The limit distribution of the maximal deviation of a spline estimate of a probability density (Q912475)

[For the entire collection of the original see Zbl 0626.00025.]. The authors define \(S_ n(\cdot)\) as the cubical spline interpolating the empirical distribution function \(F_ n\) of the sample \(X_ 1,...,X_ n\) of the distribution F. The function \(S_ n\) is built with the nodes \((x_ 1,y_ 1),...,(x_ n,y_ n)\) where \(x_ k=k/N\) and \(y_ k=F_ n(x_ k)\), \(k=0,1,...,N\). The integer is a function of n which is not defined in the paper. More, according to these notations, it seems that the distribution F is implicitly defined only on the interval (0,1). Setting \[ \xi_ n=\sqrt{nh}\max_{0\leq x\leq 1}| [S_ n'(x)-f(x)]/\sigma_ n(x)\sqrt{f(x)}| \] where \(h=1/N\), f is the density function of the X's, and \(\sigma_ n\) a convenient normalization factor, the authors derive, with very long and laborious majorizations and extensive calculus, the limit distribution of \(\xi_ n\). It is proved that this distribution is of the Gumbel extreme value distribution type. The proof uses results of \textit{S. M. Berman} [see Ann. Probab. 2, 950-953 (1974; Zbl 0292.60054), and Ann. Math. Statistics 42, 927-945 (1971; Zbl 0218.60038)].