Properties of Estimators

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Osborne Harrell
5 years ago
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1 Statistical properties of â and ˆb Mean and variance of ˆb Properties of Estimators E(ˆb) = b var(ˆb) = σ2 Recall that = n (x i x) 2 i= Distribution of ˆb ˆb N (b, σ 2 ) Mean and variance of â E(â) = a ( var(â) = n + Distribution of â ( ( â N a, n + x2 x2 ) σ 2 )σ 2 ) Inference for Regression, Mar,

2 Note that ˆb ( σ N b, 2 ). Thus ˆb b σ/ N (0, ) Substituting for σ, we obtain ˆb b / t n 2 Confidence Intervals ( α) confidence interval for b: ˆb ± tn 2,a/2 SXX Similarly â a N (0, ) σ n + X 2 Substituting for σ, we obtain â a n + x2 t n 2 ( α) confidence interval for a: â ± t n 2,α/2 n + x2 Inference for Regression, Mar,

3 Tests on the Coefficients Question: Is b equal to some value b 0? The correspoding test problem is H 0 : b = b 0 versus H a : b b 0. The test statistic is given by T b = ˆb b 0 / t n 2 The null hypothesis H 0 : b = b 0 is rejected if T > t n 2,α/2 Question: Is a equal to some value a 0? The correspoding test problem is H 0 : a = a 0 versus H a : a a 0. The test statistic is given by T a = â a 0 n + x2 t n 2 The null hypothesis H 0 : a = a 0 is rejected if T > t n 2,α/2 Inference for Regression, Mar,

4 Example: Body density Inference for the Coefficients The confidence interval for b is given by ˆb ± tn 2,α/2 SXX =.4 ± = [ 2.92, 9.90] The confidence interval for a is given by â ± t n 2,α/2 n + x2 Furthermore we find for T b = = 3.7 ± ˆb / = 5.22 > t 90,0.025 = = [2., 5.30] Thus we reject H 0 : b = 0 at significance level 0.05: The coefficient b is statistically significantly different from zero. Similarly T a = â n + x2 = 7.26 > t 90,0.025 =.99 Thus we reject H 0 : a = 0 at significance level 0.05: The coefficient a is statistically significantly different from zero. The corresponding P -values are P( T a 5.22) 0 P( T b 7.26) 0 Inference for Regression, Mar,

5 Estimating the Mean In the linear regression model, the mean of Y at x = x 0 is given by E(Y ) = a + b x 0 Our estimate for the mean of Y at X = x 0 is Ŷ x0 = â + ˆb x 0. Question: How precise is thistimate? Note that Ŷ x0 = â + ˆb x 0 = Ȳ ˆb(x 0 x). Hence we obtain E(Ŷx 0 ) = a + b x 0 ( var(ŷx 0 ) = n + (x 0 x) 2 ) σ 2 ( α) confidence interval for E(Y x0 ) (â + ˆb x 0 ) ± t n 2,α/2 n + (x 0 x) 2 Inference for Regression, Mar,

6 Estimating the Mean Example: Body density Suppose the measured skin thickness is x 0 =. mm. What is the mean body density for this value of skin thickness? Point estimate: Ŷ x0 = â + hb x 0 = =.59 The mean body density is kg/m 3. Confidence interval: (â + ˆb x 0 ) ± t n 2,α/2 n + (x 0 x) 2 = (3.7.4.) ± = [.09,.22] (..06) In STATA, the standard error for estimating the mean of Y is calculated by passing the option stdp to predict:. predict BDH. predict SE, stdp. generate low=bdh-invttail(49,.025)*se. generate high=bdh+invttail(49,.025)*se. sort SKINT. graph twoway line low high BDH SKINT, clpattern(dash dash solid) clcolor(black bla > ck black) scatter BODYD SKINT, legend(off) scheme(scolor) SKINT Inference for Regression, Mar,

7 Prediction Suppose we want to predict Y at x = x 0. Aim: ( α) confidence interval for Y Note that â + ˆb x 0 Y N (0, σ ( 2 + n + (x 0 X) 2 )) Thus the desired ( α) confidence interval for Y x0 â + ˆb x 0 ± t n 2,α/2 + n + (x 0 X) 2 is given by Inference for Regression, Mar,

8 Prediction Example: Body density Suppose the measured skin thickness is x 0 =. mm. What is the predicted body density for this value of skin thickness? Point estimate: Ŷx 0 = â + hb x 0 = =.59 The predicted body density is kg/m 3. Confidence interval: (â + ˆb x 0 ) ± t n 2,α/2 + n + (x 0 x) 2 = (3.7.4.) ± = [0.92,.40] (..06) In STATA, the standard error for predicting Y is calculated by passing the option stdf to predict:. drop SE low high. predict SE, stdf. generate low=tbillh-invttail(49,.025)*se. generate high=tbillh+invttail(49,.025)*se. graph twoway line low high BDH SKINT, clpattern(dash dash solid) clcolor(black bla > ck black) scatter BODYD SKINT, legend(off) scheme(scolor) Alternatively, we can use the following command:. twoway (lfitci BODYD SKINT, range(.) stdf) (scatter BODYD SKINT), > xtitle(skin thickness) ytitle(body density) scheme(scolor) legend(off) Body density SKINT SKin thickness Inference for Regression, Mar,

Review: General Approach to Hypothesis Testing. 1. Define the research question and formulate the appropriate null and alternative hypotheses.

Review: General Approach to Hypothesis Testing. 1. Define the research question and formulate the appropriate null and alternative hypotheses. 1 Review: Let X 1, X,..., X n denote n independent random variables sampled from some distribution might not be normal!) with mean µ) and standard deviation σ). Then X µ σ n In other words, X is approximately