On Sums of Lognormal Random Variables

Barouch, Eytan; Kaufman, Gordon M.; Glasser, M. L.

doi:10.1002/sapm198675137

Cited by 15 publications

(14 citation statements)

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“…In order to compute the distribution of undiscovered hydrocarbons in larger regions such as the aggregate for all coterminous U.S. onshore basins for example, basinal assessments were assumed to be probabilistically independent and aggregated by monte carlo simulation [Crovelli (1986[Crovelli ( , 1987]. A detailed study of asymptotic properties of unnormalized sums of lognormal random variables by Barouch, Kaufman and Glasser (1986) uncovered complicated behavior: such sums behave drastically differently in different asymptotic regimes. Nevertheless, Crovelli's monte carlo studies show that for parameter values within the range of those that appear in individual basinal assessments, a lognormal fit to the convolution of lognormal random variables is reasonable.…”

Section: The Shape Of Deposit Size Distributionsmentioning

confidence: 99%

Statistical Issues in the Assessment of Undiscovered Oil and Gas Resources

Kaufman¹

1993

The Energy Journal

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-For Dave WoodPrior to his untimely death, my friend Dave Wood gave me wise counsel about how best to organize a paper describing uses of statistics in oil and gas exploration. A preliminary reconnaissance of the literature alerted me to the enormous range of topics that might be covered. Geology, geophysics with particular attention to seismology, geochemistry, petroleum engineering and petroleum economics--each of these disciplines plays an important role in petroleum exploration and each weaves statistical thinking into its fabric in a distinctive way. An exhaustive review would be book length. Dave and I agreed that a timely review paper of reasonable length would:(1) Illustrate the range of statistical thinking of oil and gas exploratists.(2) Concentrate on topics with statistical novelty, show how statistical thinking can lead to better decision making and let the reader know about important controversies that might be resolved by better use of statistical methods.(3) Focus on topics that are directly relevant to exploration decision making and resource estimation.In response to Dave's sensible suggestions, the Department of Interior's 1989 assessment of U.S. undiscovered oil and gas will be a tour map for a short trip through a large territory of statistical methods and applications. Were he here to review this review, I know that it would be better than it is.

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Section: The Shape Of Deposit Size Distributionsmentioning

confidence: 99%

Statistical Issues in the Assessment of Undiscovered Oil and Gas Resources

Kaufman¹

1993

The Energy Journal

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“…(1) how we model measurement error, (2) what probability distributions we assume for measurement error, (3) which functional forms we investigate, (4) what type of attribute uncertainty we analyze, and (5) which question formats we consider. No doubt other researchers will have different tastes and make other tradeoffs.…”

Section: Perspectivementioning

confidence: 99%

“…For NORMAL errors we use the limited reproductive properties of the lognormal distribution (Aitchison and Brown, 1969) and some algebra. See Barouch and Kaufman (1976) for issues involving sums of lognormal random variables. Finally, substituting the appropriate f(rlX) yields the result.…”

Section: S/ Zmentioning

confidence: 99%

“…However, if the experimenter chooses his measurement design carefully, he can obtain practical analytic expressions for . In particular, for equation 16, if (1) he chooses a, , x and x 2 for the first question (review figure 4) and if (2) for every subsequent question, i, he chooses a and Xli. Then i and…”

Section: Binary Choice Among Alternatives Represented By Continuous Dmentioning

confidence: 99%

“…(R(x,r) is defined by equation (1) where we have added the unknown parameters, r, to the notation.) In particular, Keeney and Raiffa show i(xi,pi) = (1/2)v 2 iR(xi,r) the variance of the lottery,vi = (1/2)(x* -xi) 2 + (1/2)(x o -xi)2 , and indicates higher order terms that are assumed to be negligible. Rearranging terms yields:R i (xi,pi) = R(xi,r) Ri(Xi,Pi) 4(x i Pix* (1-pi)xo)/[(x*-xi) + (Xo-xi)2]is a function of known data because xi = pix* + (1-pi)x o.…”

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A Measurement Error Approach for Modeling Consumer Risk Preference

Eliashberg

Hauser

1985

Management Science

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Von Neumann-Morgenstern (vN-M) utility theory is the dominant theoretical model of risk preference. Recently, market researchers have adapted vN-M theory to model consumer risk preference. But, most applications assess utility functions by asking just n questions to specify n parameters. However, any questioning format, especially under market research conditions, introduces measurement error. This paper explores the implications of measurement error on the estimation of the unknown parameters in vN-M utility functions and provides procedures to deal with measurement error. We assume that the functional form of the utility function, but not its parameters, can be determined a priori through qualitative questioning. We then model measurement error as if question format and other influences cause the consumer to choose the unknown "risk parameter" from a probability distribution and to make his decisions accordingly. We provide procedures to estimate the unknown parameters when the measurement error is either (a) Normal or (b) Exponential. Uncertainty in risk parameters induces uncertainty in utility and expected utility, and hence uncertainty in choice outcomes. Thus, we derive the induced probability distributions of the consumer's utility and the estimators for the implied probability that an alternative is chosen. Results are obtained for both the standard decision analysis "preference indifference" question format and for a "revealed preference" format in which the consumer is asked simply to choose between two risky alternatives. Since uniattribute functions illustrate the essential risk preference properties of vN-M functions, we emphasize uniattribute results. We also provide multiattribute estimation procedures. Numerical examples illustrate the analytical results.marketing, utility theory, risk modeling

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