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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

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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