Show that the approximation to the sample covariance N-1 2 xy E (T; – m2) (yi – my) N i=0 obtained from .2 TU - Ti
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- Consider the model Y = B1 + B2X + e, which we estimate using a random sample with 12 observations. Let bj and b2 be the estimators for B1 and B2 and recall Eê, where ê; = Y; – bị – b2X;. Suppose the sample correlation between {X; }"_-1 and {Y;}"_, is 0.5 and E(Y; – Ỹ„)² = 100. What is ổ?? %3D n-2 Hint: (i) for simple regression, the regression R? is equal to the squared sample correlation between X and Y. (ii) R² = 1 – SSE where SSE = Eế and SST SST = O a. 7.5 O b. 8.5 О с. 9 O d. 10 O e. 8 O f. 7 Clear my choiceSuppose that @, and ô, are unbiased estimators of the parameter 0 and that V@,) = 15 and V(@2) = 4. What is the relative efficiency of the two estimators?EX7.8) Let Y be a random variable having a uniform normal distribution such that Y U(2,5) 2 Find the variance of random variable Y.
- 3. Let X N(u,0²). Find the MSE of the (uncorrected) sample variance 152 as an estimator for ² and compare with the MSE of the (corrected) sample variance S².Two independent samples have 28 and 19 pairs of observations with correlation co efficients 0.55 and 0.75 respectively. Are these values of r consistent with the hypothesis that both the samples are drawn from the same population?6. Show that 1 s2 E1(Xi – x)² is unbiased estimator of the population variance o? i=1 п-1
- 2) Let G and H be two independent unbiased estimators of θ. Assume that the variance of G is two times the variance of H. Find the constants a and b so that aG + bH is an unbiased estimator with the smallest possible variance for such a linear combination.A computer while calculating correlation coefficient between two variables X and Y from 25pairs of observation obtained the following results: n=25, 2X=125, EX2 = 650, Y=100,Y2= 460, 2XY=508. But on subsequent verification it was found that he had copied downtwo pairs as (6,14) and (8,6) while the correct values were (8,12) and (6,8). Obtain the correctvalue of correlation coefficient?Suppose the conditional mean function is Y = Bo + BiX + B2X² + B3X3 + U %3D where E[U]X]3D0. By mistake, a researcher omitted X^2 and X^3 terms in the regression and ran regression of Y on X and an intercept only. It turns out that covariance between X and B2X2 + B3X° is nonzero. Is the OLS estimator researcher computed consistent for beta 1? Yes, since the omitted variables are just functions of X. Yes, since the covariance between X and B2X2 + B3X is nonzero No, since the covariance between X and B2X4 + B3X° is nonzero No, since the omitted variables are negligibly small.
- B. A doctor wishes to see whether the pulse rates of smokers are higher than the pulse rates of non-smokers. Sample of 102 smokers and 100 non smokers are selected. The results are given as the average of smokers and non smokers are 89 and 86 respectively and the population variances of smokers and non smokers are as 25 and 36 respectively. Can the doctor conclude, at α = 0.01, that smokers have higher pulse rates than nonsmokers ?Suppose that X and Y are unknowns with E(X) = 10, Var (X) = 4, E(Y)=12 and Var (Y) = 100. In addition, suppose that the correlation coefficient of X and Y is .6. Then what is the variance of X-Y?B) Let X1,X2, .,Xn be a random sample from a N(u, o2) population with both parameters unknown. Consider the two estimators S2 and ô? for o? where S2 is the sample variance, i.e. s2 =E,(X, – X)² and ở² = 'E".,(X1 – X)². [X = =E-, X, is the sample mean]. %3D n-1 Li%3D1 [Hint: a2 (п-1)52 -~x~-1 which has mean (n-1) and variance 2(n-1)] i) Show that S2 is unbiased for o2. Find variance of S2. ii) Find the bias of 62 and the variance of ô2. iii) Show that Mean Square Error (MSE) of ô2 is smaller than MSE of S?. iv) Show that both S2 and ô? are consistent estimators for o?.