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3. A bivariate population of (X,Y) is sampled independently on three occasions. On the first, a random sample of size no is taken and only T = min{X, Y} is observed for each pair. On the second, a random sample of size në is taken, and only the X-marginal is observed for each pair. Finally, a random sample of size në is taken, and only the Y-marginal is observed for each pair. Therefore, the combined set of observations is of the form (T, X, Y), where T = (T₁,..., Tno), X = (X₁,..., Xn₁) and Y = (Y₁,..., Yn₂). Assume the following two-parameter probability = model for (X, Y): P(X > x, Y > y) = exp [-17 (121¹/8 +21²/616] - x >0, y > 0, 0 > 0, 0 < 6 ≤ 1 with unknown parameters 0 and 8. (a) Present the joint pdf of (T, X, Y). (b) Is the above family an exponential family? Justify your answer.

3. A bivariate population of (X,Y) is sampled independently on three occasions. On the first, a random sample of size no is taken and only T = min{X, Y} is observed for each pair. On the second, a random sample of size në is taken, and only the X-marginal is observed for each pair. Finally, a random sample of size në is taken, and only the Y-marginal is observed for each pair. Therefore, the combined set of observations is of the form (T, X, Y), where T = (T₁,..., Tno), X = (X₁,..., Xn₁) and Y = (Y₁,..., Yn₂). Assume the following two-parameter probability = model for (X, Y): P(X > x, Y > y) = exp [-17 (121¹/8 +21²/616] - x >0, y > 0, 0 > 0, 0 < 6 ≤ 1 with unknown parameters 0 and 8. (a) Present the joint pdf of (T, X, Y). (b) Is the above family an exponential family? Justify your answer.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.3: Least Squares Approximation
Problem 31EQ
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3. A bivariate population of (X,Y) is sampled independently on three occasions. On the first, a random sample of size no is taken and only T = min{X,Y} is observed for each pair. On the second, a random sample of size nį is taken, and only the X-marginal is observed for each pair. Finally, a random sample of size n2 is taken, and only the Y-marginal is observed for each pair. Therefore, the combined set of observations is of the form (T, X, Y), where T = X = (X1,..., Xn1) and Y model for (X, Y): (T1,...,Tno), (Y1, ..., Yn2). Assume the following two-parameter probability P(X > x,Y > y) = exp x > 0, y > 0, 0 > 0, 0 < 8 < 1 with unknown parameters 0 and 8. (a) Present the joint pdf of (T, X, Y). (b) Is the above family an exponential family? Justify your answer.
Transcribed Image Text:3. A bivariate population of (X,Y) is sampled independently on three occasions. On the first, a random sample of size no is taken and only T = min{X,Y} is observed for each pair. On the second, a random sample of size nį is taken, and only the X-marginal is observed for each pair. Finally, a random sample of size n2 is taken, and only the Y-marginal is observed for each pair. Therefore, the combined set of observations is of the form (T, X, Y), where T = X = (X1,..., Xn1) and Y model for (X, Y): (T1,...,Tno), (Y1, ..., Yn2). Assume the following two-parameter probability P(X > x,Y > y) = exp x > 0, y > 0, 0 > 0, 0 < 8 < 1 with unknown parameters 0 and 8. (a) Present the joint pdf of (T, X, Y). (b) Is the above family an exponential family? Justify your answer.
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