Acta Mathematica Academiae Scientiarum Hungaricae 5. (1954)

1954 / 1-2. szám - Hajós G. - Rényi A.: Elementary proofs of some basic facts concerning order statistics

ELEMENTARY PROOFS OF SOME BASIC FACTS CONCERNING ORDER STATISTICS By G. HAJÓS (Budapest), member of the Academy, and A. RÉNYI (Budapest), corresponding member of the Academy Let I denote a sample of size n from a population with the distribution function F(x). By other words, are mutually independent random variables with the common distribution function F(x). Let be the same set of variables, rearranged in increasing order of magnitude, i. e. S = where /?,,(xu...,x„) denotes the Är’th term of the sequence obtained by rearranging the numbers xu...,xn in increasing order of magnitude. The present paper deals with the order statistic £*. Some basic facts will be proved by simple methods. We aim expressively to avoid the cal­culus and at reduction of any calculation to possibly minimal extent. As consequence, our results may be easily checked by calculation in various different ways, which we are not intended to mention. Our results seem us mostly to be known, though we did not find some of them explicitly in the literature. We endeavoured to give an elementary and systematic treatment of our subject. Accordingly, our paper may be of methodical interest. As to the literature we refer to the bibliography compiled by S. S. Wilks [3] and by the second named author [4]. 1. In order to obtain distribution-free results, i. e. results independent of the distribution function F(x), we introduce rlk = F(f,k) (k=\,...,ri). If we suppose that y = F(x) is strictly increasing and continuous, the same holds for the inverse function x = F l(y) and we have1 P(r/t < x) = P(£t < F~l (x)) = F(F’ (x)) = x (Osxsl), what shows that the variables rjlf...,rjn are uniformly distributed in the interval (0,1). Putting rf — F(fk) (k=\,...,n) we have n% = F(S) = A(/?,-(£,,..., £,,)) = Rk(F(f),..., F(£„>) = /fcfai,..., rin). Consequently r\\, ...rjl are order statistics of a sample of size n from a population of uniform distribution in (0, 1). 1 P(A) denotes the probability of the event A. 1 Acta Mathematica

Next