A Note on the Nonparametric Least-squares Test for Checking a Polynomial Relationship

Abstract

Recently, Gijbels and Rousson [6] suggested a new approach, called nonparametric least-squares test, to check polynomial regression relationships. Although this test procedure is not only simple but also powerful in most cases, there are several other parameters to be chosen in addition to the kernel and bandwidth. As shown in their paper, choice of these parameters is crucial but sometimes intractable. We propose in this paper a new statistic which is based on sample variance of the locally estimated pth derivative of the regression function at each design point. The resulting test is still simple but includes no extra parameters to be determined besides the kernel and bandwidth that are necessary for nonparametric smoothing techniques. Comparison by simulations demonstrates that our test performs as well as or even better than Gijbels and Rousson's approach.Furthermore, a real-life data set is analyzed by our method and the results obtained are satisfactory.

Identification of Non-Varying Coefficients in Varying-Coefficient Models

“…The coefficients a, b and the degrees of freedom r are chosen in such a way that the first three moments of a + bχ 2 r are made to match those of the quadratic form. Both some theoretical examination [3] and extensive simulations [5,16,18] have demonstrated that the three-moment χ 2 approximation performs quite well in many situations. In our case, by neglecting the biases of the estimation and with the similar derivations in [18], we obtain…”

“…Therefore, the proposed test statistic is a ratio of quadratic forms in normal variables and the pvalue in (16) can also be transformed as the probability of a quadratic form of normal variables being greater than zero (for example, see [1,18]). Therefore, many algorithms and routines exist for computing the p-value of the test including the exact method [16] and many approximate methods (for example, see [1] for a review).…”

Application of Local Polynomial Interpola-tion in Isomagnetic Map Magnetic Survey

“…& Gijbels i. (1996) made more detailed description for the local linear estimation [3][4][5][6] . In addition to the long-term trend, the variable quantities with short-range changes are important studied in earth science.…”

Spatially Varying Coefficient Models: Testing for Spatial Heteroscedasticity and Reweighting Estimation of the Coefficients

“…For the details of the three-moment w 2 approximation, refer to Azzalini and Bowman (1993) and Leung et al (2000b). In particular, see Mei et al (2003) for dealing with the case of tr(Q 3 i ) 0 (i 1, 2).…”

“…(The reasonableness of the assumptions will be explained in remark 2). With these assumptions and under the null hypotheses H 0 , the p-value in equation ( 18) can be computed by using the three-moment w 2 approximation of the quadratic forms in normal variables (eg see Azzalini and Bowman, 1993;Leung et al, 2000b;Mei et al, 2003). Here, we give only the final computational formula.…”