The wobble in a reciprocal MoM estimate

Question

For an exponential sample the method-of-moments rate estimate is \hat\lambda = 1/\bar X. Using the delta method, find the approximate variance of \hat\lambda for large n.

Accepted Answer

Why we need an approximation. The estimate is a nonlinear function (a reciprocal) of the sample average, so its variance is not simply the average's variance. The delta method handles this: for a smooth function g, the variance of g(\bar X) is approximately \big(g'(\mu)\big)^2 times the variance of \bar X, where \mu = \mathbb{E}[X]. Set up the pieces. For the exponential, \mu = \mathbb{E}[X] = 1/\lambda and \operatorname{Var}(X) = 1/\lambda^2, so \operatorname{Var}(\bar X) = \frac{1}{n\lambda^2}. The function is g(m) = 1/m, with derivative g'(m) = -1/m^2. Evaluated at m = \mu = 1/\lambda: g'(…

The wobble in a reciprocal MoM estimate

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