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How do I find the variance of $F(x)$, the CDF, if the form of $F(x)$ is known, and it's (single) parameter was estimated by ML?

For example, suppose $X\sim Exp(\theta)$, which means $F(x)=1-exp(-\theta x)$.

Assume $\theta$ was estimated using the method of maximum likelihood (denote as $\widehat{\theta}$), and $Var(X)$ was estimated using the inverse of the Fisher information (denote as $\widehat{\sigma}^2$).

It is clear that $\widehat{F}(2)=1-exp(-2\widehat{\theta})$, for example, but how would one compute $\widehat{Var}(\widehat{F}(2))$?

The only method I can think of (for any parametric CDF) for an analytical solution is just to apply the delta method, but is there a better way?

Thanks

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  • $\begingroup$ What do you mean by the variance of a function? $\endgroup$
    – Xi'an
    Apr 18, 2018 at 11:11
  • $\begingroup$ I mean the variance (or std error) of a CDF evaluated at a certain point, like in the example in the penultimate paragraph. The CDF contains an estimated parameter, so presumably an estimate of the variance of a function of that parameter may be calculated. (PS I like your blog.) $\endgroup$
    – Anna Efron
    Apr 18, 2018 at 12:15
  • $\begingroup$ Thanks. The problem is then simply one of the variance of a transform of a random variable, which most often does not have a closed form. Delta-methods, Edgeworth expansions, Monte Carlo approximations are ways of approximating this variance. $\endgroup$
    – Xi'an
    Apr 18, 2018 at 14:10

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