composite hypothesis statistics definition

Simple Hypothesis and Composite Hypothesis

A simple hypothesis is one in which all parameters of the distribution are specified. For example, the heights of college students are normally distributed with $${\sigma ^2} = 4$$, and the hypothesis that its mean $$\mu $$ is, say, $$62”$$; that is, $${H_o}:\mu = 62$$. So we have stated a simple hypothesis, as the mean and variance together specify a normal distribution completely. A simple hypothesis, in general, states that $$\theta = {\theta _o}$$ where $${\theta _o}$$ is the specified value of a parameter $$\theta $$, ($$\theta $$ may represent $$\mu ,p,{\mu _1} – {\mu _2}$$ etc).

A hypothesis which is not simple (i.e. in which not all of the parameters are specified) is called a composite hypothesis. For instance, if we hypothesize that $${H_o}:\mu > 62$$ (and $${\sigma ^2} = 4$$) or$${H_o}:\mu = 62$$ and $${\sigma ^2} < 4$$, the hypothesis becomes a composite hypothesis because we cannot know the exact distribution of the population in either case. Obviously, the parameters $$\mu > 62”$$ and$${\sigma ^2} < 4$$ have more than one value and no specified values are being assigned. The general form of a composite hypothesis is $$\theta \leqslant {\theta _o}$$ or $$\theta \geqslant {\theta _o}$$; that is, the parameter $$\theta $$ does not exceed or does not fall short of a specified value $${\theta _o}$$. The concept of simple and composite hypotheses applies to both the null hypothesis and alternative hypothesis.

Hypotheses may also be classified as exact and inexact. A hypothesis is said to be an exact hypothesis if it selects a unique value for the parameter, such as $${H_o}:\mu = 62$$ or $$p > 0.5$$. A hypothesis is called an inexact hypothesis when it indicates more than one possible value for the parameter, such as $${H_o}:\mu \ne 62$$ or $${H_o}:p = 62$$. A simple hypothesis must be exact while an exact hypothesis is not necessarily a simple hypothesis. An inexact hypothesis is a composite hypothesis.

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Definition: Simple and composite hypothesis

Definition: Let $H$ be a statistical hypothesis . Then,

$H$ is called a simple hypothesis, if it completely specifies the population distribution; in this case, the sampling distribution of the test statistic is a function of sample size alone.

$H$ is called a composite hypothesis, if it does not completely specify the population distribution; for example, the hypothesis may only specify one parameter of the distribution and leave others unspecified.

• Wikipedia (2021): "Exclusion of the null hypothesis" ; in: Wikipedia, the free encyclopedia , retrieved on 2021-03-19 ; URL: https://en.wikipedia.org/wiki/Exclusion_of_the_null_hypothesis#Terminology .

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Lesson 10 of 24 By Avijeet Biswal

Table of Contents

In today’s data-driven world, decisions are based on data all the time. Hypothesis plays a crucial role in that process, whether it may be making business decisions, in the health sector, academia, or in quality improvement. Without hypothesis & hypothesis tests, you risk drawing the wrong conclusions and making bad decisions. In this tutorial, you will look at Hypothesis Testing in Statistics.

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What Is Hypothesis Testing in Statistics?

Hypothesis Testing is a type of statistical analysis in which you put your assumptions about a population parameter to the test. It is used to estimate the relationship between 2 statistical variables.

Let's discuss few examples of statistical hypothesis from real-life - 

• A teacher assumes that 60% of his college's students come from lower-middle-class families.
• A doctor believes that 3D (Diet, Dose, and Discipline) is 90% effective for diabetic patients.

Now that you know about hypothesis testing, look at the two types of hypothesis testing in statistics.

Hypothesis Testing Formula

Z = ( x̅ – μ0 ) / (σ /√n)

• Here, x̅ is the sample mean,
• μ0 is the population mean,
• σ is the standard deviation,
• n is the sample size.

