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Chapter 6: Q12E (page 263)

Suppose a certain type of fertilizer has an expected yield per acre of \({{\rm{\mu }}_{\rm{2}}}\)with variance \({{\rm{\sigma }}^{\rm{2}}}\)whereas the expected yield for a second type of fertilizer is with the same variance \({{\rm{\sigma }}^{\rm{2}}}\).Let \({\rm{S}}_{\rm{1}}^{\rm{2}}\) and \({\rm{S}}_{\rm{2}}^{\rm{2}}\)denote the sample variances of yields based on sample sizes \({{\rm{n}}_{\rm{1}}}\)and \({{\rm{n}}_{\rm{2}}}\),respectively, of the two fertilizers. Show that the pooled (combined) estimator

\({{\rm{\hat \sigma }}^{\rm{2}}}{\rm{ = }}\frac{{\left( {{{\rm{n}}_{\rm{1}}}{\rm{ - 1}}} \right){\rm{S}}_{\rm{1}}^{\rm{2}}{\rm{ + }}\left( {{{\rm{n}}_{\rm{2}}}{\rm{ - 1}}} \right){\rm{S}}_{\rm{2}}^{\rm{2}}}}{{{{\rm{n}}_{\rm{1}}}{\rm{ + }}{{\rm{n}}_{\rm{2}}}{\rm{ - 2}}}}\)

is an unbiased estimator of \({{\rm{\sigma }}^{\rm{2}}}\)

Short Answer

Expert verified

It is unbiased.

Step by step solution

01

Concept introduction

An accurate statistic that is used to approximate a population parameter is known as an unbiased estimator. In this context, "accurate" means neither an overestimate nor an underestimate. If an overestimate or underestimate occurs, the difference's mean is referred to as a "bias."

02

Finding whether it is unbiased or not

The estimator is unbiased if the expected value of the supplied estimator is\({{\rm{\sigma }}^{\rm{2}}}\)As a result, the following is true:

\(\begin{array}{c}{\rm{E}}\left( {\frac{{\left( {{{\rm{n}}_{\rm{1}}}{\rm{ - 1}}} \right){\rm{S}}_{\rm{1}}^{\rm{2}}{\rm{ + }}\left( {{{\rm{n}}_{\rm{2}}}{\rm{ - 1}}} \right){\rm{S}}_{\rm{2}}^{\rm{2}}}}{{{{\rm{n}}_{\rm{1}}}{\rm{ + }}{{\rm{n}}_{\rm{2}}}{\rm{ - 2}}}}} \right)\\{\rm{ = }}\frac{{{{\rm{n}}_{\rm{1}}}{\rm{ - 1}}}}{{{{\rm{n}}_{\rm{1}}}{\rm{ + }}{{\rm{n}}_{\rm{2}}}{\rm{ - 2}}}}{\rm{E}}{\left( {{{\rm{S}}_{\rm{1}}}} \right)^{\rm{2}}}{\rm{ + }}\frac{{{{\rm{n}}_{\rm{2}}}{\rm{ - 1}}}}{{{{\rm{n}}_{\rm{1}}}{\rm{ + }}{{\rm{n}}_{\rm{2}}}{\rm{ - 2}}}}{\rm{E}}\left( {{\rm{S}}_{\rm{2}}^{\rm{2}}} \right)\\\mathop {\rm{ = }}\limits^{{\rm{(1)}}} \frac{{{{\rm{n}}_{\rm{1}}}{\rm{ - 1}}}}{{{{\rm{n}}_{\rm{1}}}{\rm{ + }}{{\rm{n}}_{\rm{2}}}{\rm{ - 2}}}}{{\rm{\sigma }}^{\rm{2}}}{\rm{ + }}\frac{{{{\rm{n}}_{\rm{2}}}{\rm{ - 1}}}}{{{{\rm{n}}_{\rm{1}}}{\rm{ + }}{{\rm{n}}_{\rm{2}}}{\rm{ - 2}}}}{{\rm{\sigma }}^{\rm{2}}}\\{\rm{ = }}{{\rm{\sigma }}^{\rm{2}}}\frac{{{{\rm{n}}_{\rm{1}}}{\rm{ + }}{{\rm{n}}_{\rm{2}}}{\rm{ - 2}}}}{{{{\rm{n}}_{\rm{1}}}{\rm{ + }}{{\rm{n}}_{\rm{2}}}{\rm{ - 2}}}}\\{\rm{ = }}{{\rm{\sigma }}^{\rm{2}}}{\rm{,}}\end{array}\)

(1): the estimators \({S_1}\)and \({S_2}\)are unbiased variance estimators,

Hence, it is unbiased.

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