Exercise 1

Question 1.

Let y1,...,yn be i.i.d. from U[θ1,θ2]. Find the sufficient statistic for θ=(θ1,θ2)'.

Question 2.

Using the factorization theorem, find the sufficient statistic for a sample from the Poisson distribution. What is the distribution of the sufficient statistic? Is it minimal? Show its completeness using the definition of a complete statistic.

Question 3.

Which of the following distributions belong to the exponential family? Find their natural parameters and sufficient statistics.
  1. uniform distribution U[0,θ]
  2. Gamma(α,β) distribution f(y)=βαyα-1e-βy/G(α), y > 0
  3. Negative Binomial distribution NB(r,p) (for known r) - the number of Bernoulli trials with probability p until the r-th success
  4. Weinbull distribution f(y)=β α yα-1exp(-β yα), y>0, α>0, β>0
    1. α is known
    2. α is uknown
  5. Cauchy distribution f(y)=π-1/(1+(y-θ)2)
  6. f(y)=2(y+a)/(1+2a), 0 < y < 1, a > 0

Question 4.

Suppose we have a sample of n independent observations y1,..., yn, where each yi belongs to one of k categories: P(y belongs to the j-th category)=pj, j=1,...,k; p1+...+pk=1 (multinomial distribution). Show that this distribution belongs to the exponential family with k-1 parameters. Find the natural parameters and sufficient statistic. Is it complete? Calculate the means and the variance-covariance matrix of the sufficient statistic.

Question 5.

Let y1,...,yn be a sample of independent observations, where yi ~ fθi(y) is from the one parameter exponential family (i.e. each yi belongs to the same class of distributions fθ(y) from the exponential family but not necessarily with the same parameter), and ηi=c(θi) are the corresponding natural parameteres. The following (generalized linear regression) model describes ηi as a function of p explanatory variables x1,...,xp:

ηi=β0+β1xi1+...+βpxip

Show that the joint distribution of the data belongs to the exponential family with p+1 parameters, find the natural parameters and sufficient statistic. Write down the likelihood function as a function of β for the particular case of Poisson distribution.