Abstract:
Asymptotic power of a test for detecting outliers examined for the first time in [L. N. Bolshev, International Summer School on Probability Theory and Mathematical Statistics, Varna, 1974, pp. 1–41 (in Russian)] is considered for a two-parameter family of distributions dependent on unknown shift and scale parameters. The probability that the test wrongly identifies a "good" observation as an outlier and the probability that a contaminated observation shows up as an outlier are examined with some other useful properties of the test. To avoid the masking effect we propose using robust estimators of shift and scale parameters with a high breakdown point and a bounded influence function in Hampel's terminology [J. Galambos, The Asymptotic Theory of Extreme Order Statistics, Wiley, New York, 1978]. The rate of convergence of the normalized maximum of the independent identically distributed random variables for univariate families of distributions dependent on an unknown shift parameter is investigated in [V. I. Pagurova and S. A. Nesterova, Theory Probab. Appl., 36 (1991), pp. 176–185], [V. I. Pagurova and A. V. Shvedov, On asymptotic distribution for the maximum in a stationary Gaussian sequence, Vestnik MGU, 15, 4 (1995), pp. 33–43 (in Russian)]. Asymptotic properties of a test for detection of outliers for an elliptical family of multivariate distributions are studied in [V. I. Pagurova and I. L. Chizhikova, Theory Probab. Appl., 40 (1995), pp. 390–398].
Keywords:test for detecting outliers, asymptotic power, two-parameter family of distributions.
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