A LOCATION SHIFT PROBLEM IN NONPARAMETRIC DENSITY ESTIMATION

Let X1, X2.X„ be i.i.d. random variables having a probability density function f(x) and f,,(x) be a nonparametric density estimator of f(x). We investigate the property of a location shift random variable a„ which minimizes integrated squared error ISE„(a): ISE„(a) = f,,(x) — f(x — a)12 dx. The asymptotic normality and the order of strong convergence of the r.v. a,, and those of ISE„(a„) are studied. We also give some numerical examples and some simulations which show the effectiveness of using the a„ when one estimates f(x) by f„(x).