Quadratic deviation of penalized mean squares regression estimates (Q1186775)

Consider the regression model \(y_ i = g(t_ i) + \varepsilon_ i\) and let \(g_{n, \lambda}\) be a spline estimate of \(g\) which minimizes the penalty functional \[ n^{-1} \sum^ n_ 1 \bigl[ y_ i - f(t_ i) \bigr]^ 2 + \lambda \int \bigl[ f^{(m)} (t) \bigr]^ 2dt. \] The paper presents a CLT for the quadratic functionals \[ R_{n, \lambda} = \sum \bigl[ g_{n, \lambda} (t_ i) - g(t_ i) \bigr]^ 2 \quad \text{ and } \quad Z_{n, \lambda} = \int (g_{n, \lambda} - g)^ 2dt \] under various assumptions on the rate of \(\lambda \to 0\) and behaviour of \(g\). These statistics are suggested for goodness of fit testing purposes. The reviewer has two remarks: 1) The results on the asymptotic behaviour of \(R_{n, \lambda}\) and \(Z_{n, \lambda}\) under the hypothesis does not justify yet the use of these statistics for testing problems because under what kind of local alternatives it will be possible to reject remains unclear, 2) there are old papers by \textit{V. D. Konakov} and \textit{V. I. Piterbarg} [see Teor. Veroyatn. Primen. 27, No. 4, 707-724 (1982; Zbl 0503.60036), and ibid. 28, No. 1, 164-169 (1983; Zbl 0527.62047)] concerning \(\sup | f_ i - f |\) in density estimation problems which are closely related to what is studied in the present paper and are worthy of due reference.