The probability that the largest observation is censored

1993 ◽  
Vol 30 (3) ◽  
pp. 602-615 ◽  
Author(s):  
R. A. Maller ◽  
S. Zhou
Keyword(s):  
Survival Time ◽  
Failure Time ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Consistent Estimator ◽  
Survival Times ◽  
Practical Difficulty ◽  
Independent Censoring ◽  
The Mean ◽  
Necessary And Sufficient

Suppose n possibly censored survival times are observed under an independent censoring model, in which the observed times are generated as the minimum of independent positive failure and censor random variables. A practical difficulty arises when the largest observation is censored since then the usual non-parametric estimator of the distribution of the survival time is improper. We calculate the probability that this occurs and give necessary and sufficient conditions for this probability to converge to 0 as n →∞. As an application, we show that if this probability is 0, asymptotically, then a consistent estimator for the mean failure time can be found. An almost sure version of the problem is also considered.

The probability that the largest observation is censored

1993 ◽  
Vol 30 (03) ◽  
pp. 602-615 ◽  
Author(s):  
R. A. Maller ◽  
S. Zhou
Keyword(s):  
Survival Time ◽  
Failure Time ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Consistent Estimator ◽  
Survival Times ◽  
Practical Difficulty ◽  
Independent Censoring ◽  
The Mean ◽  
Necessary And Sufficient

Suppose n possibly censored survival times are observed under an independent censoring model, in which the observed times are generated as the minimum of independent positive failure and censor random variables. A practical difficulty arises when the largest observation is censored since then the usual non-parametric estimator of the distribution of the survival time is improper. We calculate the probability that this occurs and give necessary and sufficient conditions for this probability to converge to 0 as n →∞. As an application, we show that if this probability is 0, asymptotically, then a consistent estimator for the mean failure time can be found. An almost sure version of the problem is also considered.

scholarly journals Two probability theorems and their application to some first passage problems

1964 ◽  
Vol 4 (2) ◽  
pp. 214-222 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Rate Of Convergence ◽  
Law Of Large Numbers ◽  
Sufficient Conditions ◽  
Random Variables ◽  
First Passage ◽  
Sums Of Random Variables ◽  
Large Numbers ◽  
The Mean ◽  
Image Position ◽  
Necessary And Sufficient

Let Xi, i = 1, 2, 3,··· be a sequence of independent and identically distributed random variables and write Sn = X1+X2+…+Xn. If the mean of Xi is finite and positive, we have Pr(Sn ≦ x) → 0 as n → ∞ for all x1 – ∞ < x < ∞ using the weak law of large numbers. It is our purpose in this paper to study the rate of convergence of Pr(Sn ≦ x) to zero. Necessary and sufficient conditions are established for the convergence of the two series where k is a non-negative integer, and where r > 0. These conditions are applied to some first passage problems for sums of random variables. The former is also used in correcting a queueing Theorem of Finch [4].

scholarly journals On the law of the iterated logarithm in the infinite variance case

1980 ◽  
Vol 30 (1) ◽  
pp. 5-14 ◽  
Author(s):  
R. A. Maller
Keyword(s):  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Infinite Variance ◽  
Iterated Logarithm ◽  
The Law ◽  
Necessary And Sufficient ◽  
Independent And Identically Distributed

AbstractThe main purpose of the paper is to give necessary and sufficient conditions for the almost sure boundedness of (Sn – αn)/B(n), where Sn = X1 + X2 + … + XmXi being independent and identically distributed random variables, and αnand B(n) being centering and norming constants. The conditions take the form of the convergence or divergence of a series of a geometric subsequence of the sequence P(Sn − αn > a B(n)), where a is a constant. The theorem is distinguished from previous similar results by the comparative weakness of the subsidiary conditions and the simplicity of the calculations. As an application, a law of the iterated logarithm general enough to include a result of Feller is derived.

A Note on Normal Attraction to a Stable Law

1971 ◽  
Vol 14 (3) ◽  
pp. 451-452
Author(s):  
M. V. Menon ◽  
V. Seshadri
Keyword(s):  
Distribution Function ◽  
Sufficient Conditions ◽  
Characteristic Exponent ◽  
Random Variables ◽  
Stable Law ◽  
Common Distribution ◽  
Normal Attraction ◽  
Common Distribution Function ◽  
The Common ◽  
Necessary And Sufficient

Let X1, X2, …, be a sequence of independent and identically distributed random variables, with the common distribution function F(x). The sequence is said to be normally attracted to a stable law V with characteristic exponent α, if for some an (converges in distribution to V). Necessary and sufficient conditions for normal attraction are known (cf [1, p. 181]).

