Means and Variances of Stochastic Vector Products with Applications to Random Linear Models
Abstract
Applications in operations research often employ models which contain linear functions.
These linear functions may have some components (coefficients and variables) which are
random. (For instance, linear functions in mathematical programming often represent models
of processes which exhibit randomness in resource availability, consumption rates, and
activity levels.) Even when the linearity assumptions of these models is unquestioned, the
effects of the randomness in the functions is of concern. Methods to accomodate, or at least
estimate for a linear function the implications of randomness in its components typically
make several simplifying assumptions. Unfortunately, when components are known to be
random in a general, multivariate dependent fashion, concise specification of the randomness
exhibited by the linear function is, at best, extremely complicated, usually requiring severe,
unrealistic restrictions on the density functions of the random components. Frequent stipula-
tions include assertion of normality, or of independence-yet, observed data, accepted collat-
eral theory and common sense may dictate that a symmetric distribution with infinite domain
limits is inappropriate, or that a dependent structure is definitely present. (For example,
random resource levels may be highly correlated due to economic conditions, and non-
negative for physical reasons.) Often, an investigation is performed by discretizing the
random components at point quantile levels, or by replacing the random components by their
means-methods which give a deterministic "equivalent" model with constant terms, but
possibly very misleading results. Outright simulation can be used, but requires considerable
time investment for setup and debugging (especially for generation of dependent sequences of
pseudorandom variates) and gives results with high parametric specificity and computation
cost. This paper shows how to use elementary methods to estimate the mean and variance of
a linear function with arbitrary multivariate randomness in its components. Expressions are
given for the mean and variance and are used to make Tchebycheff-type probability state-
ments which can accomodate and exploit stochastic dependence. Simple estimation examples
are given which lead to illustrative applications with (dependent-) stochastic programming
models.
Description
Management Science, 24, 2, pp. 210-216.
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