scholarly journals On the Accuracy of the Exponential Approximation to Random Sums of Alternating Random Variables

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1917
Author(s):  
Irina Shevtsova ◽  
Mikhail Tselishchev
Keyword(s):  
Error Bounds ◽  
Random Variables ◽  
Distribution Functions ◽  
Exponential Approximation ◽  
Conditional Distributions ◽  
Random Sums ◽  
Conditional Expectations ◽  
First Moment ◽  
New Better Than Used ◽  
Better Than

Using the generalized stationary renewal distribution (also called the equilibrium transform) for arbitrary distributions with a finite non-zero first moment, we prove moment-type error-bounds in the Kantorovich distance for the exponential approximation to random sums of possibly dependent random variables with positive finite expectations, in particular, to geometric random sums, generalizing the previous results to alternating and dependent random summands. We also extend the notions of new better than used in expectation (NBUE) and new worse than used in expectation (NWUE) distributions to alternating random variables in terms of the corresponding distribution functions and provide a criteria in terms of conditional expectations similar to the classical one. As corollary, we provide simplified error-bounds in the case of NBUE/NWUE conditional distributions of random summands.

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.

Improving Poisson Approximations

1994 ◽  
Vol 8 (4) ◽  
pp. 449-462 ◽  
Author(s):  
Erol A. Peköz ◽  
Sheldon M. Ross
Keyword(s):  
Poisson Distribution ◽  
Error Bounds ◽  
Random Variables ◽  
Urn Models ◽  
Matching Problem ◽  
Bernoulli Convolutions ◽  
Poisson Approximations ◽  
Better Than

Let X1,…, Xn, be indicator random variables, and set We present a method for estimating the distribution of W in settings where W has an approximately Poisson distribution. Our method is shown to yield estimates significantly better than straight Poisson estimates when applied to Bernoulli convolutions, urn models, the circular k of n: F system, and a matching problem. Error bounds are given.

scholarly journals Generalized Variability Orderings among Nonnegative Fuzzy Random Variables

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
S. Ramasubramanian ◽  
P. Mahendran
Keyword(s):  
Distribution Function ◽  
Integral Inequality ◽  
Random Variables ◽  
Fuzzy Random Variables ◽  
Equivalent Conditions ◽  
New Better Than Used ◽  
Better Than ◽  
Fuzzy Random

The variability ordering for more and less variables of fuzzy random variables in terms of its distribution function is defined. A property of new better than used in expectation (NBUE) and new worse than used in expectation (NWUE) is derived as an application to the variability ordering of fuzzy random variables. The concept of generalized variability orderings of nonnegative fuzzy random variables representing lifetime of components is introduced. The<Pdomination is a generalized variability ordering. We proposed an integral inequality to the case of fuzzy random variables using<Pordering. The results included equivalent conditions which justify the generalized variability orderings.

A class of correlated cumulative shock models

1985 ◽  
Vol 17 (02) ◽  
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.

New better than used processes

1983 ◽  
Vol 15 (3) ◽  
pp. 601-615 ◽  
Author(s):  
Albert W. Marshall ◽  
Moshe Shared
Keyword(s):  
First Passage Time ◽  
Waiting Times ◽  
Random Variables ◽  
Passage Time ◽  
Renewal Processes ◽  
Recovery Processes ◽  
Strong Markov ◽  
New Better Than Used ◽  
Better Than ◽  
Quantities Of Interest

A stochastic process , such that P{Z(0) = 0} = 1, is said to be new better than used (NBU) if, for every x, the first-passage time Tx = inf {t: Z(t) > x} satisfies P{TX > s + t} for everys . In this paper it is shown that many useful processes are NBU. Examples of such processes include processes with shocks and recovery, processes with random repair-times, various Gaver–Miller processes and some strong Markov processes. Applications in reliability theory, queueing, dams, inventory and electrical activity of neurons are indicated. It is shown that various waiting times for clusters of events and for short and wide gaps in some renewal processes are NBU random variables. The NBU property of processes and random variables can be used to obtain bounds on various probabilistic quantities of interest; this is illustrated numerically.

ON SKEWNESS AND DISPERSION AMONG CONVOLUTIONS OF INDEPENDENT GAMMA RANDOM VARIABLES

2010 ◽  
Vol 25 (1) ◽  
pp. 55-69 ◽  
Author(s):  
Leila Amiri ◽  
Baha-Eldin Khaledi ◽  
Francisco J. Samaniego
Keyword(s):  
Residual Life ◽  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
Mean Residual Life ◽  
Exponential Random Variables ◽  
Right Spread Order ◽  
New Better Than Used ◽  
Common Shape ◽  
Better Than

