Identifiability problems in coherent systems

Given a coherent system, let Z be the age of the machine at breakdown, and I the set of parts failed by time Z. Assume that the component lifetimes are independent. Assume further that the lifetime distributions are mutually absolutely continuous and that each possesses a single positive atom at the common essential infimum. We prove that the joint distribution of (Z, I) identifies the lifetime distribution of each part if and only if there is at most one component belonging to all cut sets. If we relax the mutual absolute continuity assumption by allowing isolated intervals of constancy, then a necessary and sufficient condition for identifiability is that no two parts be in parallel.

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A characterization of the exponential distribution

Journal of Applied Probability ◽

10.1017/s0021900200095164 ◽

1972 ◽

Vol 9 (02) ◽

pp. 457-461 ◽

Cited By ~ 1

Author(s):

M. Ahsanullah ◽

M. Rahman

Keyword(s):

Probability Distribution ◽

Order Statistics ◽

Lebesgue Measure ◽

Random Variable ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Positive Random Variable ◽

Absolutely Continuous ◽

Necessary And Sufficient

A necessary and sufficient condition based on order statistics that a positive random variable having an absolutely continuous probability distribution (with respect to Lebesgue measure) will be exponential is given.

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A TIME-DEPENDENT PÓLYA URN WITH MULTIPLE DRAWINGS

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964819000093 ◽

2019 ◽

Vol 34 (4) ◽

pp. 469-483

Author(s):

May-Ru Chen

Keyword(s):

Random Variable ◽

Time Dependent ◽

Necessary And Sufficient Condition ◽

Urn Model ◽

Absolutely Continuous ◽

Pólya Urn ◽

Polya Urn ◽

Generalized Pólya Urn ◽

Necessary And Sufficient ◽

Positive Integers

In this paper, we consider a generalized Pólya urn model with multiple drawings and time-dependent reinforcements. Suppose an urn initially contains w white and r red balls. At the nth action, m balls are drawn at random from the urn, say k white and m−k red balls, and then replaced in the urn along with cnk white and cn(m − k) red balls, where {cn} is a given sequence of positive integers. Repeat the above procedure ad infinitum. Let Xn be the proportion of the white balls in the urn after the nth action. We first show that Xn converges almost surely to a random variable X. Next, we give a necessary and sufficient condition for X to have a Bernoulli distribution with parameter w/(w + r). Finally, we prove that X is absolutely continuous if {cn} is bounded.

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A non-uniform rate of convergence in a local limit theorem

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100055171 ◽

1978 ◽

Vol 84 (2) ◽

pp. 351-359 ◽

Cited By ~ 4

Author(s):

Sujit K. Basu

Keyword(s):

Limit Theorem ◽

Rate Of Convergence ◽

Local Limit Theorem ◽

Random Variables ◽

Local Limit ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Standard Normal ◽

The Common ◽

Necessary And Sufficient

AbstractLet {Xn} be a sequence of iid random variables. If the common charac-teristic function is absolutely integrable in mth power for some integer m ≥ 1, then Zn = n−½(X1 + … + Xn) has a pdf fn for all n ≥ m. Here we give a necessary and sufficient condition for sup as n. → ∞, where φ (x) is the standard normal pdf and M(x) is a non-decreasing function of x ≥ 0 such that M(0) > 0 and M(x)/xδ is non-increasing for 0 < δ ≤ 1.

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A Novel Necessary and Sufficient Condition for the Positivity of a Binary Quartic Form

Journal of Mathematics ◽

10.1155/2021/2339746 ◽

2021 ◽

Vol 2021 ◽

pp. 1-7

Author(s):

Yang Guo

Keyword(s):

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Pencil Of Conics ◽

Quartic Form ◽

The Common ◽

Necessary And Sufficient

In this paper, by considering the common points of two conics instead of the roots of the binary quartic form, we propose a novel necessary and sufficient condition for the positivity of a binary quartic form using the theory of the pencil of conics. First, we show the degenerate members of the pencil of conics according to the distinct natures of the common points of two base conics. Then, the inequalities about the parameters of the degenerate members are obtained according to the properties of the degenerate conics. Last, from the inequalities, we derive a novel criterion for determining the positivity of a binary quartic form without the discriminant.

