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### HW7

Course: STA 244, Spring 2008
School: Duke
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Word Count: 368

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7 Due STA244 4/7/2003 Homework 4/14/2001 1. The matrix X(i) X(i) can be written as X(i) X(i) = X X - xi xi where xi is the ith row of X and X(i) is the matrix X with the ith row removed. Use this to show (X(i) X(i) )-1 = (X X)-1 + (X X)-1 xi xi (X X)-1 1 - hii (1) ^ 2. The quantity Yi - xi (i) is the residual for the ith case when is estimated without the ith case. Show that ei ^ ^ (2) Yi - xi (i) = 1 - hii 3....

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7 Due STA244 4/7/2003 Homework 4/14/2001 1. The matrix X(i) X(i) can be written as X(i) X(i) = X X - xi xi where xi is the ith row of X and X(i) is the matrix X with the ith row removed. Use this to show (X(i) X(i) )-1 = (X X)-1 + (X X)-1 xi xi (X X)-1 1 - hii (1) ^ 2. The quantity Yi - xi (i) is the residual for the ith case when is estimated without the ith case. Show that ei ^ ^ (2) Yi - xi (i) = 1 - hii 3. Show that Cook's distance (page 365) can be written as 1 2 hii Di = r i p 1 - hii where ri = ei / 2 (1 - hii ) is the internally Studentized residual. ^ ^ 4. The data set in http://www.stat.duke.edu/courses/Spring03/sta244/Data/rat.dat contains results from an experiment conducted to investigate the amount of drug retained in the liver of a rat. Nineteen rats were randomly selected, weighed, placed under light anesthesia and given an oral dose of the drug. The dose an animal received was determined as approximately 40 mg of the drug per kilogram of body weight, since the liver weight is known to be strongly to related body weight and it was felt that large livers would absorb more of a given drug than smaller livers. After a fixed period of time, each rat was sacrificed, and the liver weighed, and the percent dose (Y) in the liver determined. The researchers' hypothesis was that for the method of determining the dose given to each rat, there there is no relationship between the percentage of the dose in the liver (Y) and the body weight in grams (Body), liver weight in grams (Liver), and relative dose (Dose). Fit statistical models to the data to examine their hypothesis. Write a 1 page (typed) report describing your analysis and conclusions, with 1-2 pages max of supporting figures and tables. Any figures/tables should be labeled and referenced within the body of the report. 5. Extra Credit: Write a function to implement the Bayesian outlier Diagnostics in R or another language. Verify that it works correctly with the stackloss data. http://www.stat.duke.edu/courses/Spring03/sta244/Data/stackloss.txt (3)
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Duke - STA - 244
STA2441/15/2003Homework 2Due 1/22/2003 1. Write the following two way analysis of variance (AOV) model with interactions Yijk = + i + j + ij + with i = 1, 2, 3, j = 1, 2, k = 1, 2 in matrix notation. 2. Suppose we have a k k matrix S partition
Duke - STA - 244
STA2442/28/2005Homework 5Due 3/7/2001 1. For a random vector n , is called exchangeable if has the same distribution as any permutation of the vector . If is exchangeable, prove that E( ) = 1 ( ), and that the Cov( ) = has the forma a b .
Duke - STA - 244
STA2441/15/2001Homework 1Due 1/22/20011. Assume that we have a sample of size n where Y i = 0 + 1 Xi + e i and the errors ei are iid N (0, 2 ). (a) Find the maximum likelihood estimator of 2 , 2 . Hint: let = 2 and maximize. ^ (b) Under
Duke - STA - 103
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Duke - STA - 244
STA2442/5/2005Homework 3Due 2/12/2001 1. Recall from class that a non-central 2 (m, ) can be represented as a Poisson mixture of central 2 random variables, where Y P (/2) and X|Y 2 (m + 2y, 0). Find the mean and variance of a non-central Chi-
