The Aumann Integral and the Conditional Expectation of a Set-Valued Random Variable

2002 ◽  
pp. 41-85
Author(s):  
Shoumei Li ◽  
Yukio Ogura ◽  
Vladik Kreinovich
Keyword(s):  
Conditional Expectation ◽  
Random Variable ◽  
Aumann Integral

scholarly journals Polynomial Interpolation of Normal Conditional Expectation

2019 ◽  
Vol 8 (3) ◽  
pp. 75
Author(s):  
Anis Rezgui
Keyword(s):  
Error Estimation ◽  
Conditional Expectation ◽  
Polynomial Function ◽  
Polynomial Interpolation ◽  
Random Variable ◽  
Standard Normal Distribution ◽  
Degree Polynomial ◽  
Reasonable Condition ◽  
Pointwise Error ◽  
Standard Normal

In this paper we are interested in approximating the conditional expectation of a given random variable X with respect to the standard normal distribution N(0, 1). Actually we have shown that the conditional expectation E(X|Z) could be interpolated by an N degree polynomial function of Z, φN(Z) where N is the number of observations recorded for the conditional expectation E(X|Z = z). A pointwise error estimation has been proved under reasonable condition on the random variable X.

scholarly journals A Berry-Esseen bound for the lightbulb process

2011 ◽  
Vol 43 (03) ◽  
pp. 875-898 ◽  
Author(s):  
Larry Goldstein ◽  
Haimeng Zhang
Keyword(s):  
Conditional Expectation ◽  
Spectral Decomposition ◽  
Random Variable ◽  
Stein’S Method ◽  
Normal Random Variable ◽  
The Past ◽  
Terminal Time ◽  
Standard Normal Random Variable ◽  
Standard Normal ◽  
Size Bias

In the so-called lightbulb process, on daysr= 1,…,n, out ofnlightbulbs, all initially off, exactlyrbulbs, selected uniformly and independent of the past, have their status changed from off to on, or vice versa. WithXthe number of bulbs on at the terminal timen, an even integer, and μ =n/2, σ2= var(X), we have supz∈R| P((X- μ)/σ ≤z) - P(Z≤z) | ≤nΔ̅0/2σ2+ 1.64n/σ3+ 2/σ, whereZis a standard normal random variable and Δ̅0= 1/2√n+ 1/2n+ e−n/2/3 forn≥ 6, yielding a bound of orderO(n−1/2) asn→ ∞. A similar, though slightly larger bound, holds for oddn. The results are shown using a version of Stein's method for bounded, monotone size bias couplings. The argument for evenndepends on the construction of a variableXson the same space asXthat has theX-size bias distribution, that is, which satisfies E[Xg(X)] = μE[g(Xs)] for all bounded continuousg, and for which there exists aB≥ 0, in this caseB= 2, such thatX≤Xs≤X+Balmost surely. The argument for oddnis similar to that for evenn, but one first couplesXclosely toV, a symmetrized version ofX, for which a size bias coupling ofVtoVscan proceed as in the even case. In both the even and odd cases, the crucial calculation of the variance of a conditional expectation requires detailed information on the spectral decomposition of the lightbulb chain.

scholarly journals A relation between positive dependence of signal and the variability of conditional expectation given signal

2006 ◽  
Vol 43 (04) ◽  
pp. 1181-1185 ◽  
Author(s):  
Toshihide Mizuno
Keyword(s):  
Conditional Expectation ◽  
Marginal Distribution ◽  
Stochastic Order ◽  
Random Variable ◽  
Sufficient Condition ◽  
Conditional Distributions ◽  
Convex Order ◽  
Positive Dependence ◽  
Dispersive Order ◽  
Conditional Expectations

Let S 1 and S 2 be two signals of a random variable X, where G 1(s 1 ∣ x) and G 2(s 2 ∣ x) are their conditional distributions given X = x. If, for all s 1 and s 2, G 1(s 1 ∣ x) - G 2(s 2 ∣ x) changes sign at most once from negative to positive as x increases, then the conditional expectation of X given S 1 is greater than the conditional expectation of X given S 2 in the convex order, provided that both conditional expectations are increasing. The stochastic order of the sufficient condition is equivalent to the more stochastically increasing order when S 1 and S 2 have the same marginal distribution and, when S 1 and S 2 are sums of X and independent noises, it is equivalent to the dispersive order of the noises.

scholarly journals On Characterizations of Randomly Censored Generalized Exponential Distribution

2014 ◽  
Vol 10 (2) ◽  
pp. 85-92
Author(s):  
G.G. Hamedani
Keyword(s):  
Exponential Distribution ◽  
Conditional Expectation ◽  
Hazard Function ◽  
Random Variable ◽  
Generalized Exponential Distribution ◽  
Underlying Distribution ◽  
Single Function ◽  
Randomly Censored ◽  
Model Fits

