Classes of life distributions and renewal counting process

1994 ◽  
Vol 31 (4) ◽  
pp. 1110-1115 ◽  
Author(s):  
Yi-Hau Chen
Keyword(s):  
Exponential Distribution ◽  
Counting Process ◽  
Renewal Function ◽  
Life Distribution ◽  
Renewal Counting Process ◽  
Life Distributions

We prove that if the renewal function M(t) corresponding to a life distribution F is convex (concave) then F is NBU (NWU), and hence answer two questions posed by Shaked and Zhu (1992). Moreover, based-on the renewal function, some characterizations of the exponential distribution within certain classes of life distributions are given.

Classes of life distributions and renewal counting process

1994 ◽  
Vol 31 (04) ◽  
pp. 1110-1115 ◽  
Author(s):  
Yi-Hau Chen
Keyword(s):  
Exponential Distribution ◽  
Counting Process ◽  
Renewal Function ◽  
Life Distribution ◽  
Renewal Counting Process ◽  
Life Distributions

We prove that if the renewal function M(t) corresponding to a life distribution F is convex (concave) then F is NBU (NWU), and hence answer two questions posed by Shaked and Zhu (1992). Moreover, based-on the renewal function, some characterizations of the exponential distribution within certain classes of life distributions are given.

scholarly journals Renewal theory in two dimensions: asymptotic results

1974 ◽  
Vol 6 (3) ◽  
pp. 546-562 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
Joint Distribution ◽  
Counting Process ◽  
Renewal Theory ◽  
Two Dimensions ◽  
Renewal Processes ◽  
Unified Theory ◽  
Two Dimensional ◽  
Renewal Function ◽  
Asymptotic Results ◽  
Renewal Counting Process

In an earlier paper (Renewal theory in two dimensions: Basic results) the author developed a unified theory for the study of bivariate renewal processes. In contrast to this aforementioned work where explicit expressions were obtained, we develop some asymptotic results concerning the joint distribution of the bivariate renewal counting process (Nx(1), Ny(2)), the distribution of the two-dimensional renewal counting process Nx,y and the two-dimensional renewal function &Nx,y. A by-product of the investigation is the study of the distribution and moments of the minimum of two correlated normal random variables. A comprehensive bibliography on multi-dimensional renewal theory is also appended.

Characterizations of life distributions by moments of extremes and sample mean

1994 ◽  
Vol 31 (1) ◽  
pp. 148-155 ◽  
Author(s):  
Cai Jun
Keyword(s):  
Exponential Distribution ◽  
Random Variables ◽  
Random Vectors ◽  
Life Distribution ◽  
Sample Mean ◽  
Special Cases ◽  
Partial Orderings ◽  
Life Distributions

We consider two weaker partial orderings among non-negative random variables. We derive some characterizations of the equivalence of two life distributions under the weaker orderings. These results lead to the characterizations of the equality of two non-negative random vectors in distributions by moments of extremes and sample mean. As the application of these results, we get various characterizations of the exponential distribution among HNBUE and HNWUE life distribution classes. The main results of Bhattacharjee and Sethuraman (1990), and Basu and Kirmani (1986) are special cases of these more general results.

Families of life distributions characterized by two moments

1990 ◽  
Vol 27 (03) ◽  
pp. 720-725 ◽  
Author(s):  
Manish C. Bhattacharjee ◽  
Jayaram Sethuraman
Keyword(s):  
Exponential Distribution ◽  
Partial Ordering ◽  
Life Distribution ◽  
Partial Orderings ◽  
Life Distributions ◽  
The Moment

We consider several classical notions of partial orderings among life distributions which have been used to describe ageing properties and tail domination. We show that if a distribution G dominates another distribution F in one of these partial orderings introduced here, and if two moments of G agree with those of F, including the moment that describes this partial ordering, then G = F. This leads to a characterization of the exponential distribution among HNBUE and HNWUE life distribution classes, and thus extends the results of Basu and Bhattacharjee (1984) and rectifies an error in that paper.

