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; p₁+...+p_k=1 (i.e., 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? Is it a minimal sufficient statistic? Calculate the means vector and the variance-covariance matrix of the sufficient statistic.

Question 3.

1. 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?
2. Repeat the previous paragraph, where $X_3 \sim Pois(\lambda_1+\lambda_2)$.