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*9.29 Let Y₁, Y₂,..., Y, denote a random sample of size n from a power family distribution (see Exercise 6.17). Then the methods of Section 6.7 imply that Y(n) = max(Y₁, Y₂, ..., Y₁) has the distribution function given by 0, y < 0, F(n) (y) (y/0), 0≤ y ≤0, - fre 1, y > 0. Use the method described in Exercise 9.26 to show that Y() is a consistent estimator of 0.

*9.29 Let Y₁, Y₂,..., Y, denote a random sample of size n from a power family distribution (see Exercise 6.17). Then the methods of Section 6.7 imply that Y(n) = max(Y₁, Y₂, ..., Y₁) has the distribution function given by 0, y < 0, F(n) (y) (y/0), 0≤ y ≤0, - fre 1, y > 0. Use the method described in Exercise 9.26 to show that Y() is a consistent estimator of 0.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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Answer 9.29 

 

*9.29 Let Y₁, Y₂,..., Y,, denote a random sample of size n from a power family distribution (see Exercise 6.17). Then the methods of Section 6.7 imply that Y(n) = max(Y₁, Y2, ..., Yn) has the distribution function given by y < 0, /yan F(n) (y) (y/0)an, 1, - 0 ≤ y ≤ 0, y > 0. Use the method described in Exercise 9.26 to show that Y() is a consistent estimator of 0. 0,
Transcribed Image Text:*9.29 Let Y₁, Y₂,..., Y,, denote a random sample of size n from a power family distribution (see Exercise 6.17). Then the methods of Section 6.7 imply that Y(n) = max(Y₁, Y2, ..., Yn) has the distribution function given by y < 0, /yan F(n) (y) (y/0)an, 1, - 0 ≤ y ≤ 0, y > 0. Use the method described in Exercise 9.26 to show that Y() is a consistent estimator of 0. 0,
*9.26 It is sometimes relatively easy to establish consistency or lack of consistency by appeal- ing directly to Definition 9.2, evaluating P(|ên − 0| ≤ ɛ) directly, and then showing that lim,→ P(|ên - 0 ≤ 8) = 1. Let Y₁, Y2, ..., Y₁ denote a random sample of size n from a uniform distribution on the interval (0, 0). If Y(n) = max(Y₁, Y₂, ..., Yn), we showed in Exercise 6.74 that the probability distribution function of Y(n) is given by y < 0, 0, F(n) (y) (y/0)", 0≤ y ≤0, y > 0. - fover. 1, a For each n ≥ 1 and every & > 0, it follows that P(|Y(n) − 0| ≤ ɛ) = P(0 − & ≤ Y(n) ≤ 0 + ε). If e > 0, verify that P(0 - & ≤ Y(n) ≤ 0 + ε) = 1 and that, for every positive & < 0, we obtain P(0 - & ≤Y(n) ≤ 0 + ε) = 1 - [(0 - €)/0]". b Using the result from part (a), show that Y() is a consistent estimator for by showing that, for every & > 0, lim→∞ P(|Y(n) - 0 ≤ 8) = 1.
Transcribed Image Text:*9.26 It is sometimes relatively easy to establish consistency or lack of consistency by appeal- ing directly to Definition 9.2, evaluating P(|ên − 0| ≤ ɛ) directly, and then showing that lim,→ P(|ên - 0 ≤ 8) = 1. Let Y₁, Y2, ..., Y₁ denote a random sample of size n from a uniform distribution on the interval (0, 0). If Y(n) = max(Y₁, Y₂, ..., Yn), we showed in Exercise 6.74 that the probability distribution function of Y(n) is given by y < 0, 0, F(n) (y) (y/0)", 0≤ y ≤0, y > 0. - fover. 1, a For each n ≥ 1 and every & > 0, it follows that P(|Y(n) − 0| ≤ ɛ) = P(0 − & ≤ Y(n) ≤ 0 + ε). If e > 0, verify that P(0 - & ≤ Y(n) ≤ 0 + ε) = 1 and that, for every positive & < 0, we obtain P(0 - & ≤Y(n) ≤ 0 + ε) = 1 - [(0 - €)/0]". b Using the result from part (a), show that Y() is a consistent estimator for by showing that, for every & > 0, lim→∞ P(|Y(n) - 0 ≤ 8) = 1.
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