Berkeley, UC Department of Statistics STAT 210A: Introduction to Mathematical Statistics HW#4
Fall, 2009
Due: Tuesday, September 29, 2009
4.1. Let X 1 ,.., X n be independent with X i ~ N (ti ,1) , where t1 ,..., tn are a sequence of known constants (not all zero). (a) Show that the least squares estimator, = ti X i / ti2 is complete sufficient for the family of joint distributions.
(b) Use Basus theorem to show that and
(X
i
ti ) are independent.
2
4.2. The inverse Gaussian distribution IG ( , ) has density function
1 exp( ) x 3 / 2 exp ( + x ) , 2 2x
x > 0, > 0, > 0.
(a) Show that this density constitutes an exponential family. n 1 1 1n (b) Show that the statistic T = ( X = X i , S = ( ) ) is complete and sufficient. n i =1 X i =1 X i
4.3. Determine the natural parameter space of the associated exponential family of dimension one with = , T ( x ) = x and (a) h(x) = exp(- x 2 ). (b) h(x) = exp(-|x|). (c) h(x) = exp(-|x|)/(1+ x 2 ).
4.4. Let Z be distributed according to N ( , 2 ) with and 2 unknown. For a given and
2 + , define G , ( x ) = P( Z x | Z > 0) .
2
(a) Prove G that is a cdf for any given , 2 . (b) Prove that the distribution G belongs to the exponential family. (c) Find the moment generating function of the distribution G and compute the mean and variance of G.
4.5. (a) Given a k-dimensional random vector x, prove that the distribution on {1, 2, 3, } given by p( x; ) = k 1{max x j } is not an exponential family.
j
(b) Is the uniform distribution on [0, ] an exponential family? Why or why not?
4.6. Suppose that X has density in exponential family form
p( x, ) exp
(
d i =1
i Ti ( x ) A( ) h( x ),
)
where each Ti is differentiable. (a) Prove the following form of Stein's identity. If X has support on the real line and g is any differentiable function such that E | g '( X ) | < + , then
h '( X ) d E + i Ti ' ( X ) g ( X ) = E[ g '( X )]. h ( X ) i =1 2 (b) Specializing to the N ( , ) case, use part (a) to show that Cov ( g ( X ), X ) = 2 E [ g '( X )] .
(c) Use part (b) to compute the third and fourth moments of the N ( , 2 ) distribution.
The following problems will not be graded Bickel & Doksum, 1.6.3, 1.6.5, 1.6.10 (pages 87 & 88)