On the Asymptotic Behaviour of Extremes and Near Maxima of Random Observations from the General Error Distributions

Abstract

As the name suggests, the family of general error distributions has been used to model nonnormal errors in a variety of situations. In this article we show that the asymptotic distribution of linearly normalized partial maxima of random observations from the general error distributions is Gumbel when the parameter of these distributions lies in the interval (0, 1). Our result fills a gap in the literature. We also establish the corresponding density convergence, obtain an asymptotic distribution of the partial maxima under power normalization, and state and prove a strong law. We also study the asymptotic behaviour of observations near the partial maxima and the sum of such observations.On the asymptotic behaviour of extremes and near maxima 529 distribution function (DF) of the GED. Here '∼' means 'asymptotically equal to' as x → ∞. One can see that the tail of the GED is asymptotically equal to the Weibullian tail.Nelson (1991) developed a market volatility model with the GED as the underlying distribution, instead of a normal distribution. Let ξ t , t ≥ 1, denote the prediction error at time t, and let σ 2 t be the variance of ξ t , given the information up to time t. Nelson introduced the model ξ t = σ t Z t , t ≥ 1, where {Z t , t ≥ 1} is a sequence of independent and identically distributed (i.i.d.) random variables (RVs) having the GED as the common distribution. For a data set on daily returns from CRSP (Center for Research in Security Prices-US stock market), over the period from July 1962 to December 1987, it has been shown that a GED with v = 1.58 (thicker than a normal tail) is a fairly good fit. As a result, one may observe that the conditional distribution of the prediction error ξ t is a GED with the same v.Do and Vetterli (2002) used the GED to model the distribution of wavelets. For experimental results from 640 different data sets on texture images, it was found that GEDs with v (β in their paper) ranging from 0.5 to 1 fit quite well.If {X n , n ≥ 1} denotes an i.i.d. sequence of RVs defined over a common probability space, with the GED as the common DF, certain characteristics of interest are the asymptotic behaviour of the partial sums, partial maxima, etc. Since all the moments exist, one can trivially see that the central limit theorem and the strong law of large numbers hold.In this paper we discuss the behaviour of maximal errors, as it is equally important. Given a sequence {X n , n ≥ 1} of i.i.d. RVs having the GED as the common distribution, define M n = max{X 1 , X 2 , . . . , X n }, n ≥ 1. If there exist sequences {a n , n ≥ 1} of positive constants and {b n , n ≥ 1} of real constants such that ((M n −b n )/a n ) converges weakly to a nondegenerate RV Y , then it is well known that Y is either Fréchet or Weibull or Gumbel (see, for example, Galambos (1978)). Peng et al. (2009) showed that Y is Gumbel whenever the parameter v of the GED is greater than 1, and Peng et al. (2010) studied the associated rate of convergence. When the parameter v of the underlying GED belongs to (0, 1), we ...

Uniform convergence rates of skew-normal extremes

“…Gasull et al (2015b) illustrated another application of the Lambert W function to decide on normalizing constants for gamma and other Weibull-like distributions. For other work related to convergence rates of distributions of normalized order statistics, see Liao and Peng (2012) for the log-normal distribution, and Peng et al (2010) and Vasudeva et al (2014) for the general error distribution.…”