Extremal processes and random measures

1975 ◽  
Vol 12 (2) ◽  
pp. 316-323 ◽  
Author(s):  
R. W. Shorrock
Keyword(s):  
Random Measure ◽  
Record Values ◽  
Poisson Processes ◽  
Two Dimensional ◽  
Random Measures ◽  
Record Times ◽  
Upper Record Values ◽  
Extremal Processes ◽  
Upper Record ◽  
Gamma Processes

Discrete time extremal processes with a continuous underlying c.d.f. are random measures which can be viewed as two-dimensional Poisson processes and this representation is used to obtain the conditional law of the sequence of states the process passes through (upper record values) given the sequence of holding times in states (inter-record times). In addition the Gamma processes (which lead to the Ferguson Dirichlet processes) and a random measure that arises in sampling from a biological population are discussed as two-dimensional Poisson processes.

Extremal processes and random measures

1975 ◽  
Vol 12 (02) ◽  
pp. 316-323
Author(s):  
R. W. Shorrock
Keyword(s):  
Random Measure ◽  
Record Values ◽  
Poisson Processes ◽  
Two Dimensional ◽  
Random Measures ◽  
Record Times ◽  
Upper Record Values ◽  
Extremal Processes ◽  
Upper Record ◽  
Gamma Processes

Discrete time extremal processes with a continuous underlying c.d.f. are random measures which can be viewed as two-dimensional Poisson processes and this representation is used to obtain the conditional law of the sequence of states the process passes through (upper record values) given the sequence of holding times in states (inter-record times). In addition the Gamma processes (which lead to the Ferguson Dirichlet processes) and a random measure that arises in sampling from a biological population are discussed as two-dimensional Poisson processes.

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.

Record values and inter-record times

1973 ◽  
Vol 10 (03) ◽  
pp. 543-555 ◽  
Author(s):  
R. W. Shorrock
Keyword(s):  
Road Traffic ◽  
Stochastic Order ◽  
Jump Process ◽  
Record Values ◽  
Asymptotic Results ◽  
Markov Jump ◽  
Record Times ◽  
Continuous State ◽  
Upper Record Values ◽  
Upper Record

First, asymptotic results for inter-record times when the CDF of the underlying IID process is not necessarily continuous are obtained, by a stochastic order argument, from known results for the continuous case. Then the asymptotic behaviour of the bivariate process of upper-record values and inter-record times is studied. Finally, assuming continuity of the underlying CDF, we derive the law of the process of total times spent in sets of states, viewing upper record values as states and inter-record times as times spent in a state, the process so viewed being a discrete time continuous state Markov jump process. The possible relevance of this result to single lane road traffic flow is indicated.

Closure and Monotonicity Properties of Nonhomogeneous Poisson Processes and Record Values

1988 ◽  
Vol 2 (4) ◽  
pp. 475-484 ◽  
Author(s):  
Ramesh C. Gupta ◽  
S.N.U.A. Kirmani
Keyword(s):  
Value Function ◽  
Record Values ◽  
Mean Value ◽  
Poisson Processes ◽  
Minimal Repair ◽  
Nonhomogeneous Poisson Processes ◽  
Upper Record Values ◽  
The Mean ◽  
Upper Record ◽  
Occurrence Times

Interconnections between occurrence times of nonhomogeneous Poisson processes, record values, minimal repair times, and the relevation transform are explained. A number of properties of the distributions of occurrence times and interoccurrence times of a nonhomogeneous Poisson process are proved when the mean-value function of the process is convex, starshaped, or superadditive. The same results hold for upper record values of independently identically distributed random variables from IFR, IFRA, and NBU distributions.

scholarly journals Simultaneous Joint Lower and Upper record values Probability Laws for Absolutely Continuous or Discrete Data

2021 ◽  
Vol 16 (4) ◽  
pp. 2923-2978
Author(s):  
Gane Samb Lo ◽  
◽  
Aladji Babacar Niang ◽  
Mohammad Ahsanullah
Keyword(s):  
Probability Density Function ◽  
Probability Space ◽  
Record Values ◽  
Independent Random Variables ◽  
Record Times ◽  
Absolutely Continuous ◽  
Upper Record Values ◽  
Upper Record ◽  
Complete Investigation ◽  
Probability Density Function Pdf

This paper investigates the probability density function (pdf) of the \((2n-1)\)-vector \((n\geq 1)\) of both lower and upper record values for a sequence of independent random variables with common \textit{pdf} \(f\) defined on the same probability space, provided that the lower and upper record times are finite up to \(n\). A lot is known about the lower or the upper record values when they are studied separately. When put together, the challenges are far complicated. The rare results in the literature still present some flaws. This paper begins a new and complete investigation with a few number of records: (n=2\) and \(n=3\). Lessons from these simple cases will allow addressing the general formulation of simultaneous joint lower-upper records.

Record values and inter-record times

1973 ◽  
Vol 10 (3) ◽  
pp. 543-555 ◽  
Author(s):  
R. W. Shorrock
Keyword(s):  
Road Traffic ◽  
Stochastic Order ◽  
Jump Process ◽  
Record Values ◽  
Asymptotic Results ◽  
Markov Jump ◽  
Record Times ◽  
Continuous State ◽  
Upper Record Values ◽  
Upper Record

First, asymptotic results for inter-record times when the CDF of the underlying IID process is not necessarily continuous are obtained, by a stochastic order argument, from known results for the continuous case. Then the asymptotic behaviour of the bivariate process of upper-record values and inter-record times is studied. Finally, assuming continuity of the underlying CDF, we derive the law of the process of total times spent in sets of states, viewing upper record values as states and inter-record times as times spent in a state, the process so viewed being a discrete time continuous state Markov jump process.The possible relevance of this result to single lane road traffic flow is indicated.

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.

scholarly journals Nonparametric Prediction Intervals of generalized order statistics from two independent sequences

2016 ◽  
Vol 4 (1) ◽  
pp. 36 ◽  
Author(s):  
Mostafa Mohie El-Din ◽  
Walid Emam
Keyword(s):  
Order Statistics ◽  
Record Values ◽  
Prediction Intervals ◽  
Generalized Order Statistics ◽  
Special Cases ◽  
Upper Record Values ◽  
Upper Record ◽  
Nonparametric Prediction ◽  
Nonparametric Prediction Intervals ◽  
Generalized Order

<p>This paper, discusses the problem of predicting future a generalized order statistic of an iid sequence sample was drawn from an arbitrary unknown distribution, based on observed also generalized order statistics from the same population. The coverage probabilities of these prediction intervals are exact and free of the parent distribution F(). Prediction formulas of ordinary order statistics and upper record values are extracted as special cases from the productive results. Finally, numerical computations on several models of ordered random variables are given to illustrate the proposed procedures.</p>

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