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Approximate likelihood and probability calculations based on transforms. (English) Zbl 0542.62001
From the introduction: Let X be a random variable with distribution $$P_{\theta}$$ specified up to an unknown real parameter $$\theta$$. Likelihood methods, for example, require the evaluation of the density (p(x;$$\theta))$$ based forms $$\ell(x;\theta)=\log p(x;\theta)$$, $$S(x;\theta)=(\partial \log p(x;\theta))/\partial \theta$$, and $${\mathcal I}(\theta)=var\{S(X;\theta)\}.$$
In this paper we shall explore certain transform-based approximations to these forms which facilitate likelihood estimation; and the associated asymptotic methods, when p(x;$$\theta)$$ is unavailable or intractable. Also we shall consider the use of the exponential transformation of the approximation to $$\ell(x;\theta)$$ as an alternative to more familiar density approximations, such as the Edgeworth and Gram-Charlier expansions.
Reviewer: M.M.Rao

##### MSC:
 62A01 Foundations and philosophical topics in statistics 65C99 Probabilistic methods, stochastic differential equations 60E10 Characteristic functions; other transforms 62E99 Statistical distribution theory
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