Math Problem Solver

Question 1:

Generalized linear models (GLMs) were introduced as a general class of models by Nelder and Wedderburn in 1972. This class, or family F, follow one specification, which means that GLMs can all be

expressed in the following general form:

f(x|θ, ϕ) = exp h

α(ϕ){xθ − g(θ) + h(x)} + β(ϕ, x)

i

with scalar parameters θ, ϕ, and functions α, β, g, h and the scale factor α(ϕ) > 0.

(a) Show that: F belongs to the exponential family, if ϕ is a constant.

Question 2:

ˆ Notation: Sometimes we write f(x | a, b), fa,b(x), but f(x) is possible as well

A triangular distribution on R is defined by the following density function, fa,b,c(x), with parameters

a < b < c ∈ R:

fa,b,c(x) =







2(x−a)

(c−a)(b−a)

for a ≤ x ≤ b

2(c−x)

(c−a)(c−b)

for b < x ≤ c

0 otherwise

(a) Show that the symmetrical distribution given by fa,b,c, with the constraint b − a = c − b =: d,

belongs to the location-scale family. Find the appropriate parameterization and resulting density

f0.

(b) Explain why the family of triangular distributions does not belong to the exponential family.

Question 3:

(a) Assume that for an independent and identically distributed (i.i.d.) random sample X1, . . . , Xn,

the distributions of Xi

, i = 1, ..., n, belong to an exponential family, G. Show that the joint

distribution of X1, ..., Xn also belongs to an exponential family.

(b) Let X0 be a real random variable with E[X0] = 0 and V ar[X0] = 1. We look at the family of

distributions of Xm,s = sX0 + m with s > 0, m ∈ R. Show that the distribution of Xm,s belongs

to the location and scale family, whose location parameter will correspond to the expected value

and the scale parameter will correspond to the standard devia