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I'm reading Statistical Models by A. C. Davison and I'm really confused by this section on the Delta method.

  1. It's not mentioned explicitly, but is $h(T_n)$ a consistent estimator of $h(\mu)$?

  2. In the Taylor series expansion, I'm not sure why the term inside $h'$ is $\mu+n^{-1/2}\tau W_n$ if the expansion is around $\mu$, and why the expansion is an inequality when higher order derivatives are omitted.

  3. Why can $\frac{n^{1/2}(h(T_n)-h(\mu))}{\tau h'(\mu+n^{-1/2}\tau W_n)}$ be substituted with $Z_n$?

  4. From the last line, why does $h(T_n)$ have expected value $h(\mu)$ or is the relationship approximate/asymptotic? By Jensen's inequality, if $E[T_n] = \mu$, I don't think $E[h(T_n)]$ equals $E[h(\mu)]$?

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