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Bump hunting in regression revisited Harezlak, Jaroslaw


Suppose bivariate data [formula] are observed at times a ≤ t1 ≤ t2 ≤ ... ≤ tn ≤ b. Given a nonparametric regression model [formula] mean 0, variance ϭ² , for i = 1, 2 , . . . , n, we want to estimate the number of modes of the underlying regression function m(-) or its derivative. We use the penalized least squares technique to get an estimate of m(-), i.e. the function minimizing [formula] where [formula] dt is a penalty function and ƛ is a smoothing parameter. The estimate of the derivative, m’ is simply the derivative of the estimate, (m’). A new test of multimodality is introduced and its performance is studied. Our idea is motivated by the test proposed by Silverman (1981) concerning the number of modes in the density function. He used a "critical bandwidth" as a test statistic in a kernel smoothing context. He noted that if the data are strongly bimodal, we would need a large value of a bandwidth to obtain a unimodal density estimate. In our case we define the "critical smoothing parameter" Xcrit as the smallest ƛ giving an estimate with the specified number of modes. We use Ac r ; 4 as a test statistic in our new test CriSP. We use bootstrap techniques to assess the performance of our test. We study the effects of the penalty L on the quality of our test via simulation using different regression functions and we compare it with Bowman et.al.'s monotonicity test (1998). CriSP is also applied to the children's growth data in studying the number of bumps in the derivatives of the growth functions. In a sample of 43 boys and 50 girls, our test procedure gives an automatic classification rule in about 80% of the growth curves analyzed.

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