How Hypothesis Testing Works?

An analyst performs hypothesis testing on a statistical sample to present evidence of the plausibility of the null hypothesis. Measurements and analyses are conducted on a random sample of the population to test a theory. Analysts use a random population sample to test two hypotheses: the null and alternative hypotheses.

The null hypothesis is typically an equality hypothesis between population parameters; for example, a null hypothesis may claim that the population means return equals zero. The alternate hypothesis is essentially the inverse of the null hypothesis (e.g., the population means the return is not equal to zero). As a result, they are mutually exclusive, and only one can be correct. One of the two possibilities, however, will always be correct.

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Null Hypothesis and Alternative Hypothesis

The Null Hypothesis is the assumption that the event will not occur. A null hypothesis has no bearing on the study's outcome unless it is rejected.

H0 is the symbol for it, and it is pronounced H-naught.

The Alternate Hypothesis is the logical opposite of the null hypothesis. The acceptance of the alternative hypothesis follows the rejection of the null hypothesis. H1 is the symbol for it.

Let's understand this with an example.

A sanitizer manufacturer claims that its product kills 95 percent of germs on average. 

To put this company's claim to the test, create a null and alternate hypothesis.

H0 (Null Hypothesis): Average = 95%.

Alternative Hypothesis (H1): The average is less than 95%.

Another straightforward example to understand this concept is determining whether or not a coin is fair and balanced. The null hypothesis states that the probability of a show of heads is equal to the likelihood of a show of tails. In contrast, the alternate theory states that the probability of a show of heads and tails would be very different.

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Hypothesis Testing Calculation With Examples

Let's consider a hypothesis test for the average height of women in the United States. Suppose our null hypothesis is that the average height is 5'4". We gather a sample of 100 women and determine that their average height is 5'5". The standard deviation of population is 2.

To calculate the z-score, we would use the following formula:

z = ( x̅ – μ0 ) / (σ /√n)

z = (5'5" - 5'4") / (2" / √100)

z = 0.5 / (0.045)

We will reject the null hypothesis as the z-score of 11.11 is very large and conclude that there is evidence to suggest that the average height of women in the US is greater than 5'4".

Steps in Hypothesis Testing

Hypothesis testing is a statistical method to determine if there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. Here’s a breakdown of the typical steps involved in hypothesis testing:

Formulate Hypotheses

• Null Hypothesis (H0): This hypothesis states that there is no effect or difference, and it is the hypothesis you attempt to reject with your test.
• Alternative Hypothesis (H1 or Ha): This hypothesis is what you might believe to be true or hope to prove true. It is usually considered the opposite of the null hypothesis.

Choose the Significance Level (α)

The significance level, often denoted by alpha (α), is the probability of rejecting the null hypothesis when it is true. Common choices for α are 0.05 (5%), 0.01 (1%), and 0.10 (10%).

Select the Appropriate Test

Choose a statistical test based on the type of data and the hypothesis. Common tests include t-tests, chi-square tests, ANOVA, and regression analysis. The selection depends on data type, distribution, sample size, and whether the hypothesis is one-tailed or two-tailed.

Collect Data

Gather the data that will be analyzed in the test. This data should be representative of the population to infer conclusions accurately.

Calculate the Test Statistic

Based on the collected data and the chosen test, calculate a test statistic that reflects how much the observed data deviates from the null hypothesis.

Determine the p-value

The p-value is the probability of observing test results at least as extreme as the results observed, assuming the null hypothesis is correct. It helps determine the strength of the evidence against the null hypothesis.

Make a Decision

Compare the p-value to the chosen significance level:

• If the p-value ≤ α: Reject the null hypothesis, suggesting sufficient evidence in the data supports the alternative hypothesis.
• If the p-value > α: Do not reject the null hypothesis, suggesting insufficient evidence to support the alternative hypothesis.

Report the Results

Present the findings from the hypothesis test, including the test statistic, p-value, and the conclusion about the hypotheses.

