Exercise 1
Question 1.
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 2.
Suppose we have a sample of
n independent observations $Y_1,\ldots,Y_n$, where each $Y_i$ belongs to one of
$K$ categories: P($Y$ belongs to the j-th category)=p
j, j=1,...,K; p
1+...+p
k=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 3.
Let $Y_1,\ldots,Y_n$ be a sample of independent observations, where $Y_i \sim f_{\theta_i}(y)$ is from the one parameter
exponential family (i.e. each $Y_i$ belongs to the same class of distributions $f_\theta(y)$ from the exponential family but not necessarily with
the same parameter), and $\eta_i=c(\theta_i)$ are the corresponding natural parameteres. The following (generalized linear regression) model
describes $\eta_i$ as a function of $p$ explanatory variables $x_1,\ldots,x_p$:
$$\eta_i=\beta_0+\beta_1 x_{i1}+\ldots+\beta_p x_{ip}$$
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.
Question 4.
- Let $X_1 \sim Pois(\lambda_1), X_2 \sim Pois(\lambda_2)$ and $X_3 \sim Pois(\lambda_1\cdot \lambda_2)$, where $X_1, X_2, X_3$ are independent.
Does the joint distribution of $(X_1,X_2,X_3)$ belong to an exponential family? Find the minimal sufficient statistic for $(\lambda_1,\lambda_2)$. Is it complete?
- Repeat the previous paragraph, where $X_3 \sim Pois(\lambda_1+\lambda_2)$.