BOUNDING THE L_{1} DISTANCE IN NONPARAMETRIC DENSITY ESTIMATION

SUBRATA KUNDU^{1} AND ADAM T. MARTINSEK^{2}

1Department of Statistics and Applied Probability, University of California, Santa Barbara, CA 93106-3110, U.S.A. 2Department of Statistics, University of Illinois, 101 Illini Hall, 725 South Wright Street, Champaign, IL 61820, U.S.A.

(Received October 23, 1995; revised May 13, 1996)

Abstract. Let X_{1}, X_{2}, …, X_{n} be i.i.d. random variables with common unknown density function f. We are interested in estimating the unknown density f with bounded Mean Integrated Absolute Error (MIAE). Devroye and Győrfi (1985, Nonparametric Density Estimation: The L_{1} View, Wiley, New York) obtained asymptotic bounds for the MIAE in estimating f by a kernel estimate f̂_{n}. Using these bounds one can identify an appropriate sample size such that an asymptotic upper bound for the MIAE is smaller than some pre-assigned quantity w > 0. But this sample size depends on the unknown density f. Hence there is no fixed sample size that can be used to solve the problem of bounding the MIAE. In this work we propose stopping rules and two-stage procedures for bounding the L_{1} distance. We show that these procedures are asymptotically optimal in a certain sense as w → 0, i.e., as one requires increasingly better fit.

Key words and phrases: Density estimation, mean integrated absolute error, stopping rule, sequential estimation.