Perform Post-hoc Analysis (if necessary)

Depending on the results and the study design, further analysis may be needed to explore the data more deeply or to address multiple comparisons if several hypotheses were tested simultaneously.

Types of Hypothesis Testing

To determine whether a discovery or relationship is statistically significant, hypothesis testing uses a z-test. It usually checks to see if two means are the same (the null hypothesis). Only when the population standard deviation is known and the sample size is 30 data points or more, can a z-test be applied.

A statistical test called a t-test is employed to compare the means of two groups. To determine whether two groups differ or if a procedure or treatment affects the population of interest, it is frequently used in hypothesis testing.

Chi-Square 

You utilize a Chi-square test for hypothesis testing concerning whether your data is as predicted. To determine if the expected and observed results are well-fitted, the Chi-square test analyzes the differences between categorical variables from a random sample. The test's fundamental premise is that the observed values in your data should be compared to the predicted values that would be present if the null hypothesis were true.

Hypothesis Testing and Confidence Intervals

Both confidence intervals and hypothesis tests are inferential techniques that depend on approximating the sample distribution. Data from a sample is used to estimate a population parameter using confidence intervals. Data from a sample is used in hypothesis testing to examine a given hypothesis. We must have a postulated parameter to conduct hypothesis testing.

Bootstrap distributions and randomization distributions are created using comparable simulation techniques. The observed sample statistic is the focal point of a bootstrap distribution, whereas the null hypothesis value is the focal point of a randomization distribution.

A variety of feasible population parameter estimates are included in confidence ranges. In this lesson, we created just two-tailed confidence intervals. There is a direct connection between these two-tail confidence intervals and these two-tail hypothesis tests. The results of a two-tailed hypothesis test and two-tailed confidence intervals typically provide the same results. In other words, a hypothesis test at the 0.05 level will virtually always fail to reject the null hypothesis if the 95% confidence interval contains the predicted value. A hypothesis test at the 0.05 level will nearly certainly reject the null hypothesis if the 95% confidence interval does not include the hypothesized parameter.

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Simple and Composite Hypothesis Testing

Depending on the population distribution, you can classify the statistical hypothesis into two types.

Simple Hypothesis: A simple hypothesis specifies an exact value for the parameter.

Composite Hypothesis: A composite hypothesis specifies a range of values.

A company is claiming that their average sales for this quarter are 1000 units. This is an example of a simple hypothesis.

Suppose the company claims that the sales are in the range of 900 to 1000 units. Then this is a case of a composite hypothesis.

One-Tailed and Two-Tailed Hypothesis Testing

The One-Tailed test, also called a directional test, considers a critical region of data that would result in the null hypothesis being rejected if the test sample falls into it, inevitably meaning the acceptance of the alternate hypothesis.

In a one-tailed test, the critical distribution area is one-sided, meaning the test sample is either greater or lesser than a specific value.

In two tails, the test sample is checked to be greater or less than a range of values in a Two-Tailed test, implying that the critical distribution area is two-sided.

If the sample falls within this range, the alternate hypothesis will be accepted, and the null hypothesis will be rejected.

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Right Tailed Hypothesis Testing

If the larger than (>) sign appears in your hypothesis statement, you are using a right-tailed test, also known as an upper test. Or, to put it another way, the disparity is to the right. For instance, you can contrast the battery life before and after a change in production. Your hypothesis statements can be the following if you want to know if the battery life is longer than the original (let's say 90 hours):

• The null hypothesis is (H0 <= 90) or less change.
• A possibility is that battery life has risen (H1) > 90.

The crucial point in this situation is that the alternate hypothesis (H1), not the null hypothesis, decides whether you get a right-tailed test.

Left Tailed Hypothesis Testing

Alternative hypotheses that assert the true value of a parameter is lower than the null hypothesis are tested with a left-tailed test; they are indicated by the asterisk "<".

