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Since a second-order expansion for [()] has already been derived above, it only remains to find [() ()]. Treating f ( X ) f ( Y ) {\displaystyle f(X)f(Y)} as a two-variable function, the second-order Taylor expansion is as follows:
Examples of functions that are not entire include the square root, the logarithm, the trigonometric function tangent, and its inverse, arctan. For these functions the Taylor series do not converge if x is far from b. That is, the Taylor series diverges at x if the distance between x and b is larger than the radius of convergence. The Taylor ...
Methods of complex analysis provide some powerful results regarding Taylor expansions. For example, using Cauchy's integral formula for any positively oriented Jordan curve which parametrizes the boundary of a region , one obtains expressions for the derivatives f (j) (c) as above, and modifying slightly the computation for T f (z) = f(z), one ...
Even if the PDF can be found, finding the moments (above) can be difficult. 4. The solution is to expand the function z in a second-order Taylor series; the expansion is done around the mean values of the several variables x. (Usually the expansion is done to first order; the second-order terms are needed to find the bias in the mean.
For the second-order approximations of the third central moment as well as for the derivation of all higher-order approximations see Appendix D of Ref. [3] Taking into account the quadratic terms of the Taylor series and the third moments of the input variables is referred to as second-order third-moment method. [4]
Another estimator based on the Taylor expansion is [3] = where n is the sample size, N is the population size, m x is the mean of the x variate and s x 2 and s y 2 are the sample variances of the x and y variates respectively.
Any non-linear differentiable function, (,), of two variables, and , can be expanded as + +. If we take the variance on both sides and use the formula [11] for the variance of a linear combination of variables (+) = + + (,), then we obtain | | + | | +, where is the standard deviation of the function , is the standard deviation of , is the standard deviation of and = is the ...
In fact, for a smooth enough function, we have the similar Taylor expansion (+) = | | ()! + (,), where the last term (the remainder) depends on the exact version of Taylor's formula.