Acta Mathematica Academiae Scientiarum Hungaricae 5. (1954)

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

2 G. HAJÓS and A. RÉNYI Accordingly, we confine ourselves in what succeeds to the research of the order statistics rf. Our results may be interpreted as facts concerning the original order statistics §?• 2. The variables rf are not independent, the relation rf g rf (in case j< k) contradicts the independency. The joint density function of the variables if,..rf is (1) /(x1;. = n\ (OgXi^---1). As a matter of fact, if E denotes any measurable subset ot the n-dimensional simplex defined by the inequalities 0 ^ • • • g; xn ^ 1, we have Pm,..if) iE) = Z P((%,. • - , r]in)cE), where the summation is extended over all permutations in of the indices 1 and the density function of lyj is equal to 1 at any point (Xj,..., x„) of the cube 0 gj xk ^ 1 (k n). Considering now the case that rf = ck,...,if = c„ (2 ^ к ^ n) are fixed, we state that rf,...,rf-\ are order statistics of a sample of size к—1 from a population of uniform distribution in the interval (0, ck). In fact, this is true if rf,...,rf-1 are furnished by any given к—1 variables out of i]j,..., r\n, since these к—1 variables are uniformly distributed, even within the cube 0 Ш x, ^ ck (i = \ ,к—1). Thus, by (1), the joint density function of the variables rf,...,rf-\, under condition if = ck,. .., ?]* = cn, is (2) /(x„ ..., xfc-i |cfc,..cf) = (k~~ri By the same argument, if if =C\,..., if = ck (1 Si к ^ n — 1) are fixed, rf+i, ■ ■■,rf are order statistics of a sample of size n—к from a population of uniform distribution in the interval (ck, 1) and the joint density function of the variables rf+\, ..., if, under condition rf = Ci,..., if — Cu, is (n—k)! (3)f (x,,-...i, . . ., Xn j Ci,..., cf) -(1 -cf)'1 (ftSXit 1 g-SX,il). Since (2) and (3) depend only on ck, our statements hold also under the only condition if = Ck, i. e. (2) and (3) give also the values of the functions /(xi,..., xft+i|c,£) and f(xM,..., x„jck), and the same holds under any restriction on the non-occurring variables. By the same argument, under condition if = ck, the sets of variables (rf,. .., and (if+i,..., if) are independent. By other words, order statistics form a Markov chain} 3. The joint density function of the variables rf+1,,rf (1 s / < к si n) is (4) Mxi+1, ■■■, Xk) = -Щ’%)Г х’ш (1 — Xfc)"-1 (0 - xi+i i"^xt5l). 2 A. N. Kolmogoroff [1] was the first to remark this.

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