Suppose H0: mean = 50 and H1: mean not equal to 50

According to the H1, the mean can be greater than or less than 50. This is an example of a Two-tailed test.

In a similar manner, if H0: mean >=50, then H1: mean <50

Here the mean is less than 50. It is called a One-tailed test.

Type 1 and Type 2 Error

A hypothesis test can result in two types of errors.

Type 1 Error: A Type-I error occurs when sample results reject the null hypothesis despite being true.

Type 2 Error: A Type-II error occurs when the null hypothesis is not rejected when it is false, unlike a Type-I error.

Suppose a teacher evaluates the examination paper to decide whether a student passes or fails.

H0: Student has passed

H1: Student has failed

Type I error will be the teacher failing the student [rejects H0] although the student scored the passing marks [H0 was true]. 

Type II error will be the case where the teacher passes the student [do not reject H0] although the student did not score the passing marks [H1 is true].

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Limitations of Hypothesis Testing

Hypothesis testing has some limitations that researchers should be aware of:

• It cannot prove or establish the truth: Hypothesis testing provides evidence to support or reject a hypothesis, but it cannot confirm the absolute truth of the research question.
• Results are sample-specific: Hypothesis testing is based on analyzing a sample from a population, and the conclusions drawn are specific to that particular sample.
• Possible errors: During hypothesis testing, there is a chance of committing type I error (rejecting a true null hypothesis) or type II error (failing to reject a false null hypothesis).
• Assumptions and requirements: Different tests have specific assumptions and requirements that must be met to accurately interpret results.

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After reading this tutorial, you would have a much better understanding of hypothesis testing, one of the most important concepts in the field of Data Science . The majority of hypotheses are based on speculation about observed behavior, natural phenomena, or established theories.

If you are interested in statistics of data science and skills needed for such a career, you ought to explore the Post Graduate Program in Data Science.

If you have any questions regarding this ‘Hypothesis Testing In Statistics’ tutorial, do share them in the comment section. Our subject matter expert will respond to your queries. Happy learning!

1. What is hypothesis testing in statistics with example?

Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence. An example: testing if a new drug improves patient recovery (Ha) compared to the standard treatment (H0) based on collected patient data.

2. What is H0 and H1 in statistics?

In statistics, H0​ and H1​ represent the null and alternative hypotheses. The null hypothesis, H0​, is the default assumption that no effect or difference exists between groups or conditions. The alternative hypothesis, H1​, is the competing claim suggesting an effect or a difference. Statistical tests determine whether to reject the null hypothesis in favor of the alternative hypothesis based on the data.

3. What is a simple hypothesis with an example?

A simple hypothesis is a specific statement predicting a single relationship between two variables. It posits a direct and uncomplicated outcome. For example, a simple hypothesis might state, "Increased sunlight exposure increases the growth rate of sunflowers." Here, the hypothesis suggests a direct relationship between the amount of sunlight (independent variable) and the growth rate of sunflowers (dependent variable), with no additional variables considered.

4. What are the 3 major types of hypothesis?

The three major types of hypotheses are:

• Null Hypothesis (H0): Represents the default assumption, stating that there is no significant effect or relationship in the data.
• Alternative Hypothesis (Ha): Contradicts the null hypothesis and proposes a specific effect or relationship that researchers want to investigate.
• Nondirectional Hypothesis: An alternative hypothesis that doesn't specify the direction of the effect, leaving it open for both positive and negative possibilities.

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About the Author

Avijeet is a Senior Research Analyst at Simplilearn. Passionate about Data Analytics, Machine Learning, and Deep Learning, Avijeet is also interested in politics, cricket, and football.

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Composite Hypothesis

Composite Hypothesis:

A statistical hypothesis which does not completely specify the distribution of a random variable is referred to as a composite hypothesis.

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What does "Composite Hypothesis" mean?

Definition of Composite Hypothesis in the context of A/B testing (online controlled experiments).

