Please provide complete and detail solutions. Thank you! We conduct an experimen

Question

Please provide complete and detail solutions. Thank you!

We conduct an experiment where there are only four possible outcomes: A, B, C, or D. There are four possible distributions on these outcomes corresponding to theta = 0, 1, 2, or 3 respectively. These distributions are I want a test that decides between the null hypothesis theta = 0 versus the alternative theta = 0 (or, in other words, the alternative that theta is either 1, 2, or 3). (a) Consider the test that has the critical region {B}. Calculate the level of this test, and calculate the power under each of the alternatives theta = 1, 2, or 3. (b) Find the LRT for testing H_0: theta = 0 versus H_A: theta = 0 with level alpha = 0.25. (c) Is it possible for the test with critical region {B} to be more powerful than the LRT? Explain your reasoning.

Explanation / Answer

(a) Level of the test just means that we need to find the significance level of the test i.e. value of or type I error in the queston, it is stated that the rejection region is {B} so by the defination of = pr{ rej. H0 when it is true} we can see that this probability comes out to be 0.25. Now the power of test can be given by 1- Pr. [Type II error] the defination of a type II error is = Pr{ accepting H0 when it is false}

(b) the likelihood of ratio test tries to find the best possible test procedure to increase the size of / 1- so lets try to find the test procedure given the significance level equal to 0.25

the value of which signifies the ratio of Max H0/Max H1

g(X=A ; =1,2 OR,3)= 1.42        f(X=A ; =0) = .25          = .176         

g(X=B ; =1,2 OR,3)= .48          f(X=B ; =0) = .25          = .52

g(X=C ; =1,2 OR,3)= .43         f(X=C ; =0) = .25           = .52

g(X=D ; =1,2 OR,3)= .67         f(X=D ; =0) = .25   = .377

use this table and compare it to the given value of =0.25 only X=A value corresponding in the table comes out to be less than or equal to so the new critical region will be A.

(c) the test procedure derived using LRT is having rejection region = {A} in our case the probability of a type II error is less than what was derived by using rejection region {B}

1 - 1 .25 .75 2 .13 .87 3 .10 .90

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.