Tail Behavior and Limit Distribution of Maximum of Logarithmic General Error Distribution

Distributional expansion of maximum from logarithmic general error distribution

“…Let h n (x) = n log F v (a n x + b n ) + e −x with the norming constants a n and b n given by (5), and let β = min(1, v − 1). Then,…”

“…Note that (8) In order to prove the main results, we need a more precise distributional tail representation than that of (2) due to Liao et al (2014a). The result is stated as follows.…”

“…Let F v (x) denote the cumulative distribution function (cdf) of η ∼ logGED(v). By Proposition 2.1 in Liao et al(2014a), for large x we have the following distributional tail representation of logGED(v) with v > 1:…”

“…Similarly, the so-called logarithmic general error distribution (logGED) proposed by Liao et al (2014a) is a natural extension of the log-normal distribution. Let random variable ξ follow the standard GED(v) with pdf given by (1).…”

“…Tail behavior and extreme value distribution of logGED(v) were studied by Liao et al (2014a), where its applications are illustrated by fitting the rainfall data sets. In this short note, we focus on deriving the higher-order expansion of the distribution of partial maximum of logGED(v) under optimal choice of norming constants.…”