Let Y1, Y2, . . . , Yn denote a random sample from a

QUESTION:

Let \(Y_{1}, Y_{2}, \ldots, Y_{n}\) denote a random sample from a population with mean \(\mu\) and variance \(\sigma^{2}\). Consider the following three estimators for \(\mu\):

\(\hat{\mu}_{1}=\frac{1}{2}\left(Y_{1}+Y_{2}\right), \quad \hat{\mu}_{2}=\frac{1}{4} Y_{1}+\frac{Y_{2}+\cdots+Y_{n-1}}{2(n-2)}+\frac{1}{4} Y_{n}, \quad \hat{\mu}_{3}=\bar{Y}\)

a Show that each of the three estimators is unbiased.

b Find the efficiency of \(\hat{\mu}_{3}\) relative to \(\hat{\mu}_{2}\) and \(\hat{\mu}_{1}\), respectively.

Equation Transcription:

 = ,  =   =                

Text Transcription:

Y_{1}, Y_{2}, ..., Y_{n}

mu

sigma^2

mu

hat{mu}_{1} = frac{1}{2} (Y_1 +Y_2), quad hat{mu}_2 = frac{1}{4} Y_{1} + frac{Y_{2}+ ... +Y_{n-1}}{2(n-2)} + frac{1}{4} Y_ n, quad hat{mu}_{3} = bar{Y}

hat{mu}_{3}

hat{mu}_{2}

hat{mu}_{1}