What is a Composite Hypothesis?

In hypothesis testing a composite hypothesis is a hypothesis which covers a set of values from the parameter space. For example, if the entire parameter space covers -∞ to +∞ a composite hypothesis could be μ ≤ 0. It could be any other number as well, such 1, 2 or 3,1245. The alternative hypothesis is always a composite hypothesis : either one-sided hypothesis if the null is composite or a two-sided one if the null is a point null. The "composite" part means that such a hypothesis is the union of many simple point hypotheses.

In a Null Hypothesis Statistical Test only the null hypothesis can be a point hypothesis. Also, a composite hypothesis usually spans from -∞ to zero or some value of practical significance or from such a value to +∞.

Testing a composite null is what is most often of interest in an A/B testing scenario as we are usually interested in detecting and estimating effects in only one direction: either an increase in conversion rate or average revenue per user, or a decrease in unsubscribe events would be of interest and not its opposite. In fact, running a test so long as to detect a statistically significant negative outcome can result in significant business harm.

Like this glossary entry? For an in-depth and comprehensive reading on A/B testing stats, check out the book "Statistical Methods in Online A/B Testing" by the author of this glossary, Georgi Georgiev.

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One-tailed vs Two-tailed Tests of Significance in A/B Testing blog.analytics-toolkit.com

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Statistics/Hypothesis Testing

• 1 Introduction
• 2 Basic concepts and terminologies
• 3 Evaluating a hypothesis test
• 4.1 Neyman-Pearson lemma
• 4.2 Likelihood-ratio test
• 5 Relationship between hypothesis testing and confidence intervals

Introduction

In previous chapters, we have discussed two methods for estimating unknown parameters , namely point estimation and interval estimation . Estimating unknown parameters is an important area in statistical inference, and in this chapter we will discuss another important area, namely hypothesis testing , which is related to decision making . Indeed, the concepts of confidence intervals and hypothesis testing are closely related, as we will demonstrate.

Basic concepts and terminologies

Before discussing how to conduct hypothesis testing, and evaluate the "goodness" of a hypothesis test, let us introduce some basic concepts and terminologies related to hypothesis testing first.

Definition. (Hypothesis) A (statistical) hypothesis is a statement about population parameter(s).

There are two terms that classify hypotheses:

Definition. (Simple and composite hypothesis) A hypothesis is a simple hypothesis if it completely specifies the distribution of the population (that is, the distribution is completely known, without any unknown parameters involved), and is a composite hypothesis otherwise.

Sometimes, it is not immediately clear that whether a hypothesis is simple or composite. To understand the classification of hypotheses more clearly, let us consider the following example.

• (a) and (b) are simple hypotheses, since they all completely specifies the distribution.

In hypothesis tests, we consider two hypotheses:

Example. Suppose your friend gives you a coin for tossing, and we do not know whether it is fair or not. However, since the coin is given by your friend, you believe that the coin is fair unless there are sufficient evidences suggesting otherwise. What is the null hypothesis and alternative hypothesis in this context (suppose the coin never land on edge)?

• Of course, in some other places, the saying of "accepting null hypothesis" is avoided because of these philosophical issues.

Now, we are facing with two questions. First, what evidences should we consider? Second, what is meant by "sufficient"? For the first question, a natural answer is that we should consider the observed samples , right? This is because we are making hypothesis about the population, and the samples are taken from, and thus closely related to the population, which should help us make the decision.

Let us formally define the terms related to hypothesis testing in the following.

• Graphically, it looks like

• We use the terminology "tail" since the rejection region includes the values that are located at the "extreme portions" (i.e., very left (with small values) or very right (with large values) portions) (called tails) of distributions.

• We sometimes also call upper-tailed and lower-tailed tests as one-sided tests , and two-tailed tests as two-sided tests .

Exercise. What is the type of this hypothesis test?

Right-tailed test.

