Verify whether the following belong to the exponential family and find a sufficient statistic for the unknown parameters?
- Gamma distribution with α or β or both α and β unknown
K-parameter exponential family: If a K-parameter family wth pdf/pmf can be expressed as-
then it belongs to K-parameter exponential family.
where
Verification of Gamma distribution:
The density of a random variable with distribution , can be written as
Clearly,
Hence belongs to two parameter exponential family
Sufficient Statistics:A statistic is said to be sufficient for if the conditional distribution of given T = t does not depend on θ for any value of t.
Theorem: If is a random sample from a k-parameter exponential family:
Then are complete sufficient statistics of
parameter .
Verify whether the following belong to the exponential family and find a sufficient statistic for the...
with parameters α and β. 2. Yİ,%, , Y, are a random sample from the Gamma distribution (a) Suppose that α 4 is known and β is unknown. Find a complete sufficient statistic for β. Find the MVUE of β. (Hint: What is E(Y)?) (b) Suppose that β 4 is known and α is unknown. Find a complete sufficient statistic for a. with parameters α and β. 2. Yİ,%, , Y, are a random sample from the Gamma distribution (a)...
Y1, Y2, ... Yn are a random sample from the Gamma distribution with parameters α and β (a) Suppose that α-4 is known and β is unknown. Find a complete sufficient statistic for β. Find the MVUE of β. (Hint: What is E(Y)?) (b) Suppose that β = 4 is known and a is unknown. Find a complete sufficient statistic for α.
Show that the following distributions belong to the exponential family. Find the natural parameter θ, scale parameter p and convex function b(9). Also find the E(Y) and Var(Y) as functions of the natural parameter. Specify the canonical link functions 1. Exponential distribution Bxp ), f(y:λ) λe-Ag. Binomial distribution known; f(y: π- C)π"(1-π)n-y, where n is 2. Bin(n,π). 3. Poisson distribution Pois(A), f(y:A)-e
Exercise: Let Yİ,Y2, ,, be a random sample from a Gamma distribution with parameters and β. Assume α > 0 is known. a. Find the Maximum Likelihood Estimator for β. b. Show that the MLE is consistent for β. c. Find a sufficient statistic for β. d. Find a minimum variance unbiased estimator of β. e. Find a uniformly most powerful test for HO : β-2 vs. HA : β > 2. (Assume P(Type!Error)- 0.05, n 10 and a -...
Let {} be a random sample from the distribution. (a) Find a sufficient statistic for when is known (b) Find a sufficient statistic for when is known 7l beta ( α , β ) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
(al This question asks you to consider a Bayesian approach to inference about λ, the mortality rate in an exponential model for survival time. In order to take a Bayesian . Show that the gamma distribution is a conjugate prior distribution for the distribution is also Gamma, with parameters that depend on a, P, n,y. approach, we specify a prior distribution for A which is gamma distribution exponential model, ie. if we specify that λ~Gamma (α, β) a priori, then...
1.(c) 2.(a),(b) 5. Let Xi,..., X, be iid N(e, 1). (a) Show that X is a complete sufficient statistic. (b) Show that the UMVUE of θ 2 is X2-1/n x"-'e-x/θ , x > 0.0 > 0 6. Let Xi, ,Xn be i.i.d. gamma(α,6) where α > l is known. ( f(x) Γ(α)θα (a) Show that Σ X, is complete and sufficient for θ (b) Find ElI/X] (c) Find the UMVUE of 1/0 -e λ , X > 0 2) (x...
3. Let Xi, , Xn be a random sample from a Poisson distribution with p.m.f Assume the prior distribution of Of λ is is an exponential with mean 1, i.e. the prior pdi g(A) e-λ, λ > 0 Note that the exponential distribution is a special gamma distribution; and a general gamma distribution with parameters α > 0 and β > 0 has the pd.f. h(A; α, β)-16(. otherwise Also the mean of a gamma random variable with the pd.f.h(Χα,...
Consider the binomial distribution with parameters n 10 and p (unknown) a) Is this binomial distribution an exponential family distribution? b) Find a sufficient statistic for p.
Problem 5. Find jointly sufficient statistic for the continuous uniform distribution on 01,02 (here, and θ2 are two unknown parameters). Hint: Use definition 6.7-1 (Factorization Theorem)