Consider the Following Alternative Estimator for the Population Mean Prove That

question 52

Essay

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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Definitions:

Front-end Load

A fee paid by an investor at the time of entering a fund, such as a mutual fund, which is typically used to cover the fund's sales and marketing costs.

Expense Ratio

A measure of the total cost of managing and operating a fund, including administrative and management fees, expressed as a percentage of the fund's total assets.

Mutual Fund

This is a fund that amasses capital from numerous investors to make investments in different securities like stocks, bonds, and money market assets.

Turnover Ratio

A financial metric that measures the rate at which an investment or a portfolio of investments is bought and sold over a particular period.