The asymptotic behavior of the limit distribution of the Kolmogorov- Smirnov statistic in the case of a composite hypothesis for the class of projecting estimates of an unknown parameter. (English. Russian original) Zbl 0719.62041

Makarov, A. A.

Theory Probab. Appl. 32, No. 2, 380-383 (1987); translation from Teor. Veroyatn. Primen 32, No. 2, 412-416 (1987).

For \(x_ 1,...,x_ n\) a sample from a parametric family of distributions \(\{F(x,\theta),\theta \in \Theta \subset {\mathbb{R}}^ r\}\), satisfying the Cramér-Rao regularity conditions, we define the Kolmogorov-Smirnov statistic \[ D^+_ n=\sup_{x}\sqrt{n}(F_ n(x)- F(x,\theta^*_ n)), \] where \(F_ n(x)\) is the empirical distribution function and \(\theta^*_ n\) is an estimate of the unknown parameter \(\theta^ 0\). We study the limit distribution of \(D^+_ n\) in the case where \(\theta^*_ n\) is taken to be a \(\sqrt{n}\)-consistent estimate of a special type, named a projecting estimate.

MSC:

62F10	Point estimation
62F12	Asymptotic properties of parametric estimators

Keywords:

Cramér-Rao regularity conditions; Kolmogorov-Smirnov statistic; empirical distribution; projecting estimate