Duke - STA - 103
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Duke - STA - 103
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Duke - STA - 103
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Duke - STA - 244
STA2444/9/2002Homework 7Due 4/16/2001 1. Problem 15.7 in CW. To obtain case diagnostics in S-Plus, fit a model using the QR option, i.e. mylm.obj &lt;- lm(Y X1 + X2, data=mydataframe, qr=T) To obtain the case diagnostics, use the function ls.diag(
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Duke - STA - 103
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Duke - STA - 103
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Duke - STA - 244
STA2443/19/2005Homework 6Due 3/28/2002 1. For the usual linear model Y N (X, -1 In ) with prior distributions N (bo , Vo ) independent of and p() 1/: (a) Find the posterior distribution of |. (b) Can you find a closed form expression for th
Duke - STA - 102
ill sandwich &quot;Yes&quot; &quot;Yes&quot; &quot;Yes&quot; &quot;Yes&quot; &quot;Yes&quot; &quot;Yes&quot; &quot;Yes&quot; &quot;Yes&quot; &quot;Yes&quot; &quot;Yes&quot; &quot;Yes&quot; &quot;Yes&quot; &quot;Yes&quot; &quot;Yes&quot; &quot;Yes&quot; &quot;Yes&quot; &quot;Yes&quot; &quot;Yes&quot;
Duke - STA - 102
exposure nma &quot;high&quot; 28 &quot;high&quot; 35 &quot;high&quot; 37 &quot;high&quot; 37 &quot;high&quot; 43.5 &quot;high&quot; 44 &quot;high&quot; 45.5 &quot;high&quot; 46 &quot;high&quot; 48 &quot;high&quot; 48.
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subject fev1 gender 1 2.30 0 2 2.15 1 3 3.50 1 4 2.60 0 5 2.75 0 6 2.82 1 7 4.05 1 8 2.25 1 9 2.68 0
Duke - STA - 244
x1 x2 y1 y2 y3 y4 10 8 8.04 9.14 7.46 6.58 8 8 6.95 8.14 6.77 5.76 13 8 7.58 8.74 12.74 7.71 9 8 8.81 8.77 7.11 8.84 11 8 8.33 9.26 7.81 8.47 14 8 9.96 8.1 8.84 7.04 6 8 7.24 6.13 6.08 5.25 4 19 4.26 3.1 5.39 12.5 12 8 10.84 9.13 8.15 5.56
Duke - STA - 244
D m S WS y 0 10 1 3408 623 0.04 5 1 206.8 680.2 0.1 5 1 1841.2 721.4 0.16 5 1 1223.2 750.4 0.28 5 1 861.2 789.4 0.04 5 2 2810.8 672.2 0.1 5 2 860.8 709.2 0.16 5 2 592.8 731.2 0.28 5 2 2642.8 778.2 0.04 5 3 2399.2 668.4 0.1 5 3 327.2 715.6
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FUEL/POP INC LIC/POP POP TAX VEH/POP VM/VEH 644.147 14.826 0.70923 4041 13 0.911408 11.0684 474.545 21.761 0.549091 550 8 0.669091 10.5625 552.524 16.297 0.660573 3665 18 0.777899 12.2119 683.539 14.218 0.735857 2351 18.7 0.615908 14.0981 501.34
Duke - STA - 244
Pressure Temp 20.79 194.5 20.79 194.3 22.4 197.9 22.67 198.4 23.15 199.4 23.35 199.9 23.89 200.9 23.99 201.1 24.02 201.4 24.01 201.3 25.14 203.6 26.57 204.6 28.49 209.5 27.76 208.6 29.04 210.7 29.88 211.9 30.06 212.2
Duke - STA - 244
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STA2444/18/2002Homework 8Due 4/26/2001 Refer to Exercise 11.5 in CW (page 285). Use any appropriate methods covered in class to answer the problem (Bayesian, Frequentist, or compare both). Provide a typed solution describing the problem and how
Duke - STA - 244
STA2442/14/2005Homework 4Due 2/21/2001 1. Consider the linear model Y = X 1 1 + X 2 2 + where X1 is n q and X2 is n (p q), with both matrices of full column rank. Consider the problem of testing N H : 1 = 0. Assume that N (0, 2 In ). (a) Gi
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U X1 X2 Y 0.493151 1 1 0.872302 1.40245 2 1 1.59988 2.31175 3 1 2.4019 3.22104 4 1 3.25942 4.13034 5 1 4.14616 5.03964 6 1 5.04607 5.94894 7 1 5.95154 6.85823 8 1 6.85928 7.76753 9 1 7.76795 8.67683 10 1 8.677 0.0770038 1 2 0.557762 0.986
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Homework 9a SolutionsAs yi iid Bernoullip, then E yi = = p. Setting this expression to its respective sample P y =1 ^ y moment, we obtain: = n , or p = n ^ n! y n,y where K = 8.8 c. For the Binomial experiment, the likelihood function is L = K p
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