Estimating Parameters of Derived Random Variables: Comparison of the Delta and Parametric Bootstrap Methods

Abstract

In quantitative fisheries research, analysts often need to estimate the parameters of derived random variables that cannot be observed directly and are instead computed from other observed random variables. The delta method, which is based on the Taylor series, has been widely used for approximating the mean and variance of derived variables. This paper applies a computer simulation method that involves the parametric bootstrap method for estimating the mean and variance of derived random variables. The results from these two methods are compared with the true values (when they can be computed) or with each other. Several types of commonly used functions in quantitative fisheries studies are examined, namely, the exponential, logarithmic, product, quotient, and asymptotic functions. The parametric bootstrap method leads to fairly accurate estimates for the mean and variance of the derived variables. The delta method provides reasonably good estimates when the variation (expressed as the coefficient of variation, CV, which is defined as 100·SD/mean) of the underlying variable is small; however, the deviation from the true values or the bootstrap results increases as the variations of the underlying observed variables increase. The impact is more severe for the variance estimator than for the mean estimator. The delta method under- or overestimates the variance by about 10% when the CVs of the underlying variables approach the following values: 25% for the function exp(x), 20% for log _e (x), 50% for xy, 15% for x/y and 1/x, 25% for log _e (x/y), and 30% for x(1—exp(-yt)). It is recommended that the bootstrap method be used for estimating the variance of a derived random variable when the observed variables have a large variance.