Random translations, contractions and dilations of order statistics and records

Characterization through distributional properties of dual generalized order statistics

“…where X sÀrÀj:nÀr is independent of X r:n if and only if X 1 $ Par(a), as obtained by Castan˜o-Martı´nez et al [5] and Khan and Shah Imtiyaz [7].…”

“…Further, results in terms of lower records, upper records and order statistics are deduced. One may also refer to Alzaid and Ahsanullah [2], Beutner and Kamps [3], Wesolowski and Ahsanullah [8] and Castan˜o-Martı´nez et al [5] for the related results.…”

Characterizations of Exponential Distribution Based on Two-Sided Random Shifts

“…Let k and n be fixed integers, such that 1 ≤ k ≤ n − 1. Under some regularity conditions, it is proven in [10] and [5] that each one of the following two equations characterizes the exponential distribution:…”

“…In view of Lemma 1, to prove that (15) implies (6), it is sufficient to show that it implies (5). We shall prove (5) Applying Lemma 2 we write last equation as…”

On a characterization of generalized Erlang-truncated exponential distribution through distributional properties of dual generalized order statistics

“…If support of the distribution F (x) be over (a, b) , then by convention, we will write [3] and Castaño-Martínez et al [5] for the related results.…”