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Stat 366 Lab 9 Problems (November 9, 2006) page 1 TA: Yury Petrachenko, CAB 484, [email protected], http://www.ualberta.ca/∼yuryp/
The Bias and Mean Square Error of Point Estimators
8.5 Suppose that Y 1 , Y 2 ,... , Yn constitute a random sample from a population with probability density function
f (y) =
θ + 1
e−y/(θ+1), y > 0 , θ > − 1 ,
0 , elsewhere. Suggest a suitable statistic to use as an unbiased estimator for θ.
8.6 The number of breakdowns per week for a type of minicomputer is a random variable Y with a Poisson distribution and mean λ. A random sample Y 1 , Y 2 ,... , Yn of observations on the weekly number of breakdowns is available.
(a) Suggest an unbiased estimator for λ. (b) The weekly cost of repairing these breakdowns is C = 3Y + Y 2. Show that E(C) = 4 λ + λ^2. (c) Find a function of Y 1 , Y 2 ,... , Yn that is an unbiased estimator of E(C).
8.7 Let Y 1 , Y 2 ,... , Yn denote a random sample of size n from a population with mean 3. Assume that θˆ 2 is an unbiased estimator of E[Y 2 ] and that θˆ 3 is an unbiased estimator of E[Y 3 ]. Give an unbiased estimator for the third central moment of the underlying distribution.
8.13 If Y has a binomial distribution with parameters n and p, then ˆp 1 = Y /n is an unbiased estimator of p. Another estimator of p is ˆp 2 = (Y + 1)/(n + 2).
(a) Derive the bias of ˆp 2. (b) Derive MSE(ˆp 1 ) and MSE(ˆp 2 ). (c) For what values of p is MSE(ˆp 1 )<MSE(ˆp 2 )?
Stat 366 Lab 9 Problems (November 9, 2006) page 2
8.16 Suppose that Y 1 , Y 2 , Y 3 , Y 4 denote a random sample of size four from a population with an exponential distribution with parameter θ.
(a) Let X = √Y 1 Y 2. Find a multiple of X that is an unbiased estimator for θ. (b) Let W = √Y 1 Y 2 Y 3 Y 4. Find a multiple of W that is an unbiased estimator for θ^2.
Evaluating the Goodness of a Point Estimator
8.20 Results of a public opinion poll reported in a news magazine (Time, April 5, 1993) indicated that 54% of respondents between the ages of 18 and 26 feel that religion is a “very important” part of their lives. The article states that 1013 individuals were interviewed and that the results have a sampling error of 3%. How was the 3% calculated, and how should it be interpreted? Can we conclude that a majority of the individuals in this age group feel that religion is a very important part of their lives?
8.32 If Y 1 , Y 2 ,... , Yn denotes a random sample from an exponential distribution with mean θ, then E[Yi] = θ and V [Yi] = θ^2. Thus, E[ Y¯ ] = θ and V [ Y¯ ] = θ^2 /n, or σ (^) Y¯ = θ/√n. Suggest an unbiased estimator for θ, and provide an estimate for the standard error of your estimator.
8.34 The number of persons coming through a blood bank until the first person with type A blood is a random variable Y with a geometric distribution. If p denotes the probability that any one randomly selected person will possess type A blood, then E[Y ] = 1/p and V [Y ] = (1 − p)/p^2. Find a function of Y that is an unbiased estimator of V [Y ]. Then suggest how to form a two-standard error bound on the error of estimation when Y is used to estimate 1/p.
