This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full 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 &gt; 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