{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# 5 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS...

This preview shows pages 1–3. Sign up to view the full content.

THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1302 PROBABILITY AND STATISTICS II (2010-11) EXAMPLE CLASS 5 1. Let X 1 , ..., X n be independent Poisson random variables with X j having pa- rameter , where λ > 0 is an unknown parameter. Given that the Fisher information contained in ( X 1 , ..., X n ) about λ is n ( n +1) 2 λ . Find the MLE of λ . What is its (i) Bias; (ii) Variance; (iii) Mean squared error? (iv) Is this MLE of λ efficient? 2. Let X 1 , ..., X n be i.i.d. from the uniform distribution over the interval [ θ, θ +1], where θ is unknown. (a) Find a bivariate sufficient statistic for θ . (b) Find a maximum likelihood estimator of θ . (c) Show that max { X 1 , ..., X n } - n 1+ n is an unbiased estimator of θ . 3. Let X 1 , ..., X n be i.i.d. from the Gamma distribution with parameters α and β , which f ( x ) = 1 Γ( α ) β α x α - 1 e - x/β , obtain the method-of-moments estimators for α and β . 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
EXAMPLE CLASS 5 SOLUTIONS 1. Since the likelihood function of λ is x ( λ ) = C ( X ) λ n j =1 X j exp - λ n ( n + 1) 2 ,
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}