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BOUNDING THE *L*1 DISTANCE IN NONPARAMETRIC

DENSITY ESTIMATION

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

^{1} *Department of Statistics and Applied Probability,
University of California,*

Santa Barbara, CA 93106-3110, U.S.A.

^{2} *Department 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 Györfi
(1985, *Nonparametric Density Estimation: The
L*1 *View*, Wiley, New York) obtained asymptotic bounds for the
MIAE in estimating *f* by a kernel estimate ^{^}*f*_{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.

**Source**
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