4.11 Problems

• 4.2-1 In the binomial problem, assume that at least two observations are taken. Write 0** for the point (2, 1). Estimate unbiasedly ?(1 ? ?). [Hint: we have P(0**)=2 ?(1 ? ?) and let where ?** ( n, t) denotes the number of paths from 0** to ( n, t). Then Y is an unbiased estimate of ?(1 ? ?).]

• 4.2-2 In the binomial case, check whether removal of (i) 0**, (ii) (2, 0) destroys closure. Find all unbiased estimates of ? which depend on (N, T _N ) only. Among them, there is only one bounded estimate.

• 4.2-3 Verify that the sequence of sufficient statistics for the exponential family of densities is transitive.

• 4.2-4 If X is normal ( ?, ? ²), show that { } is transitive.

• 4.3-1 Show that the exponential family of densities satisfy the regularity conditions of Theorem 4.3.2 on f( x; ?).

• 4.4-1 Is it possible to have a two-stage procedure for estimating the parameter of the exponential density where

  1. f( x; ?)=exp{ ?( x ? ?)}, x> ?,

  2. f ( x; ?)= ? ^?1 exp ( ? x/ ?), x>0 ?

• 4.4-2 (Two-stage procedure for the binomial parameter). Let X ₁, X ₂ ,... be i.i.d. Bernoulli variables with