Asked May 3, 2019

Give and explain one way to find uniformly most powerful test (UMP).


Expert Answer

Step 1

The UMP test:

While testing a null hypothesis, H0 against a composite alternative hypothesis, Ha, the test that is the most powerful against each of the simple alternatives contained in the composite alternative hypothesis, is called the UMP test or the uniformly most powerful test.

In other words, the UMP test has the highest power, (1 – β) [β = P (Type II error)] among all size α tests, that is, all tests at level of significance, α, for testing the particular null hypothesis H0 against the composite alternative hypothesis, Ha.

For a test of the parameter of interest, θ, when the composite alternative contains a set of values denoted by ω1, there may be 2 possible tests, that is, 2 possible sets of hypotheses in this case:

  • H0: θ = θ0 versus Ha: θ ϵ ω1 [Simple null versus composite alternative],
  • H0: θ ϵ ω0 versus Ha: θ ϵ ω1 [Composite null versus composite alternative].

Then, the UMP test of size α has the maximum power, say, β (θ1), among all size α tests, for each simple alternative θ1 ϵ ω1, defined in Ha.

The test function, which gives the rejection rule, when the rejection region is denoted by R, is given as:

Step 2

Neyman-Pearson Lemma:

According to Neyman-Pearson Lemma, the Likelihood Ratio Test (LRT) is the most powerful test for testing a simple null hypothesis against a simple alternative hypothesis.

The Likelihood Ratio Test:

The LRT statistic is the ratio of the supremum of likelihood function under the null model (i.e., under H0) to the supremum of likelihood function under the alternative model (i.e., under Ha).

Suppose X1, X2, …, Xn are the observations of a sample of size n. The likelihood function under the null model is denoted by L (θ | x; θ ϵ ω0) and that under the alternative model is denoted by L (θ | x; θ ϵ ω1). Then, the LRT statistic, λ is given as:

λ = sup. { L (θ | x; θ ϵ ω0)}/ sup. { L (θ | x; θ ϵ ω1)}.

The decision rule:

If λ (x) < c, reject H0. If λ (x) ≥ c, do not reject H0.

The test being a size α test, the P (Type I error) = α, that is, the probability of rejecting H0 when H0 is not false is α. The value of c can be obtained using this property as: P (λ (x) < c | H0) = α.

There may be one of the two relationships between λ (x) and T (x), where T (x) is a statistic or a function of X1, X2, …, Xn:

  • λ (x) is an increasing function of T (x), so that λ (x) < c implies and is implied by T (x) < k, such that P (T (x) < k | H0) = α.
  • λ (x) is a decreasing function of T (x), so that λ (x) < c implies...

Want to see the full answer?

See Solution

Check out a sample Q&A here.

Want to see this answer and more?

Solutions are written by subject experts who are available 24/7. Questions are typically answered within 1 hour.*

See Solution
*Response times may vary by subject and question.
Tagged in



Hypothesis Testing

Related Statistics Q&A

Find answers to questions asked by student like you
Show more Q&A

Q: Jobs and productivity! How do banks rate? One way to answer this question is to examine annual profi...

A: (a)Using the provided data, the sample mean can be calculated using Excel function “AVERAGE()”.Enter...


Q: I want you to come up with one study idea that would use a one sample chi square and one study that ...

A: The one sample Chi-square test is also called as Chi-square goodness of fit, which is used to decide...


Q: Do heavier cars really use more gasoline? Suppose a car is chosen at random. Let x be the weight of ...

A: Hello. Since your question has multiple sub-parts, we will solve the first three sub-parts for you. ...


Q: Suppose x has a distribution with μ = 60 and σ = 17. (a) If random samples of size n = 16 are select...

A: Note:Hi there! Thank you for posting the question. The Part (b) of your question asks about the “x d...


Q: Jane has stored three albums on her iPod. The first album consists of 11 songs by Frank Sinatra, the...

A: In this question, there is total 32 songs collection out of which 11 songs by Frank Sinatra, 13 song...


Q: 4 (continued). Currently, Melvin has equipped the Club of Ad Hominem. 20% of the time he swings the ...

A: The given information is not sufficient to solve all the subparts. More information is required to s...


Q: A hospital reported that the normal death rate for patients with extensive burns (more than 40% of s...

A: The provided information are:


Q: please answer the whole question  (3) Let x be per capita income in thousands of dollars. Let y be t...

A: Hi, since the problem posted by you contains multiple sub-parts, we are answering the first three su...


Q: Joe likes to go to an old arcade and play 12 games of Skeeball every day. On​ average, he wins a pri...

A: In this context, there are fixed number of trials. That is, n = 12. There are two possible outcomes ...