NONPARAMETRIC TESTS FOR BOUNDS ON THE DERIVATIVE OF A REGRESSION FUNCTION

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

1Statistics Department, University of British Columbia, Vancouver BC, Canada V6T 1Z2 2Statistics 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.