Let {x(1)≤···≤x(n)} denote the increasing arrangement of the components of a vector x=(x1, …, xn). A vector x∈Rn majorizes another vector y (written $\bf{x} \mathop{\succeq}\limits^{m} \bf{y}$) if $\sum_{i=1}^{j} x_{(i)} \le \sum_{i=1}^{j}y_{(i)}$ for j = 1, …, n−1 and $\sum_{i=1}^{n}x_{(i)} = \sum_{i=1}^{n}y_{(i)}$. A vector x∈R+n majorizes reciprocally another vector y∈R+n (written $\bf{x} \mathop{\succeq}\limits^{rm} \bf{y}$) if $\sum_{i=1}^{j}(1/x_{(i)}) \ge \sum_{i=1}^{j}(1/y_{(i)})$ for j = 1, …, n. Let $X_{\lambda_{i},\alpha},\,i=1,\ldots,n$, be n independent random variables such that $X_{\lambda_{i},\alpha}$ is a gamma random variable with shape parameter α≥1 and scale parameter λi, i = 1, …, n. We show that if $\lambda \mathop{\succeq}\limits^{rm} \lambda^{\ast}$, then $\sum_{i=1}^{n} X_{\lambda_{i},\alpha}$ is greater than $\sum_{i=1}^{n} X_{\lambda^{\ast}_{i},\alpha}$ according to right spread order as well as mean residual life order. We also prove that if $(1/ \lambda_{1}, \ldots ,1/ \lambda_{n}) \mathop{\succeq}\limits^{m} \succeq (1/ \lambda_{1}^{\ast}, \ldots , 1/ \lambda_{n}^{\ast})$, then $\sum_{i=1}^{n} X_{\lambda_{i}, \alpha}$ is greater than $\sum_{i=1}^{n} X_{\lambda^{\ast}_{i},\alpha}$ according to new better than used in expectation order as well as Lorenze order. These results mainly generalize the recent results of Kochar and Xu [7] and Zhao and Balakrishnan [14] from convolutions of independent exponential random variables to convolutions of independent gamma random variables with common shape parameters greater than or equal to 1.

scholarly journals Measures of Risk on Variability with Application in Stochastic Activity Networks

2013 ◽  
pp. 231-241
Author(s):  
Yousry H. Abdelkader
Keyword(s):  
Distribution Functions ◽  
Stochastic Activity Networks ◽  
Simple Measure ◽  
Activity Networks ◽  
Measures Of Risk ◽  
The One ◽  
New Better Than Used ◽  
Better Than ◽  
Stochastic Activity

The author proposes a simple measure of variability of some of the more commonly used distribution functions in the class of New-Better-than-Used in Expectation (NBUE). The measure result in a ranking of the distributions, and the methodology used is applicable to other distributions in NBUE class beside the one treated here. An application to stochastic activity networks is given to illustrate the idea and the applicability of the proposed measure.

New better than used processes

1983 ◽  
Vol 15 (03) ◽  
pp. 601-615 ◽  
Author(s):  
Albert W. Marshall ◽  
Moshe Shared
Keyword(s):  
First Passage Time ◽  
Waiting Times ◽  
Random Variables ◽  
Passage Time ◽  
Renewal Processes ◽  
Recovery Processes ◽  
Strong Markov ◽  
New Better Than Used ◽  
Better Than ◽  
Quantities Of Interest

A stochastic process , such that P{Z(0) = 0} = 1, is said to be new better than used (NBU) if, for every x, the first-passage time Tx = inf {t: Z(t) &gt; x} satisfies P{TX &gt; s + t} for everys . In this paper it is shown that many useful processes are NBU. Examples of such processes include processes with shocks and recovery, processes with random repair-times, various Gaver–Miller processes and some strong Markov processes. Applications in reliability theory, queueing, dams, inventory and electrical activity of neurons are indicated. It is shown that various waiting times for clusters of events and for short and wide gaps in some renewal processes are NBU random variables. The NBU property of processes and random variables can be used to obtain bounds on various probabilistic quantities of interest; this is illustrated numerically.

Poisson and compound Poisson approximations for random sums of random variables

1996 ◽  
Vol 33 (01) ◽  
pp. 127-137 ◽  
Author(s):  
P. Vellaisamy ◽  
B. Chaudhuri
Keyword(s):  
Lower Bound ◽  
Total Variation ◽  
Random Variables ◽  
Upper Bounds ◽  
Random Sums ◽  
Compound Poisson ◽  
Sums Of Random Variables ◽  
Variation Distance ◽  
Poisson Approximations ◽  
Better Than

We derive upper bounds for the total variation distance, d, between the distributions of two random sums of non-negative integer-valued random variables. The main results are then applied to some important random sums, including cluster binomial and cluster multinomial distributions, to obtain bounds on approximating them to suitable Poisson or compound Poisson distributions. These bounds are generally better than the known results on Poisson and compound Poisson approximations. We also obtain a lower bound for d and illustrate it with an example.

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