To prove: The Kruskal-Wallis statistic, B, as defined in Theorem 14.4.1 can also be written B = ∑ j = 1 K ( n − n j n ) − Z j 2

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Answer 1

Question

Chapter 14.4, Problem 8Q

To determine

To prove: The Kruskal-Wallis statistic, B, as defined in Theorem 14.4.1 can also be written

B=∑j=1K(n−njn)−Zj2

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Let X1, ...., Xn be a random sample from a population with θ unknown and given by the density f(x; θ) = ( 1 2θ √2 x e − √2 x θ if x > 0 0 if x ≤ 0 1. Show that E(X) = 2θ 2 and E( √2 X) = θ (Hint: you may use that R ∞ 0 e −z z α−1dz = (α − 1)! for every α ∈ N). 2. Show that the statistic θbn := 1 n Xn i=1 p2 Xi (1) is an unbiased estimator of θ. 3. Give the definition of a consistent estimator. 4. Show that the estimator θbn given in relation (1) is a consistent estimator of θ. 5. Show that the estimator θbn is a minimum variance estimator of θ. (Hint: use the Cramer-Rao inequality given by var(θb) ≥ 1 nE ∂ ln(f(X;θ) ∂θ 2

Answer 2

Question

Chapter 14.4, Problem 8Q

To determine

To prove: The Kruskal-Wallis statistic, B, as defined in Theorem 14.4.1 can also be written

B=∑j=1K(n−njn)−Zj2

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Answer 3

Question

Chapter 14.4, Problem 8Q

To determine

To prove: The Kruskal-Wallis statistic, B, as defined in Theorem 14.4.1 can also be written

B=∑j=1K(n−njn)−Zj2

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Answer 4

Question

Chapter 14.4, Problem 8Q

To determine

To prove: The Kruskal-Wallis statistic, B, as defined in Theorem 14.4.1 can also be written

B=∑j=1K(n−njn)−Zj2

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Answer 5

Chapter 14.4, Problem 8Q

To determine

To prove: The Kruskal-Wallis statistic, B, as defined in Theorem 14.4.1 can also be written

B=∑j=1K(n−njn)−Zj2

Answer 6

Chapter 14.4, Problem 8Q

To determine

To prove: The Kruskal-Wallis statistic, B, as defined in Theorem 14.4.1 can also be written

B=∑j=1K(n−njn)−Zj2

Answer 7

Chapter 14.4, Problem 8Q

To determine

To prove: The Kruskal-Wallis statistic, B, as defined in Theorem 14.4.1 can also be written

B=∑j=1K(n−njn)−Zj2

Answer 8

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Answer 9

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Answer 10

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Answer 11

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To prove: The Kruskal-Wallis statistic, B, as defined in Theorem 14.4.1 can also be written B = ∑ j = 1 K ( n − n j n ) − Z j 2

Solution Summary: The author explains the Kruskal-Wallis statistic, B, as defined in Theorem 14.4.1.

To prove: The Kruskal-Wallis statistic, B, as defined in Theorem 14.4.1 can also be written

B=∑j=1K(n−njn)−Zj2

Want to see the full answer?

Chapter 14 Solutions

To prove: The Kruskal-Wallis statistic, B, as defined in Theorem 14.4.1 can also be written B = ∑ j = 1 K ( n − n j n ) − Z j 2

Solution Summary: The author explains the Kruskal-Wallis statistic, B, as defined in Theorem 14.4.1.

To prove: The Kruskal-Wallis statistic, B, as defined in Theorem 14.4.1 can also be written B=∑j=1K(n−njn)−Zj2

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Chapter 14 Solutions

To prove: The Kruskal-Wallis statistic, B, as defined in Theorem 14.4.1 can also be written

B=∑j=1K(n−njn)−Zj2