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State-Transition Computation Models and Program Correctness Thereon

Journal of Advanced Computational Intelligence and Intelligent Informatics ◽

10.20965/jaciii.2007.p1250 ◽

2007 ◽

Vol 11 (10) ◽

pp. 1250-1261 ◽

Cited By ~ 2

Author(s):

Kiyoshi Akama ◽

◽

Ekawit Nantajeewarawat ◽

Keyword(s):

General Theory ◽

State Transition ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Program Correctness ◽

Common Framework ◽

The Common ◽

Correct Program ◽

Necessary And Sufficient ◽

Mapping Concept

The common framework for formalizing state-transition computation models we present is based on a general theory for studying the interrelationship of specifications, programs, computation, and program correctness. We establish a necessary and sufficient condition for program correctness for this class of computation models and demonstrate framework application by formalizing, as its instances, two concrete examples of state-transition computation models – NAT and D-rule. We compare their correct-program spaces by introducing the embedding mapping concept.

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A necessary and sufficient condition for the strong consistency of a family of estimators of the common odds ratio

Canadian Journal of Statistics ◽

10.2307/3315446 ◽

1995 ◽

Vol 23 (2) ◽

pp. 215-225

Author(s):

Kai Fun Yu

Keyword(s):

Odds Ratio ◽

Strong Consistency ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Common Odds Ratio ◽

The Common ◽

Necessary And Sufficient

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On identifiability in the autopsy model of reliability theory

Journal of Applied Probability ◽

10.1017/s0021900200044661 ◽

1993 ◽

Vol 30 (04) ◽

pp. 913-930 ◽

Cited By ~ 1

Author(s):

Robin Antoine ◽

Hani Doss ◽

Myles Hollander

Keyword(s):

Failure Time ◽

Sufficient Conditions ◽

Necessary Condition ◽

Coherent System ◽

System Failure ◽

Absolutely Continuous ◽

Failure Times ◽

Life Distributions ◽

Parametric Families ◽

Necessary And Sufficient

A coherent system is observed until it fails. At the instant of system failure, the set of failed components and the failure time of the system are noted. The failure times of the components are not known. We consider whether the component life distributions can be determined from the distributions of the observed data. Meilijson (1981) gave a condition on the structure of the system that was sufficient for the identifiability of the component distributions, under the assumption that the component life distributions are continuous and have common essential extrema. Nowik (1990) gave necessary and sufficient conditions for identifiability under the more restrictive condition that the component distributions have atoms at their common essential infimum and are mutually absolutely continuous. We give a necessary condition for identifiability, which we show to be equivalent to Nowik's condition, under the assumption that the distributions are continuous and strictly increasing. We derive a sufficient condition for identifiability, more general than Meilijson's, for the case in which the component distributions are assumed to be analytic. We also show that our necessary condition for identifiability is both necessary and sufficient when the component life distributions are assumed to belong to certain parametric families.

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A characterization of the exponential distribution

Journal of Applied Probability ◽

10.2307/3212817 ◽

1972 ◽

Vol 9 (2) ◽

pp. 457-461 ◽

Cited By ~ 8

Author(s):

M. Ahsanullah ◽

M. Rahman

Keyword(s):

Probability Distribution ◽

Order Statistics ◽

Lebesgue Measure ◽

Random Variable ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Positive Random Variable ◽

Absolutely Continuous ◽

Necessary And Sufficient

A necessary and sufficient condition based on order statistics that a positive random variable having an absolutely continuous probability distribution (with respect to Lebesgue measure) will be exponential is given.

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A remark on asymptotic regularity and fixed point property

Filomat ◽

10.2298/fil1914665b ◽

2019 ◽

Vol 33 (14) ◽

pp. 4665-4671

Author(s):

Ravindra Bisht

Keyword(s):

Fixed Point ◽

Common Fixed Point ◽

Complete Metric Space ◽

Fixed Point Property ◽

Point Property ◽

Necessary And Sufficient Condition ◽

Asymptotic Regularity ◽

The Common ◽

Necessary And Sufficient ◽

Commuting Mappings

In this paper, we show that orbital continuity of a pair of non-commuting mappings of a complete metric space is equivalent to fixed point property under the Proinov type condition. Furthermore, we establish a situation in which orbital continuity turns out to be a necessary and sufficient condition for the existence of a common fixed point of a pair of mappings yet the mappings are not necessarily continuous at the common fixed point.

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