Question

Suppose X1, X2, X3, ..., XN is a random sample of N observations; that is, each X is independent of the others and is distributed with the same mean µ and variance σ^2. Consider the following estimators of the population mean: X* = X̄ + 3, X̂ = X̄ + (80/N), X̃ = N*X̄/(N + 1), where X̄ = Σxi/N is the sample mean of X. 1. Which, if any, of these estimators is unbiased? Show your work. 2. Derive the variances of these estimators. Rank these estimators in terms of these variances. 3. Consider the large sample properties of these estimators. In particular, determine what happens to the mean and variance of each estimator as N tends to infinity? Are these estimators consistent?

          Suppose X1, X2, X3, ..., XN is a random sample of N observations; that is, each X is independent of the others and is distributed with the same mean µ and variance σ^2. Consider the following estimators of the population mean:
X* = X̄ + 3,
X̂ = X̄ + (80/N),
X̃ = N*X̄/(N + 1),
where X̄ = Σxi/N is the sample mean of X.

1. Which, if any, of these estimators is unbiased? Show your work.
2. Derive the variances of these estimators. Rank these estimators in terms of these variances.
3. Consider the large sample properties of these estimators. In particular, determine what happens to the mean and variance of each estimator as N tends to infinity? Are these estimators consistent?

Added by Daniel A.

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