# Hypothesis testing  (Page 2/2)

 Page 2 / 2

## Tests and decision regions

Consider the general hypothesis testing problem where we have $N$ $d$ -dimensional observations ${x}_{1},,{x}_{N}$ and $M$ hypotheses. If the data are real-valued, for example, then a hypothesistest is a mapping $(:\mathbb{R}^{d}^{N}, \{1, , M\})$ For every possible realization of the input, the test outputs a hypothesis. The test  partitions the input space into a disjoint collection ${R}_{1},,{R}_{M}$ , where ${R}_{k}=\{\left({x}_{1},,{x}_{N}\right)|({x}_{1}, , {x}_{N})=k\}$ The sets ${R}_{k}$ are called decision regions . The boundary between two decision regions is a decision boundary . depicts these concepts when $d=2$ , $N=1$ , and $M=3$ .

## Simple versus composite hypotheses

If the distribution of the data under a certain hypothesis is fully known, we call it a simple hypothesis. All of the hypotheses in the examples above are simple. In many cases, however, we onlyknow the distribution up to certain unknown parameters. For example, in a Gaussian noise model we may not know thevariance of the noise. In this case, a hypothesis is said to be composite .

Consider the problem of detecting the signal ${s}_{n}=\cos (2\pi {f}_{0}(n-k))\forall n\colon n=\{1, , N\}$ where $k$ is an unknown delay parameter. Then ${H}_{0}:x=w$ ${H}_{1}:x=s+w$ is a binary test of a simple hypothesis ( ${H}_{0}$ ) versus a composite alternative. Here we are assuming $({w}_{n}, (0, ^{2}))$ , with $^{2}$ known.

Often a test involving a composite hypothesis has the form ${H}_{0}:={}_{0}$ ${H}_{1}:\neq {}_{0}$ where ${}_{0}$ is fixed. Such problems are called two-sided because the composite alternative "lies on both sides of ${H}_{0}$ ." When  is a scalar, the test ${H}_{0}:\le {}_{0}$ ${H}_{1}:> {}_{0}$ is called one-sided . Here, both hypotheses are composite.

Suppose a coin turns up heads with probability $p$ . We want to assess whether the coin is fair( $p=\frac{1}{2}$ ). We toss the coin $N$ times and record ${x}_{1},,{x}_{N}$ ( ${x}_{n}=1$ means heads and ${x}_{n}=0$ means tails). Then ${H}_{0}:p=\frac{1}{2}$ ${H}_{1}:p\neq \frac{1}{2}$ is a binary test of a simple hypothesis ( ${H}_{0}$ ) versus a composite alternative. This is also a two-sided test.

## Errors and probabilities

In binary hypothesis testing, assuming at least one of the two models does indeed correspond to reality, thereare four possible scenarios:

• ## Case 1

${H}_{0}$ is true, and we declare ${H}_{0}$ to be true
• ## Case 2

${H}_{0}$ is true, but we declare ${H}_{1}$ to be true
• ## Case 3

${H}_{1}$ is true, and we declare ${H}_{1}$ to be true
• ## Case 4

${H}_{1}$ is true, but we declare ${H}_{0}$ to be true
In cases 2 and 4, errors occur. The names given to these errors depend on the area of application. In statistics, theyare called type I and type II errors respectively, while in signal processing they are known as a false alarm or a miss .

Consider the general binary hypothesis testing problem ${H}_{0}:(x, {f}_{}(x)),\in {}_{0}$ ${H}_{1}:(x, {f}_{}(x)),\in {}_{1}$ If ${H}_{0}$ is simple, that is, ${}_{0}=\{{}_{0}\}$ , then the size (denoted  ), also called the false-alarm probability ( ${P}_{F}$ ), is defined to be $={P}_{F}=({}_{0}, \text{declare}{H}_{1})$ When ${}_{0}$ is composite, we define $={P}_{F}={\mathrm{sup}}_{{}_{0}}((, \text{declare}{H}_{1}))$ For $\in {}_{1}$ , the power (denoted  ), or detection probability ( ${P}_{D}$ ), is defined to be $={P}_{D}=(, \text{declare}{H}_{1})$ The probability of a type II error, also called the miss probability , is ${P}_{M}=1-{P}_{D}$ If ${H}_{1}$ is composite, then $=()$ is viewed as a function of  .

