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NONPARAMETRIC TESTS FOR BOUNDS ON THE DERIVATIVE

OF A REGRESSION FUNCTION

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NANCY E. HECKMAN^{ 1} AND
BING
LI^{ 2}

^{1} *Statistics Department, University of British Columbia,
Vancouver BC, Canada V6T 1Z2*

^{2} *Statistics Department, Pennsylvania State University,
University Park, PA 16802, U.S.A.*
(Received August 17, 1994; revised July 20, 1995)

**Abstract.**
We consider two tests of the null hypothesis
that the *k*-th derivative of a regression function is uniformly
bounded by a specified constant. These tests can be used to study the
shape of the regression function. For instance, we can test for
convexity of the regression function by setting *k* = 2 and the
constant equal to zero. Our tests are based on *k*-th order divided
difference of the observations. The asymptotic distribution and
efficacies of these tests are computed and simulation results
presented.

*Key words and phrases*:
Derivative of a regression
function, convexity, divided differences.

**Source**
( TeX ,
DVI ,
PS )