On record values and record times

1972 ◽  
Vol 9 (2) ◽  
pp. 316-326 ◽  
Author(s):  
R. W. Shorrock
Keyword(s):  
Asymptotic Behaviour ◽  
Point Processes ◽  
Independent Increment ◽  
Record Values ◽  
Record Times ◽  
Record Time ◽  
Special Cases ◽  
Record Value

A correspondence between record values and independent increment point processes is established. The asymptotic behaviour of record value sequences is studied, and results on the asymptotic behaviour of record times (for continuous F) are obtained as special cases. The joint law of the kth record value and the kth record time is also derived.

On record values and record times

1972 ◽  
Vol 9 (02) ◽  
pp. 316-326 ◽  
Author(s):  
R. W. Shorrock
Keyword(s):  
Asymptotic Behaviour ◽  
Point Processes ◽  
Independent Increment ◽  
Record Values ◽  
Record Times ◽  
Record Time ◽  
Special Cases ◽  
Record Value

A correspondence between record values and independent increment point processes is established. The asymptotic behaviour of record value sequences is studied, and results on the asymptotic behaviour of record times (for continuous F) are obtained as special cases. The joint law of the kth record value and the kth record time is also derived.

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>

Unification of Software Reliability Models by Self-Exciting Point Processes

1997 ◽  
Vol 29 (2) ◽  
pp. 337-352 ◽  
Author(s):  
Yiping Chen ◽  
Nozer D. Singpurwalla
Keyword(s):  
Software Reliability ◽  
Point Processes ◽  
Computer Software ◽  
Probability Models ◽  
Reliability Models ◽  
Common Structure ◽  
Software Reliability Models ◽  
Special Cases ◽  
Active Debate ◽  
The Subject

Assessing the reliability of computer software has been an active area of research in computer science for the past twenty years. To date, well over a hundred probability models for software reliability have been proposed. These models have been motivated by seemingly unrelated arguments and have been the subject of active debate and discussion. In the meantime, the search for an ideal model continues to be pursued. The purpose of this paper is to point out that practically all the proposed models for software reliability are special cases of self-exciting point processes. This perspective unifies the very diverse approaches to modeling reliability growth and provides a common structure under which problems of software reliability can be discussed.

Asymptotic analysis for affine point processes with large initial intensity

2018 ◽  
Vol 17 (01) ◽  
pp. 117-143
Author(s):  
Nian Yao ◽  
Mingqing Xiao
Keyword(s):  
Asymptotic Behavior ◽  
Point Processes ◽  
Law Of Large Numbers ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
General Structure ◽  
Large Numbers ◽  
Special Cases ◽  
Classical Models ◽  
Initial Intensity

In this paper, we consider a generalized stochastic model associated with affine point processes based on several classical models. In particular, we study the asymptotic behavior of the process when the initial intensity is large, i.e. the intensity of arriving events observed initially is considerably larger, which appears in many real applications. For our generalized model, we establish (i) the large deviation principle; (ii) the corresponding functional law of large numbers; (iii) the corresponding central limit theorem, that reflect the fundamentals of the process asymptotic behavior. Our obtained results include existing results as special cases with a more general structure.

Stochastic comparison of repairable systems by coupling

1998 ◽  
Vol 35 (2) ◽  
pp. 348-370 ◽  
Author(s):  
Günter Last ◽  
Ryszard Szekli
Keyword(s):  
Point Processes ◽  
Stochastic Comparison ◽  
Repairable Systems ◽  
Imperfect Repair ◽  
Coupling Methods ◽  
Comparison Results ◽  
Special Cases ◽  
Repair Models

Stochastic comparison results for replacement policies are shown in this paper using the formalism of point processes theory. At each failure moment a repair is allowed. It is performed with a random degree of repair including (as special cases) perfect, minimal and imperfect repair models. Results for such repairable systems with schemes of planned replacements are also shown. The results are obtained by coupling methods for point processes.

Embedding sequences of successive maxima in extremal processes, with applications

1987 ◽  
Vol 24 (4) ◽  
pp. 827-837 ◽  
Author(s):  
Rocco Ballerini ◽  
Sidney I. Resnick
Keyword(s):  
Population Size ◽  
Limit Theorems ◽  
Random Variables ◽  
Record Values ◽  
Record Times ◽  
Extremal Processes ◽  
Sample Record

Consequences of embedding sequences {Mn} in an extremal-F process are discussed where Mn represents the maximum of n independent (but not necessarily identically distributed) random variables. Various limit theorems are proved for the sample record rate, record times, inter-record times, and record values. These results are illustrated with applications to three particular record models: the Yang (1975) record model where population size increases geometrically, a record model where linear improvement is present, and a record model incorporating features of the previous two.

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.

scholarly journals Survival Analysis on Pedigrees: A Marked Point Process Model

2010 ◽  
Vol 40 (1) ◽  
pp. 35-64
Author(s):  
Angus S. Macdonald
Keyword(s):  
Survival Analysis ◽  
Process Model ◽  
Point Processes ◽  
Marked Point ◽  
Age At Onset ◽  
Parametric Models ◽  
Marked Point Processes ◽  
Inherited Disorders ◽  
Partial Model ◽  
Special Cases

AbstractRegulation of insurers' use of genetic information means actuaries are interested in age-at-onset of genetic disorders. Arjas & Haara (1984) suggested marked point processes (MPPs) as useful models for life history data with complex covariates. Age-at-onset distributions (or equivalently, hazard rates) in respect of inherited disorders are often estimated from pedigrees, which are life histories with unusually complex covariates, as well as strong dependencies induced by shared genes. Since Elston (1973) parametric models have often been used, conditioning the likelihood on known genotypes. However, a genotype identii ed by a presymptomatic genetic test is a form of internal covariate (Kalbfleisch & Prentice, 2002). We propose a very general MPP model of a pedigree, including presymptomatic genetic testing, (‘the full model’) and show under what circumstances the partial model leading to Elston's likelihood is valid. In practice, pedigrees are often ascertained retrospectively. Many such events can be modelled by augmenting the natural filtration of the MPP. We show that, except in simple special cases, the partial model is no longer valid, and the resulting likelihoods appear to be intractable. In particular, ascertainment interacts even with independent censoring so that likelihoods no longer factorize. For one simple special case — studies of sibships — we generalise a classical result to age-at-onset data. We conclude that the study of genetic conditions with variable age at onset gains insights from the underlying principles of survival analysis in their modern form, but that great care is needed in translating epidemiological studies into actuarial models.

scholarly journals On the number and sum of near-record observations

2005 ◽  
Vol 37 (03) ◽  
pp. 765-780 ◽  
Author(s):  
N. Balakrishnan ◽  
A.G. Pakes ◽  
A. Stepanov
Keyword(s):  
Distribution Function ◽  
Asymptotic Properties ◽  
Continuous Distribution ◽  
Random Variables ◽  
Continuous Distribution Function ◽  
Record Time ◽  
Record Value ◽  
Independent And Identically Distributed

Let X 1,X 2,… be a sequence of independent and identically distributed random variables with some continuous distribution function F. Let L(n) and X(n) denote the nth record time and the nth record value, respectively. We refer to the variables X i as near-nth-record observations if X i ∈(X(n)-a,X(n)], with a&gt;0, and L(n)&lt;i&lt;L(n+1). In this work we study asymptotic properties of the number of near-record observations. We also discuss sums of near-record observations.

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