1. Suppose we would like to estimate the parameter θ(> 0) of a Uniform (0,θ) population based on a random sample X1,...,Xn from the population. In the class, we have discussed two estimators for θ — the maximum likelihood estimator, θˆ1 = X(n), where X(n) is the maximum of the sample, and the method of moments
estimator, θˆ2 = 2X, where X is the sample mean. The goal of this exercise is to compare the mean squared errors of the two estimators to determine which estimator is better. Recall that the mean squared error of an estimator θˆ of a parameter θ is defined as E{(θˆ−θ)2}. For the comparison, we will focus on n = 1,2,3,5,10,30 and θ = 1,5,50,100.
1
(a) Explain how you will compute the mean squared error of an estimator usingMonte Carlo simulation.
(b) For a given combination of (n,θ), compute the mean squared errors of both θˆ1 and θˆ2 using Monte Carlo simulation with N = 1000 replications. Be sure to compute both estimates from the same data.
(c) Repeat (b) for the remaining combinations of (n,θ). Summarize your results graphically.
(d) Based on (c), which estimator is better? Does the answer depend on n or θ? Explain. Provide justification for all your conclusions.
2. Suppose the lifetime, in years, of an electronic component can be modeled by a continuous random variable with probability density function
,
where θ > 0 is an unknown parameter. Let X1,...,Xn be a random sample of size n from this population.
(a) Derive an expression for maximum likelihood estimator of θ.
(b) Suppose n = 5 and the sample values are x1 = 21.72,x2 = 14.65,x3 = 50.42,x4 = 28.78,x5 = 11.23. Use the expression in (a) to provide the maximum likelihood estimate for θ based on these data.
(c) Even though we know the maximum likelihood estimate from (b), use the data in(b) to obtain the estimate by numerically maximizing the log-likelihood function using optim function in R. Do your answers match?
(d) Use the output of numerical maximization in (c) to provide an approximatestandard error of the maximum likelihood estimate and an approximate 95% confidence interval for θ. Are these approximations going to be good? Justify your answer.
2
