4. We conduct an experiment where there are only four possible outcomes:A, B, C,

Question

4. We conduct an experiment where there are only four possible outcomes:A, B, C, or D. There 0,1,2, or 3 respectively. are four possible distributions on these outcomes corresponding to These distributions are 0 A 0.25 0.5 0.12 0.8 B0.25 0.250.13 0.1 C 0.25 0.13 0.25 0.05 D 0.250.12 0.5 0.05 I want a test that decides between the null hypothesis ?-0 versus the alternative ? in other words, the alternative that ? is either 1, 2, or 3). 0 (or, (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 ? = 1, 2, or 3. (b) Find the LRT for testing Ho : ? 0 versus HA : ??0 with level a (c) Is it possible for the test with critical region {B to be more powerful than the LRT? 0.25. 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

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