The Annals of Mathematical Statistics

Minimum Variance Order when Estimating the Location of an Irregularity in the Density

Thomas Polfeldt

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Abstract

Let $f(y)$ be a probability density on the real line, with \begin{equation*}\tag{1} f(y) = ^+R(y) (y > 0),\quad f(y) = ^-R(|y|) (y < 0)\end{equation*} where $^+R$ and $^-R$ are normalized slowly varying functions as $y \rightarrow 0$ (cf. [6] chapter 8, sect. 8). Let $\theta$ be a location parameter. Denote by $X$ a sample $(x_1, \cdots, x_n)$ of $n$ independent observations from the distribution defined by $f(x - \theta)$. By $t = t(X)$ we denote an unbiased estimator of $\theta$. In this paper we study lower bounds for the variances $V_\theta(t)$, with special reference to their order, in $n$. Considering only densities with regular variation at $\theta$, (1) includes all cases where $\theta$ is the location either of a cusp or of a discontinuity with finite and positive values of $f(0 -)$ and $f(0 +)$. Under some conditions on $^+R$ and $^-R$, we calculate a function $\psi(h)$ such that \begin{equation*}\tag{2} V_\theta(t) \geqq K(\psi^{-1}(n^{-1}))^2\quad (\text{all} t) (0 < K < \infty).\end{equation*} It is surmised that this lower variance bound is of the best possible order of $n$. The bound is sometimes $o(n^{-1})$; hyperefficient estimators should then be possible. It is found that $\psi(h)$ depends heavily on the function $^+ \varepsilon(s)$ defined by $^+ R(y) = ^+ A \exp \{-\int^1_{y^+} \varepsilon(s)/s ds\}$, and on the corresponding function $^- \varepsilon(s)$. In view of [6] chapter 8, sect. 9 (or (10) below), this form of $^+R(y)$ is not a constraint on $f(y)$. The estimators $t_0$ constructed by Daniels [5] and Prakasa Rao [10] (for particular $^+R$ and $^-R$) have variances of the order of (2). This order is thus optimal with the densities considered: we have $V_\theta(t_0) \geqq \inf V_\theta(t) \geqq K(\psi^{-1}(n^{-1}))^2$. The Prakasa Rao estimators are hyperefficient. The calculations are based, partly, on ideas from the author's paper [9]. Since a cusp may be a mode, the results of this paper contribute to the discussion on the estimation of the mode (see [4], [11] and references therein). The generalization of (1) to regularly varying $f(y)$ as $y \downarrow 0$ and $y \uparrow 0$ will be treated elsewhere ([8]).--The generalization of (2) to biased estimators $t$ (or mean square error) is straightforward, but some conditions on the bias function will be necessary. Notation. $K$ and $K'$ denote positive, finite constants. If there exist $K, K'$ such that $K < a(x)/b(x) < K'$ for all $x, |x| < x_0$, we shall write $a(x) = \Omega(b(x)) (x \rightarrow 0)$; sometimes we omit $(x \rightarrow 0)$.

Article information

Source
Ann. Math. Statist., Volume 41, Number 2 (1970), 673-679.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177697112

Digital Object Identifier
doi:10.1214/aoms/1177697112

Mathematical Reviews number (MathSciNet)
MR256500

Zentralblatt MATH identifier
0198.23402

JSTOR
links.jstor.org

Citation

Polfeldt, Thomas. Minimum Variance Order when Estimating the Location of an Irregularity in the Density. Ann. Math. Statist. 41 (1970), no. 2, 673--679. doi:10.1214/aoms/1177697112. https://projecteuclid.org/euclid.aoms/1177697112


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