Effect of Intravenous Iron in Experimental Traumatic Shock

1956 ◽  
Vol 186 (3) ◽  
pp. 554-556
Author(s):  
Donn L. Smith ◽  
Irvin I. Kibbey ◽  
Max E. Bierwagen ◽  
J. R. Cruse
Keyword(s):  
Heavy Metal ◽  
Survival Time ◽  
Metal Toxicity ◽  
Intravenous Iron ◽  
Traumatic Shock ◽  
Survival Times ◽  
Albino Rat ◽  
Liver Iron ◽  
Mean Survival Time ◽  
The Mean

Intravenous administration of colloidal saccharated iron oxide prior to intestinal traumatization in the albino rat resulted in a significant reduction of the mean survival time. Sodium gold thiosulfate and colloidal manganese hydroxide employed in the same manner did not significantly alter mean survival times. ACTH and cortisone did not modify the deleterious effects of iron in experimental traumatic shock. A decrease in soluble liver iron was observed when traumatization followed the injection of iron. It was concluded that the reduction of mean survival time in iron injected, traumatized animals was due to a specific action of iron and is not the result of generalized heavy metal toxicity.

Order statistics, Poisson processes and repairable systems

1976 ◽  
Vol 13 (03) ◽  
pp. 519-529 ◽  
Author(s):  
Douglas R. Miller
Keyword(s):  
Order Statistics ◽  
Point Processes ◽  
Failure Time ◽  
Sufficient Conditions ◽  
Renewal Processes ◽  
Repairable Systems ◽  
Homogeneous Poisson Process ◽  
Necessary And Sufficient ◽  
Non Homogeneous Poisson Process ◽  
Weibull Process

Necessary and sufficient conditions are presented under which the point processes equivalent to order statistics of n i.i.d. random variables or superpositions of n i.i.d. renewal processes converge to a non-degenerate limiting process as n approaches infinity. The limiting process must be one of three types of non-homogeneous Poisson process, one of which is the Weibull process. These point processes occur as failure-time models in the reliability theory of repairable systems.

A class of correlated cumulative shock models

1985 ◽  
Vol 17 (2) ◽  
pp. 347-366 ◽  
Author(s):  
Ushio Sumita ◽  
J. George Shanthikumar
Keyword(s):  
Failure Time ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Random Variables ◽  
System Failure ◽  
Shock Models ◽  
The Times ◽  
New Better Than Used ◽  
Better Than

In this paper we define and analyze a class of cumulative shock models associated with a bivariate sequence {Xn, Yn}∞n=0 of correlated random variables. The {Xn} denote the sizes of the shocks and the {Yn} denote the times between successive shocks. The system fails when the cumulative magnitude of the shocks exceeds a prespecified level z. Two models, depending on whether the size of the nth shock is correlated with the length of the interval since the last shock or with the length of the succeeding interval until the next shock, are considered. Various transform results and asymptotic properties of the system failure time are obtained. Further, sufficient conditions are established under which system failure time is new better than used, new better than used in expectation, and harmonic new better than used in expectation.

Mean and variance of vacancy for distribution of k-dimensional spheres within k-dimensional space

1984 ◽  
Vol 21 (4) ◽  
pp. 738-752 ◽  
Author(s):  
Peter Hall
Keyword(s):  
Dimensional Space ◽  
Sufficient Conditions ◽  
Unit Cube ◽  
Necessary And Sufficient Conditions ◽  
Dimensional Sphere ◽  
Asymptotic Formulae ◽  
Dimensional Unit ◽  
Mean And Variance ◽  
The Mean ◽  
Necessary And Sufficient

Let n points be distributed independently within a k-dimensional unit cube according to density f. At each point, construct a k-dimensional sphere of content an. Let V denote the vacancy, or ‘volume' not covered by the spheres. We derive asymptotic formulae for the mean and variance of V, as n → ∞and an → 0. The formulae separate naturally into three cases, corresponding to nan → 0, nan → a (0 < a <∞) and nan →∞, respectively. We apply the formulae to derive necessary and sufficient conditions for V/E(V) → 1 in L2.

Relationship Between Survival Rate and Survival-Time of Rabbits, Oryctolagus-Cuniculus (L), Challenged With Myxoma Virus

1995 ◽  
Vol 43 (3) ◽  
pp. 303 ◽  
Author(s):  
I Parer
Keyword(s):  
Survival Time ◽  
Small Groups ◽  
Regression Line ◽  
Survival Rates ◽  
Myxoma Virus ◽  
Survival Times ◽  
Mean Survival Time ◽  
The Past ◽  
Survival Percentage ◽  
The Mean

The mean survival times of small groups of rabbits challenged with myxoma virus have been used to estimate survival rates and to allocate virulence grades to field strains of myxoma virus. The slope of the regression Line relating survival percentage to mean survival time in days was shown to be less steep than has been previously estimated. This overestimation of the regression slope has, in the past, resulted in most field strains of myxoma virus being allocated to the Grade III level of virulence when allocation to Grade I would have been more appropriate.

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