On testing composite statistical hypotheses. I. (Russian) Zbl 0644.62054

Bondarev, B. V.

Teor. Veroyatn. Mat. Stat., Kiev 36, 3-10 (1987).

Let \(X_ 1,X_ 2,...,X_ n\) be a sample from a cdf F and let \(F_ n\) be the empirical cdf. To test the composite hypothesis that \(F\in \{F(x,\theta):\theta \in \Theta \subset R^ m\), \(x\in R\}\), the statistic \(\omega^ 2_ n(\theta_ n)\) is used, where \[ \omega^ 2_ n(\theta)= n\int^{+\infty}_{-\infty}[F_ n(x)-F(x,\theta)]^ 2 dF_ n(x) \] and \(\theta_ n\) minimizes \(\omega^ 2_ n(\theta)\). An estimate for the asymptotic (as \(n\to \infty)\) distribution of \(\omega^ 2_ n(\theta_ n)\) is presented.

Reviewer: R.Zielinski

MSC:

62G10	Nonparametric hypothesis testing
62E20	Asymptotic distribution theory in statistics

Keywords:

empirical distributions; Kolmogorov-Smirnov statistic with estimated parameters; minimum distance estimators; composite hypothesis