Question

Plug-in estimates: Unbiased estimates are not transformation invariant, Show that:

a.) In N(μ, σ²), x̄² is not an unbiased estimate of μ²

b.) In Exponential(λ), e^(-x̄ ) is not an unbiased estimate of λ.

c.) In Poisson(λ), e^(-x̄ ) is not an unbiased estimate of e^λ.

d.) s is not an unbiased estimate of σ

Answer 1

If then is an unbiased estimator of θ.

a)

Here we have

Let n be the sample size. So,

So x̄2 is not an unbiased estimate of μ2.

b)

Here we have

Let n be the sample size. So,

So e^(-x̄ ) is not an unbiased estimate of λ.

c)

Here we have

Let n be the sample size. So,

d)

Let be a random sample from a distribution having finite variance . Then we need to shows is an unbiased estimator of Sigma^2.

Here we will use:

And

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Now

So s is an unbiased estimate of σ