Consider the following alternative estimator for the population mean: $$\tilde { Y } = \frac { 1 } { n } \left( \frac { 1 } { 4 } Y _ { 1 } + \frac { 7 } { 4 } Y _ { 2 } + \frac { 1 } { 4 } Y _ { 3 } + \frac { 7 } { 4 } Y _ { 4 } + \ldots + \frac { 1 } { 4 } Y _ { n - 1 } + \frac { 7 } { 4 } Y _ { n } \right)$$
Prove that $\tilde { Y }$ is unbiased and consistent, but not efficient when compared to $\bar { Y }$.

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The Answer of Consider the following alternative estimator for the population mean: $$\tilde { Y } = \frac { 1 } { n } \left( \frac { 1 } { 4 } Y _ { 1 } + \frac { 7 } { 4 } Y _ { 2 } + \frac { 1 } { 4 } Y _ { 3 } + \frac { 7 } { 4 } Y _ { 4 } + \ldots + \frac { 1 } { 4 } Y _ { n - 1 } + \frac { 7 } { 4 } Y _ { n } \right)$$
Prove that $\tilde { Y }$ is unbiased and consistent, but not efficient when compared to $\bar { Y }$.

Consider the Following Alternative Estimator for the Population Mean Prove That