Limit theorems for the number and sum of near-maxima for medium tails

Hu, Zhishui; Su, Chun

doi:10.1016/s0167-7152(03)00085-3

Cited by 22 publications

(10 citation statements)

References 7 publications

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“…Binomial limits for K + n n − k + 1 a have been obtained recently in Balakrishnan and Stepanov (2005). Other results, related to those, quickly described above, can be also found in Khmaladze et al (1997), Li andPakes (1998), Li (1999), Hu and Su (2003), and Balakrishnan and Stepanov (2004).…”

Section: Introductionsupporting

confidence: 69%

How Many Observations Fall in a Neighborhood of an Order Statistic?

Dembińska

Stepanov

Wesołowski

2007

Communications in Statistics - Theory and Methods

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Section: Introductionsupporting

confidence: 69%

How Many Observations Fall in a Neighborhood of an Order Statistic?

Dembińska

Stepanov

Wesołowski

2007

Communications in Statistics - Theory and Methods

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“…Pakes and Steutel [1] and Li [2] studied the behaviour of the number of observations near the sample maximum, K − (n, n, a). Generalizations of this result to the number of near k-extremes, K − (n, n − k, a), can be found in [3][4][5][6]. The latter authors also considered the joint behaviour of the rv's K − (n, n − k, a) and K + (n, n − k, b) and established that they are asymptotically independent under suitable conditions.…”

Section: Introductionmentioning

confidence: 81%

Asymptotic properties of numbers of observations near sample quantiles

Iliopoulos

Dembińska

2012

Statistics

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In this paper, we show that the proportions of observations falling in the left and right vicinity of the k n th order statistic converge in probability to some population quantities. We then prove that suitably normalized versions of these variables are jointly asymptotically normal under some conditions. A generalization of this result to the case of proportions of observations in the vicinity of two or more central order statistics is established next. Some concluding remarks and a potential statistical application of these results are finally made.

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“…Pakes (2000) and Hu and Su (2003) studied the asymptotic behaviour of S n (a) for distributions with r F = ∞. Pakes (2000) and Hu and Su (2003) studied the asymptotic behaviour of S n (a) for distributions with r F = ∞.…”

Section: Theorem 35 Let F Be the Df Of A Ged With Parametermentioning

confidence: 99%

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

2014

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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 ...

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Limit theorems for the number and sum of near-maxima for medium tails

Cited by 22 publications

References 7 publications

How Many Observations Fall in a Neighborhood of an Order Statistic?

How Many Observations Fall in a Neighborhood of an Order Statistic?

Asymptotic properties of numbers of observations near sample quantiles

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

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