A class of asymptotically efficient estimators based on sample spacings
- PDF / 456,033 Bytes
- 20 Pages / 439.37 x 666.142 pts Page_size
- 70 Downloads / 159 Views
A class of asymptotically efficient estimators based on sample spacings M. Ekström1,2
· S. M. Mirakhmedov3 · S. Rao Jammalamadaka4
Received: 1 October 2017 / Accepted: 30 January 2019 © The Author(s) 2019
Abstract In this paper, we consider general classes of estimators based on higher-order sample spacings, called the Generalized Spacings Estimators. Such classes of estimators are obtained by minimizing the Csiszár divergence between the empirical and true distributions for various convex functions, include the “maximum spacing estimators” as well as the maximum likelihood estimators (MLEs) as special cases, and are especially useful when the latter do not exist. These results generalize several earlier studies on spacings-based estimation, by utilizing non-overlapping spacings that are of an order which increases with the sample size. These estimators are shown to be consistent as well as asymptotically normal under a fairly general set of regularity conditions. When the step size and the number of spacings grow with the sample size, an asymptotically efficient class of estimators, called the “Minimum Power Divergence Estimators,” are shown to exist. Simulation studies give further support to the performance of these asymptotically efficient estimators in finite samples and compare well relative to the MLEs. Unlike the MLEs, some of these estimators are also shown to be quite robust under heavy contamination. Keywords Sample spacings · Estimation · Asymptotic efficiency · Robustness Mathematics Subject Classification 62F10 · 62Fl2
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11749019-00637-7) contains supplementary material, which is available to authorized users.
B
M. Ekström [email protected]; [email protected]
1
Department of Statistics, USBE, Umeå University, Umeå, Sweden
2
Department of Forest Resource Management, Swedish University of Agricultural Sciences, Umeå, Sweden
3
Department of Probability and Statistics, Institute of Mathematics, Academy of Sciences of Uzbekistan, Tashkent, Uzbekistan
4
Department of Statistics and Applied Probability, University of California, Santa Barbara, USA
123
M. Ekström et al.
1 Introduction Let {Fθ , θ ∈ Θ} be a family of absolutely continuous distribution functions on the real line and denote the corresponding densities by { f θ , θ ∈ Θ}. For any convex function φ on the positive half real line, the quantity Sφ (θ ) =
∞ −∞
φ
f θ (x) f θ0 (x)
f θ0 (x)dx
is called the φ-divergence between the distributions Fθ and Fθ0 . The φ-divergences, introduced by Csiszár (1963) as information-type measures, have several statistical applications including estimation. Although Csiszár (1977) describes how this measure can also be used for discrete distributions, we are concerned with the case of absolutely continuous distributions in the present paper. Let ξ1 , . . . , ξn−1 be a sequence of independent and identically distributed (i.i.d.) random variables (r.v.’s) from Fθ0 , θ0 ∈ Θ. The goal is to esti
Data Loading...
