Exact extreme value, product, and ratio distributions under non-standard assumptions

Exact distributions of order statistics from ln,p-symmetric sample distributions

“…As it is established in [20] for the case of order statistics of two jointly l ,p -symmetrically distributed random variables, the approach in [19] and others which is based upon the single-out decomposition of A (t), see Section 3.2, yields the same result as applying the geometric measure representation in [30] directly, see [24], and transforming the results into terms of skewed distributions, see [10]. In higher dimensional cases, analogous relations are not yet investigated.…”

“…For a visualization of the set A n k (t) for some low-dimensional cases, see [24,25]. Furthermore, A n k (t) is an open cone with vertex in t n.…”

“…Looking through the proof of Theorem 1 in [24] once again, there appear representations of the cdf Fn:n of the maximum statistic Xn:n in ln,p-symmetrically distributed populations not used so far, see Lemma 2.…”

“…As stated in [24], because of the exchangeability of the components of ln,p-symmetrically distributed random vectors, the additional parameter i has no numerical impact when evaluating the cdf Fn:n. Nevertheless, these alternative representations may be helpful in identifying a maximum distribution if particular skewed ln,psymmetric distributions occur in other contexts.…”

“…The consideration of order statistics in ln,p-symmetrically distributed populations started in [17] by obtaining the joint pdf of the vector of order statistics. A versatile study of di erent methods for deriving exact extreme value distributions in di erent ways can be found in [10] and [24][25][26]. More generally, [8] considers multivariate order statistics from dependent and nonidentically distributed random variables.…”