Distributional Properties of Order Statistics and Record Statistics

Distributional properties of the order statistics, upper and lower record statistics have been utilized to characterize distributions of interest. Further, one sided random dilation and contraction are utilized to obtain the distribution of non-adjacent ordered statistics and also their important deductions are discussed.

For the proof of sufficiency part, note that the ‫݂݀‬ of ܺ ሺሻ by the convolution method is Differentiating both the sides ‫.ݓ‬ ‫.ݎ‬ ‫.ݐ‬ ‫,ݔ‬ we have for ‫ݏ‬ ‫ݎ‬ and therefore, for ‫ݏ‬ ‫ݎ‬ and the result follows in view of (1.13), (1.14), (1.23) and (1.25).

Characterization results based on upper records
Theorem 3.1.Let ܺ ሺሻ be the ‫ݎ‬ ௧ upper record statistic from a continuous population with the ‫݂݀‬ ݂ሺ‫ݔ‬ሻ and the ݂݀ ‫ܨ‬ሺ‫ݔ‬ሻ, then where ܺ ሺሻ is independent of ܺ ሺሻ if and only if ܺ ଵ ~݁‫ݔ‬ሺߙሻ.
Proof.The necessary part can be proved easily using mgf.
To prove the sufficiency part, we proceed here as explained in the Theorem 2.1, to get Differentiating both the sides of (3.
Proof.This can be proved by noting that logܺ in view of (1.11) and (1.24).
Corollary 3.2.Let ܺ ሺሻ be the ‫ݎ‬ ௧ lower record from a continuous population with the pdf ݂ሺ‫ݔ‬ሻ and the df ‫ܨ‬ሺ‫ݔ‬ሻ, then where ܺ ሺሻ is independent of ܺ ሺሻ if and only if ܺ ଵ ~ powሺߙሻ.
Proof.The Corollary can be proved by considering with an appeal to (1.15) and (1.25).

Characterizing results based on order statistics
Theorem 4.1.Let ܺ : be the ‫ݎ‬ ௧ order statistic from a sample of size ݊ drawn from a continuous population with the ‫݂݀‬ ݂ሺ‫ݔ‬ሻ and the ݂݀ ‫ܨ‬ሺ‫ݔ‬ሻ, then where ܺ :ି is independent of ܺ : if and only if ܺ ଵ ~ ‫ݔ݁‬ ሺߙሻ.
Proof.The proof of the necessary part is easy.
To prove the sufficiency part, we have Differentiating both the sides of (4. and the result follows in view of (1.13) and (1.25).This is case of random contraction.
Theorem 4.2.Let ܺ : be the ‫ݎ‬ ௧ order statistic from a sample of size ݊ drawn from a continuous population with the ‫݂݀‬ ݂ሺ‫ݔ‬ሻ and the ݂݀ ‫ܨ‬ሺ‫ݔ‬ሻ, then Proof.The proof of the necessary part is easy.
The Corollary is proved by an appeal of (1.13) and (1.25).as given in Wesolowski and Ahsanullah (2004).

2. Characterizing results based on lower records Theorem
Alzaid and Ahsanullah (2003)have characterized distributions using distributional properties of adjacent lower records, upper records and order statistics.Castaño-Martínez et al. (2010) have extended the result for non-adjacent case using integral equations rather than differential equations, emphasizing that it is almost impossible to find the solution using differential equations.We, in this paper, have considered the cases for non-adjacent records and order statistics, which Castaño-Martínez et al. (2010) claim that it is still an open problem, with simpler proofs using differential equations and have also considered random dilation and contraction to obtain the distributions of lower records, upper records and order statistics.We assume that the df is differentiable ‫.ݓ‬ ‫.ݎ‬ ‫.ݐ‬ its arguments.Further, it is noted that if ܻ is a measurable function of ܺ with the relation 2.1.Let ܺ ሺ௦ሻ be the ‫ݏ‬ ௧ lower record statistic from a continuous population with the ‫݂݀‬ ݂ሺ‫ݔ‬ሻ and the ݂݀ ‫ܨ‬ሺ‫ݔ‬ሻ , then Let ܺ ሺ௦ሻ be the ‫ݏ‬ ௧ lower record from a continuous population with the pdf ݂ሺ‫ݔ‬ሻ and the df ‫ܨ‬ሺ‫ݔ‬ሻ, then‫ݏ‬ െ ‫ݎ‬ െ 1, ‫ݏ‬ െ ‫ݎ‬ ; 1 ‫ݎ‬ ൏ ‫ݏ‬ (2.7)where ܻ :௦ିଵ is the ݆ ௧ order statistic from a sample of size ሺ‫ݏ‬ െ 1ሻ drawn from the Parሺߙሻ distribution and is independent of ܺ ሺ௦ሻ if and only if ܺ ଵ ~ inWሺߙሻ.Let ܺ ሺ௦ሻ be the ‫ݏ‬ ௧ upper record from a continuous population with the pdf ݂ሺ‫ݔ‬ሻ and the df ‫ܨ‬ሺ‫ݔ‬ሻ, then 1, ݂ ಽሺሻ ሺ‫ݔ‬ሻ ൌ ‫ܨ‪ሾ‬ݎߙ‬ ಽሺశభሻ ሺ‫ݔ‬ሻ െ ‫ܨ‬ ಽሺሻ ሺ‫ݔ‬ሻሿ (2.6)This in view of (1.1) and (1.2) reduces to, ୀ ௗ where ܸ ~ expሺ‫ݎ‬ሻ if and only if ܺ ଵ ~ ‫݉ݑܩ‬ሺߙሻ.The expression for the ݂݀ in Alzaid and Ahsanullah [2003, (1.5)] should have been as given here in (1.2).Corollary 2.1., ݆ ൌ ‫ݏ‬ െ ‫ݎ‬ െ 1, ‫ݏ‬ െ ‫ݎ‬ ; 1 ‫ݎ‬ ൏ ‫ݏ‬ (2.8) where ܻ ௦ି:௦ିଵ is the ሺ‫ݏ‬ െ ݆ሻ ௧ order statistic from a sample of size ሺ‫ݏ‬ െ 1ሻ from powሺߙሻ distribution and is independent of ܺ ሺ௦ሻ if and only if ܺ ଵ ~ Weiሺߙሻ.Proof.Here the product ܺ ሺ௦ሻ • ܻ ௦ି:௦ିଵ in (2.8) is called random contraction of ܺ ሺ௦ሻ (Beutner and Kamps, 2008).