Abstract The problem of characterizing a distribution is an important problem which has recently attracted the attention of many researchers. Thus, various characterizations have been established in many different directions. An investigator will be vitally interested to know if their model fits the requirements of a particular distribution. To this end, one will depend on the characterizations of this distribution which provide conditions under which the underlying distribution is indeed that particular distribution. In this work, several characterizations of Randomly Censored Generalized Exponential (RCGE) distribution are presented. These characterizations are based on: ( i) conditional expectation of certain functions of the random variable , (ii ) a single function of the random variable, ( iii) the hazard function of the random variable.

scholarly journals Characterizations of Discrete Weibull Distributions

2020 ◽  
pp. 599-607
Author(s):  
G.G. Hamedani ◽  
Nadeem Shafique Butt
Keyword(s):  
Weibull Distribution ◽  
Conditional Expectation ◽  
Hazard Function ◽  
Random Variable ◽  
Weibull Distributions

Seven versions of the discrete Weibull distribution are characterized via conditional expectation of function of the random variable as well as based on the hazard or reverse hazard function.

Estimation of unknown parameters using partially observed data

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Eunji Lim
Keyword(s):  
Conditional Expectation ◽  
Unknown Parameter ◽  
Stochastic Systems ◽  
Random Variable ◽  
Computer Time ◽  
Conditional Probabilities ◽  
Unknown Parameters ◽  
Content Type ◽  
True Value ◽  
Partially Observed

Purpose This paper considers the complex stochastic systems such as supply chains, whose dynamics are controlled by an unknown parameter such as the arrival or service rates. The purpose of this paper is to provide a simulation-based estimator of the unknown parameter when only partially observed data on the underlying system is available. Design/methodology/approach The proposed method treats the unknown parameter as a random variable and estimates the parameter by computing the conditional expectation of the random variable given the partially observed data. This study then express the conditional expectation as a weighted sum of reverse conditional probabilities using Bayes’ rule. The reverse conditional probabilities are estimated using simulation. Findings The simulation studies indicate that the proposed estimator converges to the true value of the conditional expectation as the computer time allocated to the simulation increases. The proposed estimator is computed within a few seconds in all of the numerical examples, which demonstrates its time efficiency. Originality/value Most of the existing methods for estimating an unknown parameter require a significant amount of simulation, causing long computation delays. The proposed method requires a single simulation run for each candidate of the unknown parameter. Thus, it is designed to carry a significantly reduced computational burden. This feature will enable managers to use the proposed method when making real-time decisions.

scholarly journals A Berry-Esseen bound for the lightbulb process

2011 ◽  
Vol 43 (3) ◽  
pp. 875-898 ◽  
Author(s):  
Larry Goldstein ◽  
Haimeng Zhang
Keyword(s):  
Conditional Expectation ◽  
Spectral Decomposition ◽  
Random Variable ◽  
Stein’S Method ◽  
Normal Random Variable ◽  
The Past ◽  
Terminal Time ◽  
Standard Normal Random Variable ◽  
Standard Normal ◽  
Size Bias

In the so-called lightbulb process, on days r = 1,…,n, out of n lightbulbs, all initially off, exactly r bulbs, selected uniformly and independent of the past, have their status changed from off to on, or vice versa. With X the number of bulbs on at the terminal time n, an even integer, and μ = n/2, σ2 = var(X), we have supz ∈ R | P((X - μ)/σ ≤ z) - P(Z ≤ z) | ≤ nΔ̅0/2σ2 + 1.64n/σ3 + 2/σ, where Z is a standard normal random variable and Δ̅0 = 1/2√n + 1/2n + e−n/2/3 for n ≥ 6, yielding a bound of order O(n−1/2) as n → ∞. A similar, though slightly larger bound, holds for odd n. The results are shown using a version of Stein's method for bounded, monotone size bias couplings. The argument for even n depends on the construction of a variable Xs on the same space as X that has the X-size bias distribution, that is, which satisfies E[Xg(X)] = μE[g(Xs)] for all bounded continuous g, and for which there exists a B ≥ 0, in this case B = 2, such that X ≤ Xs ≤ X + B almost surely. The argument for odd n is similar to that for even n, but one first couples X closely to V, a symmetrized version of X, for which a size bias coupling of V to Vs can proceed as in the even case. In both the even and odd cases, the crucial calculation of the variance of a conditional expectation requires detailed information on the spectral decomposition of the lightbulb chain.

scholarly journals Some Characterization and Relations Based on K-th Lower Record Values

2016 ◽  
Vol 7 (1) ◽  
pp. 36-44
Author(s):  
Ali A. Al-Shomrani
Keyword(s):  
Conditional Expectation ◽  
Recurrence Relations ◽  
Sufficient Conditions ◽  
Random Variable ◽  
Record Values ◽  
Distribution Functions ◽  
Conditional Expectations ◽  
Lower Record ◽  
Lower Record Values ◽  
Necessary And Sufficient

In this paper, we obtain certain expressions and recurrence relations for two general classes of distributions based on some conditional expectations of k-th lower record values. We consider the necessary and sufficient conditions such that these conditional expectations hold for some distribution functions. Furthermore, an expression of conditional expectation of other general class of distributions through truncated moments of some random variable is considered. Some distributions as examples of these general classes are shown in Tables1and2accordingly.

Sign in / Sign up

Export Citation Format

Share Document