Linear Estimation of the Scale Parameter of the First Asymptotic Distribution of Extreme Values

1973 ◽  
Vol R-22 (5) ◽  
pp. 259-264 ◽  
Author(s):  
J. William Shelnutt ◽  
Albert H. Moore ◽  
H. Leon Harter
Keyword(s):  
Asymptotic Distribution ◽  
Scale Parameter ◽  
Extreme Values ◽  
Linear Estimation

Limit distributions for compounded sums of extreme order statistics

1992 ◽  
Vol 29 (03) ◽  
pp. 557-574 ◽  
Author(s):  
Jan Beirlant ◽  
Jozef L. Teugels
Keyword(s):  
Order Statistics ◽  
Asymptotic Distribution ◽  
Counting Process ◽  
Extreme Values ◽  
Random Variables ◽  
Limit Distributions ◽  
Extreme Order Statistics ◽  
Renewal Counting Process ◽  
Independent Identically Distributed

Let X (1) ≦ X (2) ≦ ·· ·≦ X (N(t)) be the order statistics of the first N(t) elements from a sequence of independent identically distributed random variables, where {N(t); t ≧ 0} is a renewal counting process independent of the sequence of X's. We give a complete description of the asymptotic distribution of sums made from the top kt extreme values, for any sequence kt such that kt → ∞, kt /t → 0 as t → ∞. We discuss applications to reinsurance policies based on large claims.

Extreme values of phase-type and mixed random variables with parallel-processing examples

1999 ◽  
Vol 36 (01) ◽  
pp. 194-210 ◽  
Author(s):  
Sungyeol Kang ◽  
Richard F. Serfozo
Keyword(s):  
Parallel Processing ◽  
Asymptotic Distribution ◽  
Extreme Values ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Independent Random Variables ◽  
Phase Type

A basic issue in extreme value theory is the characterization of the asymptotic distribution of the maximum of a number of random variables as the number tends to infinity. We address this issue in several settings. For independent identically distributed random variables where the distribution is a mixture, we show that the convergence of their maxima is determined by one of the distributions in the mixture that has a dominant tail. We use this result to characterize the asymptotic distribution of maxima associated with mixtures of convolutions of Erlang distributions and of normal distributions. Normalizing constants and bounds on the rates of convergence are also established. The next result is that the distribution of the maxima of independent random variables with phase type distributions converges to the Gumbel extreme-value distribution. These results are applied to describe completion times for jobs consisting of the parallel-processing of tasks represented by Markovian PERT networks or task-graphs. In these contexts, which arise in manufacturing and computer systems, the job completion time is the maximum of the task times and the number of tasks is fairly large. We also consider maxima of dependent random variables for which distributions are selected by an ergodic random environment process that may depend on the variables. We show under certain conditions that their distributions may converge to one of the three classical extreme-value distributions. This applies to parallel-processing where the subtasks are selected by a Markov chain.

scholarly journals Impact of Dependence on Parameter Estimates of Autoregressive Process with Gumbel Distributed Innovation

MATEMATIKA ◽  
2018 ◽  
Vol 34 (2) ◽  
pp. 365-380
Author(s):  
Sunday Samuel Bako ◽  
Mohd Bakri Adam ◽  
Anwar Fitrianto
Keyword(s):  
Scale Parameter ◽  
Extreme Values ◽  
Real Life ◽  
Location Parameter ◽  
Autoregressive Process ◽  
Parameter Estimates ◽  
Distributed Data ◽  
Original Series ◽  
Real Life Data ◽  
Ar Process

Recent studies have shown that independent identical distributed Gaussian random variables is not suitable for modelling extreme values observed during extremal events. However, many real life data on extreme values are dependent and stationary rather than the conventional independent identically distributed data. We propose a stationary autoregressive (AR) process with Gumbel distributed innovation and characterise the short-term dependence among maxima of an (AR) process over a range of sample sizes with varying degrees of dependence. We estimate the maximum likelihood of the parameters of the Gumbel AR process and its residuals, and evaluate the performance of the parameter estimates. The AR process is fitted to the Gumbel-generalised Pareto (GPD) distribution and we evaluate the performance of the parameter estimates fitted to the cluster maxima and the original series. Ignoring the effect of dependence leads to overestimation of the location parameter of the Gumbel-AR (1) process. The estimate of the location parameter of the AR process using the residuals gives a better estimate. Estimate of the scale parameter perform marginally better for the original series than the residual estimate. The degree of clustering increases as dependence is enhance for the AR process. The Gumbel-AR(1) fitted to the Gumbel-GPD shows that the estimates of the scale and shape parameters fitted to the cluster maxima perform better as sample size increases, however, ignoring the effect of dependence lead to an underestimation of the parameter estimates of the scale parameter. The shape parameter of the original series gives a superior estimate compare to the threshold excesses fitted to the Gumbel-GPD.

