On efficient estimation of densities for sums of squared observations

Anton Schick and Wolfgang Wefelmeyer

Statistics & Probability Letters, 2012, vol. 82, issue 9, 1637-1640

Abstract: Densities of functions of independent and identically distributed random observations can be estimated by using a local U-statistic. Under an appropriate integrability condition, this estimator behaves asymptotically like an empirical estimator. In particular, it converges at the parametric rate. The integrability condition is rather restrictive. It fails for the sum of powers of two observations when the exponent is at least 2. We have shown elsewhere that for the exponent equal to 2 the rate of convergence slows down by a logarithmic factor in the support of the squared observation. Here we show that the estimator is efficient in the sense of Hájek and LeCam. In particular, the convergence rate is optimal.

Keywords: Local asymptotic normality; Regular estimator; Nonstandard convergence rates (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1016/j.spl.2012.04.021

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