# Asymptotic approximation of log-probability using first four moments

Consider a random variable $$X \sim p_{n,\theta}$$ where the first four moments are given by known functions:

$$\begin{matrix} \ \ \ \ \ \ \mathbb{E}(X) \equiv \mu(n,\theta) & & & \ \ \ \ \ \ \ \mathbb{V}(X) \equiv \sigma^2(n,\theta), \\[6pt] \mathbb{Skew}(X) \equiv \gamma(n,\theta) & & & \mathbb{Kurt}(X) \equiv \kappa(n,\theta). \\[6pt] \end{matrix}$$

Suppose that it is known that as $$n \rightarrow \infty$$ the distribution $$p_{n,\theta}$$ converges to the normal distribution (which also implies that $$\gamma(n,\theta) \rightarrow 0$$ and $$\kappa(n,\theta) \rightarrow 3$$). In this case we can use the asymptotic approximation given by the normal log-density:

$$\log p_{n,\theta}(x) \approx -\frac{1}{2} \log(2 \pi) - \log(\sigma(n,\theta)) -\frac{1}{2} \Bigg( \frac{x-\mu(n,\theta)}{\sigma(n,\theta)} \Bigg)^2.$$

This normal approximation takes account of the mean and variance, but it does not take account of the known form for the skewness and kurtosis. (By using the normal density it implicitly takes these moments to be equal to their asymptotic values.) Although this is not a terrible approximation, it is a shame to lose the information in the skewness and kurtosis functions. I would like to get a better approximation, that still converges to the normal log-density as $$n \rightarrow \infty$$, but which conforms to the known skewness and kurtosis for all finite values of $$n$$.

(Note that I am aware of the Gram-Charlier series and the Edgeworht expansion as methods of approximation, but the approximate probabilities in these expansions are not guaranteed to be positive, so the log-density is not guaranteed to exists. Therefore I am seeking an alternative approximation.)

Question: What is an appropriate approximation for the log-density that matches the first four moments (i.e., including the known skewness and kurtosis) while still converging to the normal log-density as $$n \rightarrow \infty$$.