As we have mentioned, the decisions made by hypothesis test should not be perfect, and errors occur. Indeed, when we think carefully, there are actually two types of errors, as follows:

We can illustrate these two types of errors more clearly using the following table.

• The power function will be our basis in evaluating the goodness of a test or comparing two different tests.

You notice that the type II error of this hypothesis test can be quite large, so you want to revise the test to lower the type II error.

To describe "control the type I error probability at this level" in a more precise way, let us define the following term.

• Intuitively, we choose the maximum probability of type I error to be the size so that the size can tell us how probable type I error occurs in the worst situation , to show that how "well" can the test control the type I error [ 4 ] .

Exercise. Calculate the type I error probability and type II error probability when the sample size is 12 (the rejection region remains unchanged).

• Case 3 : The test is two-tailed.

• For case 3 subcase 1 , consider the following diagram:
• For case 3 subcase 2 , consider the following diagram:

Evaluating a hypothesis test

After discussing some basic concepts and terminologies, let us now study some ways to evaluate goodness of a hypothesis test. As we have previously mentioned, we want the probability of making type I errors and type II errors to be small, but we have mentioned that it is generally impossible to make both probabilities to be arbitrarily small. Hence, we have suggested to control the type I error, using the size of a test, and the "best" test should the one with the smallest probability of making type II error, after controlling the type I error.

These ideas lead us to the following definitions.

Using this definition, instead of saying "best" test (test with the smallest type II error probability), we can say "a test with the most power", or in other words, the "most powerful test".

Constructing a hypothesis test

Neyman-pearson lemma.

For the case where the underlying distribution is discrete, the proof is very similar (just replace the integrals with sums), and hence omitted.

• In fact, the MP test constructed by Neyman-Pearson lemma is a variant from the likelihood-ratio test , which is more general in the sense that the likelihood-ratio test can also be constructed for composite null and alternative hypotheses, apart from simple null and alternative hypotheses directly. But, the likelihood-ratio test may not be (U)MP. We will discuss likelihood-ratio test later.

• This rejection region has appeared in a previous example.

Now, let us consider another example where the underlying distribution is discrete.

Exercise. Calculate the probability of making type II error for the above test.

Likelihood-ratio test

Previously, we have suggested using the Neyman-Pearson lemma to construct MPT for testing simple null hypothesis against simple alternative hypothesis. However, when the hypotheses are composite, we may not be able to use the Neyman-Pearson lemma. So, in the following, we will give a general method for constructing tests for any hypotheses, not limited to simple hypotheses. But we should notice that the tests constructed are not necessarily UMPT.

• When the simple and alternative hypotheses are simple, the likelihood ratio test will be the same as the test suggested in the Neyman-Pearson lemma.

Relationship between hypothesis testing and confidence intervals

We have mentioned that there are similarities between hypothesis testing and confidence intervals. In this section, we will introduce a theorem suggesting how to construct a hypothesis test from a confidence interval (or in general, confidence set ), and vice versa.

• ↑ Thus, a natural measure of "goodness" of a hypothesis test is its "size of errors". We will discuss these later in this chapter.

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Lesson 27: likelihood ratio tests.

In this lesson, we'll learn how to apply a method for developing a hypothesis test for situations in which both the null and alternative hypotheses are composite. That's not completely accurate. The method, called the likelihood ratio test , can be used even when the hypotheses are simple, but it is most commonly used when the alternative hypothesis is composite. Throughout the lesson, we'll continue to assume that we know the the functional form of the probability density (or mass) function, but we don't know the value of one (or more) of its parameters. That is, we might know that the data come from a normal distrbution, but we don't know the mean or variance of the distribution, and hence the interest in performing a hypothesis test about the unknown parameter(s).

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simple vs composite hypothesis doubt

So if we have

$H_0 :\theta=\theta_0$ vs $H_1 :\theta=\theta_1$

It is easy to see that this is a case of simple vs simple hypothesis (assuming that $\theta$ is the only unknown parameter of our distribution)

$H_0 :\theta\leq\theta_0$ vs $H_1 :\theta>\theta_0$

Is this composite vs composite or simple vs composite?