## Criteria in hypothesis testing

The design of a hypothesis test/detector often involves constructing the solution to an optimizationproblem. The optimality criteria used fall into two classes: Bayesian and frequent.

Representing the former approach is the Bayes Risk Criterion . Representing the latter is the Neyman-Pearson Criterion . These two approaches are developed at length in separate modules.

## Statistics versus engineering lingo

The following table, adapted from Kay, p.65 , summarizes the different terminology for hypothesis testing from statistics and signal processing:

Statistics Signal Processing
Hypothesis Test Detector
Null Hypothesis Noise Only Hypothesis
Alternate Hypothesis Signal + Noise Hypothesis
Critical Region Signal Present Decision Region
Type I Error False Alarm
Type II Error Miss
Size of Test (  ) Probability of False Alarm ( ${P}_{F}$ )
Power of Test (  ) Probability of Detection ( ${P}_{D}$ )

what are the important of economic to accounting students with references
Economics is important because it helps people understand how a variety of factors work with and against each other to control how resources such as labor and capital get used, and how inflation, supply, demand, interest rates and other factors determine how much you pay for goods and services.
explain the steps taken by the government in developing rural market?
contribution of Adam smith in economics
I will join
Dexter
I will join
Patrick
Hey
Fatima
Hey
Amir
Hello
AS
hey
Umarou
I love this book and i need extra Economic book
Amir
Hey
Amir
what's happening here
AS
I love this book and i need extra Economic book
Amir
what is the meaning of function in economics
Pls, I need more explanation on price Elasticity of Supply
Is the degree to the degree of responsiveness of a change in quantity supplied of goods to a change in price
Afran
Discuss the short-term and long-term balance positions of the firm in the monopoly market?
hey
Soumya
hi
Mitiku
how are you?
Mitiku
can you tell how can i economics honurs(BSC) in reputed college?
Soumya
through hard study and performing well than expected from you
Mitiku
what should i prepare for it?
Soumya
prepare first, in psychologically as well as potentially to sacrifice what's expected from you, when I say this I mean that you have to be ready, for every thing and to accept failure as a good and you need to change them to potential for achievement of ur goals
Mitiku
parna kya hai behencho?
Soumya
Hallo
Rabindranath
Hello, dear what's up?
Mitiku
cool
Momoh
good morning
Isaac
pls, is anyone here from Ghana?
Isaac
Afran
Afran
Hello
OLANIYI
pls anyone from Nigeria
OLANIYI
am a new candidate here, can someone put me 2ru
OLANIYI
hello
OLANIYI
Pls economic A level exam tomorrow pls help me
akinwale
am from Ghana
Jacob
hi
Charles
Pls economic A level exam tomorrow pls help me
akinwale
Hi
Dev
bol Diya discuss ab krega v
Dev
hello Mr. Rabindranath
Dev
what do you want Dimlare
Dev
yes tell me your desire to have it
Dev
to have what?
OLANIYI
Good luck
JOSEPH
I want to know about economic A level tomorrow pls help
Lerato
okay
Umarou
okay
Umarou
what is firms
A firm is a business entity which engages in the production of goods and aimed at making profit.
What is autarky in Economics.
what is choice
So how is the perfect competition different from others
what is choice
Tia
1
Naziru
what is the difference between short run and long run?
It just depends on how far you would like to run!!!🤣🤣🤣
Anna
meaning? You guys need not to be playing here; if you don't know a question, leave it for he that knows.
Ukpen
pls is question from which subject or which course
Is this not economics?
Ukpen
This place is meant to be for serious educational matters n not playing ground so pls let's make it a serious place.
Docky
Is there an economics expert here?
Docky
Okay and I was being serous
Anna
The short run is a period of time in which the quantity of at least one inputs is fixed...
Anna
that is the answer that I found online and in my text book
Anna
Elacisity
salihu
Meaning of economics
It will creates rooms for an effective demands.
different between production and supply
babsnof
Hii
Suraj
hlo
eshita
What is the economic?
Suraj
Economics is a science which study human behavior as a relationship between ends and scarce means which has an alternative use.
Mr
what is supply
babsnof
what is different between demand and supply
Demand refers to the quantity of products that consumers are willing to purchase at various prices per time while Supply has to do with the quantity of products suppliers are willing to supply at various prices per time. find the difference in between
Saye
Please what are the effects of rationing Effect of black market Effects of hoarding
Got questions? Join the online conversation and get instant answers!