Optimal robust estimators for families of distributions on the integers
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Optimal robust estimators for families of distributions on the integers Ricardo A. Maronna1 · Victor J. Yohai2 Received: 11 November 2019 / Revised: 15 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Let Fθ be a family of distributions with support on the set of nonnegative integers Z 0 . In this paper we derive the M-estimators with smallest gross error sensitivity (GES). We start by defining the uniform median of a distribution F with support on Z 0 (umed(F)) as the median of x + u, where x and u are independent variables with distributions F and uniform in [-0.5,0.5] respectively. Under some general conditions we prove that the estimator with smallest GES satisfies umed(Fn ) =umed(Fθ ), where Fn is the empirical distribution. The asymptotic distribution of these estimators is found. This distribution is normal except when there is a positive integer k so that Fθ (k) = 0.5. In this last case, the asymptotic distribution behaves as normal at each side of 0, but with different variances. A simulation Monte Carlo study compares, for the Poisson distribution, the efficiency and robustness for finite sample sizes of this estimator with those of other robust estimators. Keywords Gross-error sensitivity · Uniform median · Contamination bias
1 Introduction Consider a one-parameter family of distributions Fθ . An important problem in the theory of robust estimation is the study of estimators which in some sense optimize their bias under contamination. The gross-error sensitivity (GES) is defined as the
This research was partially supported by Grants X-094 and 20020170100022BA from Universidad de Buenos Aires, PID 5505 from CONICET and PAV 120 and PICT 21407 from ANPCYT, Argentina.
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Ricardo A. Maronna [email protected]
1
Mathematics Department, Faculty of Exact Sciences, University of La Plata, Calle 50 y 115, 1900 La Plata, Argentina
2
Departamento de Matemática and Instituto de Calculo, CONICET, Facultad de Ciencias Exactas y Naturales, Ciudad Universitaria, 1428 Buenos Aires, Argentina
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R. A. Maronna, V. J. Yohai
maximum of the absolute values of the influence function. It gives an approximation to the maximum bias produced by an outlier contamination of rate ε, when ε is ”small”. Hampel (1974) dealt with M-estimators defined as solutions of equations of the form n ψ (xi , θ ) = 0, i=1
and considered the problem of minimizing the asymptotic variance among Fisherconsistent M-estimators which satisfy a bound on the GES. Alternatively, this problem can be stated as minimizing the GES under a bound on the asymptotic variance. Details are given in Sect. 3.1. In this paper we consider minimizing the GES without any restrictions on the asymptotic variance. Let Fθ be a family of continuous distributions with densities p(x, θ ) and score function ψ0 (x, θ ) =
∂ log p(x, θ ) . ∂θ
(1)
Maronna et al. (2019, p. 68) show that if ψ0 (x, θ ) is strictly monotone on x, the M-estimator with smallest GES is obtained by solving med (Fn ) = med F θ ,
(2)
where “med” stands for th
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