Explanation Calculation: Linear regression model: A linear regression model is given as y = b 0 + b 1 x where y ^ be the predicted values of response variable and x be the predictor variable. The estimated least square line is y ^ = β ^ 0 + β ^ 1 x where y ^ be the predicted values of response variable and x be the predictor variable. The β ^ 1 be the estimate of slope and β ^ 0 be the estimate of intercept of the line. Least squares estimate of intercept β ^ 0 : By the principle of estimation of least squares, the value of b 0 which minimizes the error function f ( b 0 , b 1 ) = ∑ i = 1 n ( y i − ( b 0 + b 1 x i ) ) 2 is, β ^ 0 = y ¯ − β ^ 1 x ¯ . Here, β ^ 1 = ∑ i ( x i − x ¯ ) ( y i − y ¯ ) ∑ i ( x i − x ¯ ) 2 is the least square estimate of slope coefficient. Unbiasedness: In general unbiasedness means having no bias. That is, the bias of the particular estimator must be “0”. A sample statistic is an unbiased estimator of the population parameter if the expected value of the statistic is equal to the population parameter. If the average of all possible sample statistics (of a given sample size n) is equal to the population parameter then the sample is said to have unbiasedness. From the properties of the least square estimates, β ^ 1 is an unbiased estimator β 1 . That is, E ( β ^ 1 ) = β 1 . Where β ^ 1 be the slope of the sample regression line and β 1 is the slope of the population regression line. To prove that β ^ 0 is an unbiased estimator of β 0 , it is enough to prove that E ( β ^ 0 ) = β 0 . Where β ^ 0 be the intercept of the sample regression line and β 0 is the intercept of the population regression line. Expectation of a linear function of random variables: The expectation of a linear function, f ( x , y ) , of 2 random variables X and Y , having joint pmf p ( x , y ) is, E ( f ( X , Y ) ) = ∑ ∑ ( f ( x , y ) ) p ( x , y )

PROBABILITY & STATS FOR ENGINEERING &SCI

9th Edition

ISBN: 9781285099804

Author: DEVORE

Publisher: CENGAGE L

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Chapter 12.3, Problem 40E

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Prove that β^0 is an unbiased estimator of β0.

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Chapter 12.3, Problem 39E

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Chapter 12 Solutions

PROBABILITY & STATS FOR ENGINEERING &SCI

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