The Point Evaluation Method for Approximating Function Means, Variances and Covariances
Bryan F. J. Manly · Biometrical Journal · 1986
Abstract A perennial problem in statistics is the determination of biases, variances and covariances for functions of random variables X1, X2, …, Xn which themselves have a known distribution. A common approach is through equations based upon Taylor series approximations but a “point evaluation” method may sometimes be a useful alternative. This involves approximating the multivariate distribution of the X variables by the 2n points given by X1=μ1±1, X2 = μ2 ±2, …, Xn = = μn μn, where μi is the mean and σi the standard deviation of Xi, with appropriate point weights. An advantage over the Taylor series approach is that function derivatives do not have to be explicitely calculated. The point evaluation method is particularly useful in cases where the X variables are uncorrelated. Then the evaluation of the 2n points can be replaced by the evaluation of 2n points. The point evaluation method is illustrated with powers of a normally distributed variable, and with estimation of gene frequencies from ABO blood group frequencies.