Elementary approach to higher order asymptotics I am trying to understand “higher order asymptotics”. I find several texts on Likelihood asymptotics, nothing’s easy to read... if you have any nice pointers on this direction, I’ll be interested; however my main question follows. The following ”roadmap“ to higher order asymptotics seems natural to me, and seems something to do and understand before looking at likelihood theory, but I don’t find anything on these lines:
Consider $T = f( X )$ where $X$ is a random vector of known distribution; let's say, for simplicity, that $X$ is normal, or multinomial (or the concatenation of several independent multinomials). The Delta method tells me that asymptotically, $T$ is normal, and how to compute its mean and variance (using linear approximation of $f$). I think one can find a better approximation of the distribution of $T$ by


*

*computing the first moments of $T$ (using linear, or quadratic, or higher order approximations of $f$)

*”finding“ a distribution with the same first moments plus some other “parsimony” criteria to ensure uniqueness (”finding” is not well-defined: at least, being able to evaluate it numerically)


Is that possible? Do you know any textbooks/lecture/article going in that direction?
Edit fg nu gave me some pointers for the second step, that lead me to Edgeworth series. A few references:


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*Rothenberg 1984 (given by fg nu)

*Blinnikov 1998 (from the wikipedia article)

*Chapter 16 of Feller’s Introduction to Probability 


The first step is rather elementary, however any good pointer is still appreciated.
 A: The higher order asymptotics books that I have on my shelf are Barndord-Nielsen and Cox, Brazzale, Davison and Reid, and Young and Smith (the very latter was my dissertation adviser, and I think he has a great ability to explain very complex concepts in a reasonably understandable way; his review of non-standard problems in likelihood inference is a must second reading in asymptotics, although it is nearly impossible to get short of asking him directly). The ultimate reference on Edgeworth expansions is probably Peter Hall's bootstrap book (1995). I would have to say that Rothenberg's chapter recommended by fg nu may be beating any of them in terms of clarity. Some books, like Rencher's Multivariate Analysis, just put Bartlett corrections everywhere without explaining much of them, but they motivate it as a small sample correction.
A: In addition to the other answer, a simpler book to read about this (and newer) is maybe Ronald Butler: "Saddlepoint approximations with applications".  I guess it would be difficult to find a more elementary treatment than this!
As others mentioned Edgeworth approximations:  Saddlepoint approximations can be seen as a tilted version of the Edgeworth approximation. The edgeworth approximations is more exact at the mean of the distribution we want to approximate. If we want to approximate the distribution at some point $x$ other than the mean, we can "tilt" the distribution by multiplying it with some exponential factor, so moving the mean to $x$, and using the Edgeworth approximation there. That is the saddlepoint approximation.  
For more about the saddlepoint approximation see this post:  How does saddlepoint approximation work?