Limit distributions for compounded sums of extreme order statistics

1992 ◽  
Vol 29 (3) ◽  
pp. 557-574 ◽  
Author(s):  
Jan Beirlant ◽  
Jozef L. Teugels
Keyword(s):  
Order Statistics ◽  
Asymptotic Distribution ◽  
Counting Process ◽  
Extreme Values ◽  
Random Variables ◽  
Limit Distributions ◽  
Extreme Order Statistics ◽  
Renewal Counting Process ◽  
Independent Identically Distributed

LetX(1)≦X(2)≦ ·· ·≦X(N(t))be the order statistics of the firstN(t) elements from a sequence of independent identically distributed random variables, where {N(t);t≧ 0} is a renewal counting process independent of the sequence ofX's. We give a complete description of the asymptotic distribution of sums made from the topktextreme values, for any sequencektsuch thatkt→ ∞,kt/t→ 0 ast→ ∞. We discuss applications to reinsurance policies based on large claims.

Extreme values of phase-type and mixed random variables with parallel-processing examples

1999 ◽  
Vol 36 (1) ◽  
pp. 194-210 ◽  
Author(s):  
Sungyeol Kang ◽  
Richard F. Serfozo
Keyword(s):  
Parallel Processing ◽  
Asymptotic Distribution ◽  
Extreme Values ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Independent Random Variables ◽  
Phase Type

A basic issue in extreme value theory is the characterization of the asymptotic distribution of the maximum of a number of random variables as the number tends to infinity. We address this issue in several settings. For independent identically distributed random variables where the distribution is a mixture, we show that the convergence of their maxima is determined by one of the distributions in the mixture that has a dominant tail. We use this result to characterize the asymptotic distribution of maxima associated with mixtures of convolutions of Erlang distributions and of normal distributions. Normalizing constants and bounds on the rates of convergence are also established. The next result is that the distribution of the maxima of independent random variables with phase type distributions converges to the Gumbel extreme-value distribution. These results are applied to describe completion times for jobs consisting of the parallel-processing of tasks represented by Markovian PERT networks or task-graphs. In these contexts, which arise in manufacturing and computer systems, the job completion time is the maximum of the task times and the number of tasks is fairly large. We also consider maxima of dependent random variables for which distributions are selected by an ergodic random environment process that may depend on the variables. We show under certain conditions that their distributions may converge to one of the three classical extreme-value distributions. This applies to parallel-processing where the subtasks are selected by a Markov chain.

scholarly journals Seismicity assessment for the different regions in and around Turkey based on instrumental data: Gumbel first asymptotic distribution and Gutenberg-Richter cumulative frequency law

2008 ◽  
Vol 8 (1) ◽  
pp. 109-122 ◽  
Author(s):  
Y. Bayrak ◽  
S. Öztürk ◽  
G. Ch. Koravos ◽  
G. A. Leventakis ◽  
T. M. Tsapanos
Keyword(s):  
Asymptotic Distribution ◽  
Goodness Of Fit ◽  
Extreme Values ◽  
Data Set ◽  
B Values ◽  
Whole Process ◽  
The North ◽  
Aegean Arc ◽  
Seismic Regions ◽  
Frequency Magnitude

Abstract. The evaluation of the seismicity in 24 seismic regions, in which Turkey and adjacent areas divided, is carried out. For this purpose two methods are adopted. The first is the "whole process" which follows the Gutenberg and Richter distribution frequency-magnitude law, while the second one is the "part process" which is well known as the theory of extreme values. This theory was developed by Gumbel in order to solve many geophysical problems. The first asymptotic distribution of extremes was used in the present study. The advantage of the method is that it does not required analysis of the whole data set. It uses, instead, the sequence of earthquakes with the largest magnitudes in a set of predetermined equal-time intervals. The parameters a and b were estimated from both methods. For the goodness of fit, to the Gutenberg-Richter frequency-magnitude law, the maximum likelihood approach is applied. The b-values calculated from Gutenberg and Richter frequency-magnitude law, reveal a better fit to the tectonic environment of the 24 seismic regions of Turkey and its surroundings examining in this study. On the other hand b-values evaluated from Gumbel's first distribution, do not adjust to the particular tectonics of the 24 seismic regions. The modal value a1/b adopted from Gutenberg-Richter for the 24 seismic regions were calculated, as well. An effort made to correlate the tectonics of the area with the spatial distribution of the various computed seismic parameter, while maps were produced for this purpose. These maps provide a detail image of seismicity and local tectonics for the whole investigated area. The results showed that the Aegean arc and the North Anatolian fault zone ranked among to the first positions between the 24 seismic regions researched.

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