EXTREMES: A CONTINUOUS-TIME PERSPECTIVE

2005 ◽  
Vol 19 (3) ◽  
pp. 289-308 ◽  
Author(s):  
Iddo Eliazar
Keyword(s):  
Order Statistics ◽  
Continuous Time ◽  
Time Perspective ◽  
Record Values ◽  
Extreme Value ◽  
Extreme Value Statistics ◽  
Record Times ◽  
Multidimensional Distribution ◽  
Simple Simulation ◽  
Entire Sequence

We consider a generic continuous-time system in which events of random magnitudes occur stochastically and study the system's extreme-value statistics. An event is described by a pair (t,x) of coordinates, wheretis the time at which the event took place andxis the magnitude of the event. The stochastic occurrence of the events is assumed to be governed by a Poisson point process.We study various issues regarding the system's extreme-value statistics, including (i) the distribution of the largest-magnitude event, the distribution of thenth “runner-up” event, and the multidimensional distribution of the “topn” extreme events, (ii) the internal hierarchy of the extreme-value events—how large are their magnitudes when measured relative to each other, and (iii) the occurrence of record times and record values. Furthermore, we unveil a hidden Poissonian structure underlying the system's sequence of order statistics (the largest-magnitude event, the second largest event, etc.). This structure provides us with a markedly simple simulation algorithm for the entire sequence of order statistics.

On discrete time extremal processes

1974 ◽  
Vol 6 (3) ◽  
pp. 580-592 ◽  
Author(s):  
R. W. Shorrock
Keyword(s):  
Population Growth ◽  
Discrete Time ◽  
Continuous Time ◽  
Record Values ◽  
Record Times ◽  
State Spaces ◽  
Upper Record Values ◽  
Extremal Processes ◽  
Upper Record ◽  
Continuous Time Processes

Upper record values and times and inter-record times are studied in their rôles as embedded structures in discrete time extremal processes. Various continuous time approximations to the discrete-time processes are analysed, especially as processes over their state spaces. Discrete time processes, suitably normalized after crossing a threshold T, are shown to converge to limiting continuous time processes as T → ∞ under suitable assumptions on the underlying CDF F, for example, when 1 — F varies regularly at ∞, and more generally. Discrete time extremal processes viewed as processes over their state spaces are noted to have an interesting interpretation in terms of processes of population growth.

On discrete time extremal processes

1974 ◽  
Vol 6 (03) ◽  
pp. 580-592 ◽  
Author(s):  
R. W. Shorrock
Keyword(s):  
Population Growth ◽  
Discrete Time ◽  
Continuous Time ◽  
Record Values ◽  
Record Times ◽  
State Spaces ◽  
Upper Record Values ◽  
Extremal Processes ◽  
Upper Record ◽  
Continuous Time Processes

Upper record values and times and inter-record times are studied in their rôles as embedded structures in discrete time extremal processes. Various continuous time approximations to the discrete-time processes are analysed, especially as processes over their state spaces. Discrete time processes, suitably normalized after crossing a threshold T, are shown to converge to limiting continuous time processes as T → ∞ under suitable assumptions on the underlying CDF F, for example, when 1 — F varies regularly at ∞, and more generally. Discrete time extremal processes viewed as processes over their state spaces are noted to have an interesting interpretation in terms of processes of population growth.

scholarly journals Interval prediction of order statistics based on records by employing inter-record times

2016 ◽  
Vol 13 (1) ◽  
Author(s):  
Morteza Amini ◽  
Seyed MirMostafaee2
Keyword(s):  
Order Statistics ◽  
Exponential Distribution ◽  
Record Values ◽  
Record Times ◽  
Underlying Distribution ◽  
Additional Information ◽  
Interval Prediction ◽  
Two Parameter

In this note, we propose a parametric inferential procedure for predicting future order statistics, based on record values, which takes inter-record times into account. We utilize the additional information contained in inter-record times for predicting future order statistics on the basis of observed record values from an independent sample. The two parameter exponential distribution is assumed to be the underlying distribution.

Stochastic comparison of parallel systems with Pareto components

2021 ◽  
pp. 1-13
Author(s):  
Sameen Naqvi ◽  
Weiyong Ding ◽  
Peng Zhao
Keyword(s):  
Order Statistics ◽  
Extreme Value Theory ◽  
Pareto Distribution ◽  
Parallel Systems ◽  
Value Theory ◽  
Extreme Value ◽  
Stochastic Comparison ◽  
Shape Parameters ◽  
Scale Parameters ◽  
Dispersive Ordering

Abstract Pareto distribution is an important distribution in extreme value theory. In this paper, we consider parallel systems with Pareto components and study the effect of heterogeneity on skewness of such systems. It is shown that, when the lifetimes of components have different shape parameters, the parallel system with heterogeneous Pareto component lifetimes is more skewed than the system with independent and identically distributed Pareto components. However, for the case when the lifetimes of components have different scale parameters, the result gets reversed in the sense of star ordering. We also establish the relation between star ordering and dispersive ordering by extending the result of Deshpande and Kochar [(1983). Dispersive ordering is the same as tail ordering. Advances in Applied Probability 15(3): 686–687] from support $(0, \infty )$ to general supports $(a, \infty )$ , $a > 0$ . As a consequence, we obtain some new results on dispersion of order statistics from heterogeneous Pareto samples with respect to dispersive ordering.

scholarly journals Some Characterizations of the Distribution of the Condition Number of a Complex Gaussian Matrix

2020 ◽  
Vol 8 (1) ◽  
pp. 22-35
Author(s):  
M. Shakil ◽  
M. Ahsanullah
Keyword(s):  
Order Statistics ◽  
Condition Number ◽  
Record Values ◽  
Distributional Properties ◽  
Upper Record Values ◽  
Upper Record

AbstractThe objective of this paper is to characterize the distribution of the condition number of a complex Gaussian matrix. Several new distributional properties of the distribution of the condition number of a complex Gaussian matrix are given. Based on such distributional properties, some characterizations of the distribution are given by truncated moment, order statistics and upper record values.

On the spectral shape of SEP events: An extreme value statistics approach

2013 ◽  
Author(s):  
M. Laurenza ◽  
G. Consolini ◽  
M. Storini ◽  
A. Damiani
Keyword(s):  
Extreme Value ◽  
Spectral Shape ◽  
Extreme Value Statistics
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