Curve and Periodogram Smoothing

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Abstract
Summary: The difficulty in constructing smoothing formulae is to express quantitatively the type of smoothness one expects of the curve one is estimating. An argument is given in Sections 1 and 3 for formulating this “smoothness hypothesis” in terms of the properties of a population of curves of which the curve being estimated is a member. In equation (20) we obtain a solution for the matrix of optimum weighting coefficients in terms of certain “population moments” of the ordinates of the curve. Explicit formulae based on special assumptions are deduced in equations (34), (56)–(58). General information is gained on the way the optimum smoothing function and the variance of the smoothed estimate vary with the sample size and with the assumed degree of smoothness of the parent curve.