Horvath, L. (1986). An Asymptotic Theory for Reliability and Concentration Processes, Lecture Notes in Statistics, 33, Springer, Heidelberg, W. Germany. ) 16. Epstein, L. and Sobel, M. (1953). J. Amer. Statist. , 48, 486–502. 17. Hollander, M. and Proschan, F. (1982). Nonparametric concepts and methods in reliability. In Handbook of Statistics, Vol. 4, P. R. T3 PLOT Krishnaiah and P. K. Sen, eds. Elsevier, Amsterdam, The Netherlands, pp.

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Encyclopedia of Statistical Scis. [Vol. 14] by S. Kotz


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