By Otto J.W.F. Kardaun

ISBN-10: 3540211152

ISBN-13: 9783540211150

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**Extra resources for Classical Methods of Statistics: With Applications in Fusion-oriented Experimental Plasma Physics**

**Example text**

Yf ∼ E(λ) = Γ(1, λ), with Y1 , Y2 , . , Yf independent, then f Yi ∼ Γf,λ . 19. (Beta distribution ) If Y1 ∼ Γ(f, λ) = Γ(g, λ), with Y1 and Y2 independent, then λ 2 2 χ2f and Y2 ∼ Y1 ∼ Bef,g . 20. (Student’s t distribution ) If X ∼ N (0, 1) and Z ∼ χ2f , with X and Z independent, then X Z/f ∼ tf . 21. (F distribution ) If Y1 ∼ χ2f and Y2 ∼ χ2g , with Y1 and Y2 independent, then Y1 /f ∼ Ff,g . 22. (‘Beta logistic’ distribution ) If X ∼ Bef,g , then Y = log X ∼ BeLof,g , 1−X which we call here the Beta logistic distribution.

The Beta function 1 f −1 (f, g) = 0 x (1 − x)g−1 dx can alternatively be deﬁned as (f, g) = Γ(f )Γ(g)/Γ(f + g), and ψ(f ) = (∂/∂f ) log Γ(f ) is the digamma function. 06 + f12 (f > 23 ). and the negative integers. See Sect. 2 for simple analytic approximations such as ψ (f ) = (∂/∂f )ψ(f ) f f+ 1 Ê 2 ¡ In many practical applications, f and g can be restricted to the set of positive integers and are called degrees of freedom. In the table, the excess of kurtosis for the hypergeometric distribution contains the factor Ca,n,N = of the F –distribution the factor Df,g = g−2 f +g−2 distribution with mean µ and variance σ 2 .

This is sometimes called the convolution property of an exponential family. We restrict now, for simplicity, the attention to the univariate case, x ∈ X ⊂ IR, which covers a number of practical situations. ,m on µ = E t1 (X), . . , tm (X) , is called the variance function. Under quite general regularity conditions, the variance function characterises a sub-family of distributions within the (semi-parametric) class of exponential families, see [428]. Therefore, it plays a prominent role in ﬁtting distributions and in parameter estimation theory, among others in the context of generalised linear models, see [455].

### Classical Methods of Statistics: With Applications in Fusion-oriented Experimental Plasma Physics by Otto J.W.F. Kardaun

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