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Weak and universal consistency of moving weighted averages. (English) Zbl 0596.62040

Consider the fixed design regression model y i,n =g(t i,n )+ϵ i,n , 1in, where the random variables ϵ i,n form a triangular array and are independent for fixed n, and identically distributed with zero mean, t i,n [0,1] are points where the measurements y i,n are taken, and g is a smooth regression function to be estimated. For moving weighted averages

g ^ (ν) (t)= i=1 n w i,n (ν) (t)y i,n ,

results on weak consistency g ^ (ν) (t) P g (ν) (t) for some ν0 are derived.

Mofifying the definition of universal consistency given by C. J. Stone [Ann. Stat. 5, 595-645 (1977; Zbl 0366.62051)], for the fixed design case, conditions for fixed design universal consistency are given. The results are then shown to apply to kernel estimators and local least squares estimators which are special cases of moving weighted averages.

62G05Nonparametric estimation
62G20Nonparametric asymptotic efficiency
62J02General nonlinear regression
[1]J. K. Benedetti, On the nonparametric estimation of regression functions,J. Roy. Statist. Soc. Ser. B 39 (1977), 248–253.MR 58: 13480
[2]W. S. Cleveland, Robust locally weighted regression and smoothing scatterplots,J. Amer. Statist. Assoc. 74 (1979), 829–836.MR 81i: 62127 · Zbl 0423.62029 · doi:10.2307/2286407
[3]Th. Gasser andH.-G. Müller, Estimating regression functions and their derivatives by the kernel method,Scand. J. Statist. 11 (1984), 171–185.MR 86h: 62056
[4]Th. Gasser, H.-G. Müller, W. Köhler, L. Molinari andA. Prader, Nonparametric regression analysis of growth curves,Ann. Statist. 12 (1984), 210–229.MR 86e: 62057 · Zbl 0535.62088 · doi:10.1214/aos/1176346402
[5]E. A. Nadaraya, On estimating regression,Theory Probab. Appl. 9 (1964), 141–142.MR 29: 4147 · doi:10.1137/1109020
[6]M. B. Priestley andM. T. Chao, Nonparametric function fitting,J. Roy. Statist. Soc. Ser. B 34 (1972), 385–392.MR 48: 9948
[7]W. E. Pruitt, Summability of independent random variables,J. Math. Mech. 15 (1966), 769–776.MR 33: 3338
[8]C. H. Reinsch, Smoothing by spline functions,Numer. Math. 10 (1967), 177–183.MR 45: 4598 · Zbl 0161.36203 · doi:10.1007/BF02162161
[9]E. Schuster andS. Yakowitz, Contributions to the theory of nonparametric regression, with applications to system identification,Ann. Statist. 7 (1979), 139–149.MR 80d: 62032 · Zbl 0401.62033 · doi:10.1214/aos/1176344560
[10]C. J. Stone, Consistent nonparametric regression,Ann. Statist. 5 (1977), 505–545.MR 56: 1574 · Zbl 0366.62051 · doi:10.1214/aos/1176343886
[11]G. Wahba, Smoothing noisy data with spline functions,Numer. Math. 24 (1975), 383–393.MR 53: 9587 · Zbl 0299.65008 · doi:10.1007/BF01437407
[12]G. S. Watson, Smooth regression analysis,Sankhyā Ser. A 26 (1964), 359–372.MR 32: 3226