Since it is somewhat equivalent to

$H_0 :\theta=\theta_0$ vs $H_1 :\theta>\theta_0$

Which I guess it's a simple vs composite hypothesis

And last, if we have two unkown parameters, is $H_0 :\alpha=\alpha_0 , \beta\geq\beta_0$

Simple or composite?

• hypothesis-testing

$H_0 :\theta=\theta_0$ vs $H_1 :\theta>\theta_0$ is a composite hypothesis since for $H_1$ you can have many different $\theta$s.

You can check these links the explanations are pretty clear.

http://www.emathzone.com/tutorials/basic-statistics/simple-hypothesis-and-composite-hypothesis.html

http://isites.harvard.edu/fs/docs/icb.topic1383356.files/Lecture%2014%20-%20Intro%20to%20Hypothesis%20Testing%20-%204%20per%20page.pdf

• 4 $\begingroup$ What matters most are the properties of the null hypothesis, because it determines the sampling distribution of the test statistic used to evaluate the null. When the null is composite, the situation is tricky because the test statistic does not have a definite distribution. $H_0:\theta=\theta_0$ might or might not be composite, depending on what other parameters might be in play and how they affect the test statistic's distribution, but at face value most would consider this to be a simple hypothesis. The nature of $H_1$ plays no role in classifying hypotheses into simple or composite. $\endgroup$ –  whuber ♦ Commented Aug 20, 2015 at 1:30

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Uniformly Most Powerful (UMP) Test: Definition

Hypothesis Testing > Uniformly Most Powerful (UMP) Test

What is a Uniformly Most Powerful Test?

A Uniformly Most Powerful (UMP) test has the most statistical power from the set of all possible alternate hypotheses of the same size α . The UMP doesn’t always exist, especially when the test has nuisance variables (variables that are irrelevant to your study but that have to be be accounted for). However, if the UMP does exist, you can use the Neyman-Pearson lemma (NPL) to find it.

A UMP test is usually defined in terms of a uniformly most powerful rejection region (UMPCR) (also called a “critical region”); A region C of size α is the UMPCR for testing a simple null hypothesis against a set of alternate hypotheses if it is the “best” critical region. The “best” critical region is one that minimizes the probability of making a Type I or a Type II error . In other words, the UMPCR is the region that gives the smallest chance of making a Type I or II error . It is also the region that gives a UMP test the largest (or equally largest) power function .

UMP and the Neyman-Pearson Lemma

The Neyman-Pearson lemma can tell you the best hypothesis test if you have a simple null hypothesis and a simple alternative hypothesis. If you have multiple hypotheses (also called a composite hypothesis ), the NPL can be extended to all individual alternate hypotheses. Composite hypotheses have multiple options for solutions. For example, H 0 :σ 2 > 8 is a composite hypothesis because it doesn’t specify a value for σ 2 ; The solution could be anything over 8. This contrasts to H 0 : μ = 0, which specifies the single value of zero.

The basic idea is that you test each simple hypothesis in turn to see if it is the UMP out of all possibilities.

Definitions using UMP and Likelihood-Ratio

Casella and Berger (2002) define a UMP test as follows:

“Let C be a class of tests for testing H 0 : θ ∈ Θ 0 versus H 1 : θ ∈ Θ c 1 . A test in class C, with power function β(θ), is a uniformly most powerful (UMP) class C test if β(θ) ≥ β′(θ) for every θ ∈ Θ 0 c and every β′(θ) that is a power function of a test in class C.

If you have an entire set of possibilities (which would be the case with composite hypotheses ), each test should be tested individually using the above criteria.

Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences , Wiley. Everitt, B. S.; Skrondal, A. (2010), The Cambridge Dictionary of Statistics , Cambridge University Press.