By Leonhard Held

This booklet covers statistical inference in accordance with the possibility functionality. Discusses frequentist likelihood-based inference from a Fisherian standpoint, Bayesian inference options together with element and period estimates, version selection and prediction and extra.

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**Extra info for Applied Statistical Inference: Likelihood and Bayes**

**Example text**

Exp(−λ) (nλ)t t! exp(−nλ) t t! n i=1 xi ! 3. A sufficient statistic T contains all relevant information from the sample X1:n with respect to θ . To show that a certain statistic is sufficient, the following result is helpful. 2 (Factorisation theorem) Let f (x1:n ; θ ) denote the probability mass or density function of the random sample X1:n . A statistic T = h(X1:n ) with realisation t = h(x1:n ) is sufficient for θ if and only if there exist functions g1 (t; θ ) and g2 (x1:n ) such that for all possible realisations x1:n and all possible parameter values θ ∈ Θ, f (x1:n ; θ ) = g1 (t; θ ) · g2 (x1:n ).

The corresponding likelihood functions are displayed here. The vertical line at α = 1 corresponds to the exponential model in both cases For α = 1, we obtain the exponential distribution with expectation μ = 1/λ as a special case. A contour plot of the Weibull likelihood, a function of two parameters, is displayed in Fig. 4a. 19, μ = 1195. The assumption of exponentially distributed survival times does not appear to be completely unrealistic, but the likelihood values for α = 1 are somewhat lower.

51 55 56 56 59 63 65 70 75 78 Maximum likelihood estimation has been introduced as an intuitive technique to derive the “most likely” parameter value θ for the observation x. But what properties does this estimate have?