ISSN:
1435-926X
Keywords:
Exponential distribution
;
characterization
;
order statistics
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract We considern independent and identically distributed random variables with common continuous distribution functionF concentrated on (0, ∞). LetX 1∶n≤X2∶n...≤Xn∶n be the corresponding order statistics. Put $$d_s \left( x \right) = P\left( {X_{k + s:n} - X_{k:n} \geqslant x} \right) - P\left( {X_{s:n - k} \geqslant x} \right), x \geqslant 0,$$ and $$\delta _s \left( {x, \rho } \right) = P\left( {X_{k + s:n} - X_{k:n} \geqslant x} \right) - e^{ - \rho \left( {n - k} \right)x} ,\rho 〉 0,x \geqslant 0.$$ Fors=1 it is well known that each of the conditions d1(x)=O ∀x≥0 and δ1 (x, p) = O ∀x≥0 implies thatF is exponential; but the analytic tools in the proofs of these two statements are radically different. In contrast to this in the present paper we present a rather elementary method which permits us to derive the above conclusions for somes, 1≤n —k, using only asymptotic assumptions (either forx→0 orx→∞) ond s(x) and δ1 (x, p), respectively.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01895297