Characterizations of life distributions by moments of extremes and sample mean

1994 ◽  
Vol 31 (01) ◽  
pp. 148-155 ◽  
Author(s):  
Cai Jun
Keyword(s):  
Exponential Distribution ◽  
Random Variables ◽  
Random Vectors ◽  
Life Distribution ◽  
Sample Mean ◽  
Special Cases ◽  
Partial Orderings ◽  
Life Distributions

We consider two weaker partial orderings among non-negative random variables. We derive some characterizations of the equivalence of two life distributions under the weaker orderings. These results lead to the characterizations of the equality of two non-negative random vectors in distributions by moments of extremes and sample mean. As the application of these results, we get various characterizations of the exponential distribution among HNBUE and HNWUE life distribution classes. The main results of Bhattacharjee and Sethuraman (1990), and Basu and Kirmani (1986) are special cases of these more general results.

Families of life distributions characterized by two moments

1990 ◽  
Vol 27 (3) ◽  
pp. 720-725 ◽  
Author(s):  
Manish C. Bhattacharjee ◽  
Jayaram Sethuraman
Keyword(s):  
Exponential Distribution ◽  
Partial Ordering ◽  
Life Distribution ◽  
Partial Orderings ◽  
Life Distributions ◽  
The Moment

We consider several classical notions of partial orderings among life distributions which have been used to describe ageing properties and tail domination. We show that if a distribution G dominates another distribution F in one of these partial orderings introduced here, and if two moments of G agree with those of F, including the moment that describes this partial ordering, then G = F. This leads to a characterization of the exponential distribution among HNBUE and HNWUE life distribution classes, and thus extends the results of Basu and Bhattacharjee (1984) and rectifies an error in that paper.

scholarly journals Renewal theory in two dimensions: asymptotic results

1974 ◽  
Vol 6 (03) ◽  
pp. 546-562 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
Joint Distribution ◽  
Counting Process ◽  
Renewal Theory ◽  
Two Dimensions ◽  
Renewal Processes ◽  
Unified Theory ◽  
Two Dimensional ◽  
Renewal Function ◽  
Asymptotic Results ◽  
Renewal Counting Process

In an earlier paper (Renewal theory in two dimensions: Basic results) the author developed a unified theory for the study of bivariate renewal processes. In contrast to this aforementioned work where explicit expressions were obtained, we develop some asymptotic results concerning the joint distribution of the bivariate renewal counting process (N x(1), N y(2)), the distribution of the two-dimensional renewal counting process N x,y and the two-dimensional renewal function &N x,y. A by-product of the investigation is the study of the distribution and moments of the minimum of two correlated normal random variables. A comprehensive bibliography on multi-dimensional renewal theory is also appended.

Minimal distributions in a stochastic partial ordering and bounds for Gaussian processes

1983 ◽  
Vol 20 (03) ◽  
pp. 529-536
Author(s):  
W. J. R. Eplett
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Partial Ordering ◽  
Boundary Crossing ◽  
Conditional Probabilities ◽  
Life Distribution ◽  
Life Distributions ◽  
Bounded Below ◽  
Crossing Probabilities

A natural requirement to impose upon the life distribution of a component is that after inspection at some randomly chosen time to check whether it is still functioning, its life distribution from the time of checking should be bounded below by some specified distribution which may be defined by external considerations. Furthermore, the life distribution should ideally be minimal in the partial ordering obtained from the conditional probabilities. We prove that these specifications provide an apparently new characterization of the DFRA class of life distributions with a corresponding result for IFRA distributions. These results may be transferred, using Slepian's lemma, to obtain bounds for the boundary crossing probabilities of a stationary Gaussian process.

Characterizing the Exponential Law Under Laplace Order Domination

1995 ◽  
Vol 45 (3-4) ◽  
pp. 171-178 ◽  
Author(s):  
Murari Mitra ◽  
Sujit K. Basu ◽  
M. C. Bhattacharjee
Keyword(s):  
Exponential Distribution ◽  
Reliability Theory ◽  
Subject Classification ◽  
Exponential Law ◽  
Life Distributions

Interesting characterizations of the exponential distribution have been obtained in classes of life distributions important in reliability theory. The results strengthen some of the analogous conclusions already existing in the literature. AMS (1991) Subject Classification No. Primary 62NOS: Secondaey 90825. 60F99.

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