soln2MT2 - MATH 306, MIDTERM EXAM II-SPRING 2004-2005, May...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 306, MIDTERM EXAM II-SPRING 2004-2005, May 18, 2005 5 questions, total 105 pts. Time allowed: 2 hrs Question 1. Let Y 1 , Y 2 , ··· , Y n be a random sample from a population with the density functiom f ( x ; Θ) = Θ 2 ye- Θ y if y > otherwise a) Find the maximum likelihood estimator (MLE) of Θ . 17pts b) Is the MLE you found in a) also a consistent estimator of Θ ? Give reasons. 5pts Solution. a ) L ( y 1 , y 2 , ···| Θ) = Q n j =1 Θ 2 y j e- Θ y j = Θ 2 n Q n j =1 y j e- Θ ∑ n j =1 y j , ln L ( y 1 , ··· , y n | Θ) = 2 ln Θ + ln n Y j =1 y j- Θ n X 1 y j , d ln L d Θ = 2 n Θ- n X 1 y j = 0-→ ˆ Θ = 2 n ∑ y j = 2 ¯ Y . b) The given density is a Gamma density with α = 2 , β = 1 Θ . Thus E ( Y ) = αβ = 2 Θ , so ¯ Y P → 2 Θ , the sample mean is always a consistent estimator of the population mean ( ¯ Y converges to the mean in probability). Then by the properties of convergence in probability: 1 ¯ Y P → Θ 2 , 2 ¯ Y P → 2( Θ 2 ) = Θ . So Θ = 2 ¯ Y is a consistent estimator of Θ . It is also acceptable if you only state the theorem ”Under certain conditions which are satisfied in almost all of the practical cases, the MLE’ s are consistent”....
View Full Document

Page1 / 3

soln2MT2 - MATH 306, MIDTERM EXAM II-SPRING 2004-2005, May...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online