The Greatest of a Finite Set of Random Variables

By C. E. Clark. 1961

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Citations in the corpus

Natural-Parameter Networks: A Class of Probabilistic Neural Networks (2016, methodology): “If the ReLU activation v ( x ) = max(0 , x ) is used, we can use the techniques in [6] to obtain the ﬁrst two moments of max( z 1 , z 2 ) where z 1 and z 2 are Gaussian random variables.”
Extended discrete choice models : integrated framework, flexible error structures, and latent variables (2001, methodology): “Much of the research on probit is in the areas of estimation (for example , Clark, 1961, developed an early used approximation; Lerman and Manski, 1981, pioneered the use of simulation for econometric models; and Geweke, Hajivassiliou, and Keane developed the now common GHK simulator4, which made…”
Biases and Variance in Value Function Estimates (2004, background): “random variables has a positive mean (Clark, 1961).”
Statistical timing analysis considering spatial correlations using a single PERT-like traversal (2003, methodology): “' / can be approximated using the closed-form formula provided in [12].”
A COMPUTATIONALLY EFFICIENT QUADRATURE PROCEDURE FOR THE ONE-FACTOR MULTINOMIAL PROBIT MODEL (1982, methodology): “6The starting values used were obtained by estimating the equation on only five periods of data, but using the approximation of Clark [4] for the evaluation of the integrals.”
Balancing stochastic two-sided assembly lines (2016, methodology): “…t Pzð Þ and max t Pxð Þ; t Py is formulated as: qz;max ¼ ½rxqzxU bð Þ þ ryqzyU bð Þ =rmax (E8) Note that max t Pxð Þ; t Py is not normally distributed, but can be well approximated by a normal variable with mean lmax, and variance r 2 max; the approximation error is discussed by Clark (1961).”
Integrated driving behavior modeling (2007, methodology): “Under these assumptions, Clark's equation (Clark 1961), which calculates the expected value of the greatest of a set of normally distributed random variables, can be used to calculate the expected maximum utility of the gap accepting decision with respect to the g (lead or lag) gap: ( ) ( )g gg gg…”
Bias and Variance Approximation in Value Function Estimates (2007, methodology): “We also refer readers to Clark (1961), which presents a procedure to approximate moments of the maximum of a finite number of correlated Gaussian random variables.”
A Knowledge-Gradient Policy for Sequential Information Collection (2008, background): “This expectation is the expectation of the maximum of a constant and a normal random variable, for which we have an analytical expression from [13].”
Expensive Multiobjective Optimization by MOEA/D With Gaussian Process Model (2010, methodology): “In the case of m = 3 , note that 3 ) } , we can, following [48], ﬁrst build an approximate Gaussian distribution for max { λ i ( f } as in the case of m = 2 , and then estimate an approximate Gaussian distribution for g te ( x | λ ) by treating it as the maximum of two independent Gaussian random…”
A Tutorial on Bayesian Optimization (2018, methodology): “The expected improvement can be evaluated in closed form using integration by parts, as described in Jones et al. (1998) or Clark (1961).”
Advances in Computation of the Maximum of a Set of Gaussian Random Variables (2007, methodology): “Since the correlation between Z and an arbitrary Gaussian X is identical to the correlation between Z G and X [2], we do not explicitly consider correlation in Ξ (Z)(Z G ) .”
VARIUS: A Model of Process Variation and Resulting Timing Errors for Microarchitects (2008, methodology): “There is no exact analytical solution for the distribution of the maximum of n normally distributed variables, but we use an approximation from [6].”