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STAT2006 Assignment 3

Due Date: 15:00, 24th March, 2015 1. (a) Let Y be an exponential random variable with mean λ and X θ1 + θ2 Y, θ2 > 0. Find the pdf of X and remember to state the support of X. X is said to follow a shifted exponential distribution with location parameter θ1 and scale parameter θ2 . (b) Let X1 , X2 , ..., Xn be a random sample which Xi are identically distributed as X. Find the method-of-moments estimator for θ1 and θ2 . (c) When θ2 is fixed, show that the likelihood function is strictly increasing in θ1 when θ1 ≤ x(1) and is equal to zero when θ1 > x(1) , where x(1) min{x1 , x2 , ..., xn } is the sample minimum. Hence find the maximum likelihood estimator of θ1 and θ2 . 2. The independent random variables X1 , ..., Xn have the common distribution  0 if x β where the parameters α and β are positive. (a) Assume α and β are both unknown, find the MLEs of α and β. (b) The length of cuckoos’s eggs found in hedge sparrow nests can be modeled with this distribution. For the data 22.0, 23.9, 20.9, 23.8, 25.0, 24.0, 21.7, 23.8, 22.8, 23.1, 23.1, 23.5, 23.0, 23.0 find the MLEs of α and β. (c) If α is a known constant, α0 , find an upper confidence limit for β with confidence coefficient 0.95. (d) Use the data in (b) to construct an interval estimate for β. Assume the α is known and equal to its MLE. 3. Let X1 , X2 , . . . , Xn be a random sample of size n from an exponential distribution with unknown mean θ. (a) Show that the distribution of the random variable W = (2/θ) (b) Use W to construct a 100(1 − α)% confidence interval for θ. (c) If n = 6 and x = 80.7, give the endpoints for a 90% confidence interval for the mean θ. ¯ 4. Let X1 , X2 , ..., Xn be a random sample from N (µ, σ 2 ), then the pivotal quantity (n − 1)S 2 ∼ χ2 (n − 1), and we can make use of its quantiles a, b to construct a 100(1 − α)% σ2 confidence interval for σ. The quantiles a, b need to satisfy the constraint G(b) − G(a) = Pr a ≤ (n − 1)S 2 ≤b σ2 =1−α

n i=1

Xi is χ2 (2n).

where G is...