Likelihood Ratio Tests For Two Simple Hypotheses

Question 1

What is the difference between a simple hypothesis and a composite one?

Answer 1

A hypothesis H is simple if it completely specifies the probability distribution of the data X, otherwise it is simple.

Question 2

What are the two mistakes that the likelihood ratio test can make and with what probabilities.

Answer 2

Question 3

Given H₀ and H₁ what are the probability distributions of X?

Answer 3

• H₀ - f₀(x)
• H1 - f1​(x)

Question 4

The likelihood ratio test accepts H₀ or H₁ based on the value of what?

Answer 4

T(x) which is the test statistic

Question 5

Define a likelihood ratio (LR).

Answer 5

The likelihood ratio (LR) for the sample x is

Question 6

What is the basic form of the likelihood ratio test?

Answer 6

The likelihood ratio tests accepts H₁ if LR(x) ≧ c, otherwise we accept H₀. Where c is a value we can choose, changng c changes the values of α₀ and α₁.

Question 7

Question 8

Prove the following Lemma

Answer 8

Question 9

What is H₀ often called?

Answer 9

The null hypothesis

Question 10

What is H₁ often called?

Answer 10

The alternative hypothesis

Question 11

What is the significance level of the likelihood ratio test?

Answer 11

α(δ) is the probability that the test rejects H₀ given that H₀ is true

Question 12

What is the power of the likelihood ratio test?

Answer 12

π(δ) is the probability that the test rejects H₀ given that H₁ is true

Question 13

What does α(δ) equal in terms of α₀(δ) and α₁(δ)?

Answer 13

α(δ) = α₀(δ)

Question 14

What does π(δ) equal in terms of α0(δ) and α1(δ)?

Answer 14

π(δ) = 1 - α₁(δ)

Question 15

What is the Neyman Pearson (NP) Lemma?

Answer 15

Question 16

Prove the Neyman Pearson (NP) Lemma.

Answer 16

Need to do.

Question 17

Often we fix the significance level and the find the test with?

Answer 17

The maximum power for this significance level - which must be the corresponding likelihood ratio test.