Suppose we have a random sample of n observations {x1, x2, x3,…xn}. Consider the following estimator...

Suppose we have a random sample of n observations {x₁, x₂, x₃,…x_n}. Consider the following estimator of µ_x, the population mean.

Z = 12x₁ + 14x₂ + 18x₃ +…+ 12n-1x_n−1 + 12nx_n

Verify that for a finite sample size, Z is a biased estimator.
Recall that Bias(Z) = E(Z) − µ_x. Write down a formula for Bias(Z) as a function of n and µ_x.
Is Z asymptotically unbiased? Explain.
Use the fact that for 0 < r < 1,

limn→∞i=1nri = r / 1−r

to show that

limn→∞Var(Z)= σX2/3

Hint: Use Var as an operator to solve for Var(Z), factor out σX2, and find the r in your infinite series of coefficients. Use the formula to compute the limit of the infinite series.

Is Z a consistent estimator? Explain.

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Rahul Sunny answered 1 year ago

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