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  • 1
    Electronic Resource
    Electronic Resource
    On Mad and Comedians (1997)
    Springer
    Annals of the Institute of Statistical Mathematics 49 (1997), S. 615-644 
    Details
    ISSN: 1572-9052
    Keywords: Median absolute deviation from the median ; robust measure of correlation ; comedian ; breakdown point ; covariance ; correlation coefficient ; bivariate normal vectors ; strong consistency ; asymptotic normality
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A popular robust measure of dispersion of a random variable (rv) X is the median absolute deviation from the median med(|X - med(X)|), MAD for short, which is based on the median med(X) of X. By choosing Y = X, the MAD turns out to be a special case of the comedian med((X - med(X))(Y - med(Y))), which is a robust measure of covariance between rvs X and Y. We investigate the comedian in detail, in particular in the normal case, and establish strong consistency and asymptotic normality of empirical counterparts. This leads to a robust competitor of the coefficient of correlation as an asymptotic level-α-statistic for testing independence of X and Y. An example shows the weird fact that knowledge of the population med(X) does not necessarily pay (in the sense of asymptotic relative efficiency) when estimating the MAD.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1003258024248
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  • 2
    Electronic Resource
    Electronic Resource
    Lan of Thinned Empirical Processes with an Application to Fuzzy Set Density Estimation (1999)
    Springer
    Extremes 1 (1999), S. 323-349 
    Details
    ISSN: 1572-915X
    Keywords: thinned empirical process ; point process ; loglikelihood ratio ; local asymptotic normality ; central sequence ; regular estimators ; asymptotic efficiency ; fuzzy set density estimator
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We establish local asymptotic normality of thinned empirical point processes, based on n i.i.d. random elements, if the probability $${\alpha }_{n}$$ of thinning satisfies $${\alpha }_n \to _{n \to \infty } 0,n{\alpha }_n \to _{n \to \infty } \infty$$ . It turns out that the central sequence is determined by the limit of the coefficient of variation of the tangent function. The central sequence depends only on the total number $${\tau }\left( n \right)$$ of nonthinned observations if and only if this limit is 1 or −1. In this case under suitable regularity conditions, an asymptotically efficient estimator of the underlying parameter can be based on $${\tau }\left( n \right)$$ only. An application to density estimation leads to a fuzzy set density estimator, which is efficient in a parametric model. In a nonparametric setup, it can also outperform the usual kernel density estimator, depending on the values of the density and its second derivative.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1009981817526
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    Paper (German National Licenses)
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