Method of differential equations for the Renyi statistics (Q1116222)

\textit{A. Rényi} [Acta Math. Acad. Sci. Hung. 4, 191-229 (1953; Zbl 0052.142)] proposed the statistics \[ R^+_ n(a,1)=\sup_{F(x)\geq a}[(F_ n(x)-F(x))/F(x)],\quad a>0, \] where \(F_ n(x)\) denotes the empirical distribution constructed from a sample of size n from a distribution with continuous function F(x). \(F_ n(x)\) is defined as the proportion of the number of observations which do not exceed the value x. This statistic may be used, along with the Kolmogorov-Smirnov statistic, in verifying simple hypotheses on the probability distribution of a random variable. Renyi found the limiting distribution of \(R^+_ n(a,1)\) for \(n\to \infty,\) \(G(\beta)=\lim_{n\to \infty}P(\sqrt{n}R^+_ n(a,1)>\beta)=2[1-\Phi (\beta \sqrt{a/(1-a)})],\quad \beta >0,\)thereby reducing the investigation of the limiting distribution of the quantity \(R^+_ n(a,1)\), which depends on the terms of the variational series, to the determination of the distribution of a function of a sum of independent random variables, using the fact that the terms of the variational series form a Markov chain. In this paper we turn to the method of differential equations because Renyi's method does not apply to samples from parametric equations.