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Suppose you measure a collection of scalars x 1 , , x N . You believe the data is distributed in one of two ways. Your first model, call it H 0 , postulates the data to be governed by the density f 0 x (some fixed density). Your second model, H 1 , postulates a different density f 1 x . These models, termed hypotheses , are denoted as follows: H 0 : x n f 0 x , n = 1 N H 1 : x n f 1 x , n = 1 N A hypothesis test is a rule that, given a measurement x , makes a decision as to which hypothesis best "explains" the data.

Suppose you are confident that your data is normally distributed with variance 1, but you are uncertain aboutthe sign of the mean. You might postulate H 0 : x n -1 1 H 1 : x n 1 1 These densities are depicted in .

Assuming each hypothesis is a priori equally likely, an intuitively appealing hypothesis test is to compute the sample mean x 1 N n 1 N x n , and choose H 0 if x 0 , and H 1 if x 0 . As we will see later, this test is in fact optimal under certain assumptions.

Generalizations and nomenclature

The concepts introduced above can be extended inseveral ways. In what follows we provide more rigorous definitions, describe different kinds of hypothesis testing, andintroduce terminology.


In the most general setup, the observation is a collection x 1 , , x N of random vectors. A common assumption, which facilitates analysis, is that the data are independent and identicallydistributed (IID). The random vectors may be continuous, discrete, or in some cases mixed. It is generally assumedthat all of the data is available at once, although for some applications, such as Sequential Hypothesis Testing , the data is a never ending stream.

Binary versus m-ary tests

When there are two competing hypotheses, we refer to a binary hypothesis test. When the number of hypotheses is M 2 , we refer to an M-ary hypothesis test. Clearly, binary is a special case of M -ary, but binary tests are accorded a special status for certain reasons. These includetheir simplicity, their prevalence in applications, and theoretical results that do not carry over to the M -ary case.

Phase-shift keying

Suppose we wish to transmit a binary string of length r over a noisy communication channel. We assign each of the M 2 r possible bit sequences to a signal s k , k 1 M where s n k 2 f 0 n 2 k 1 M This symboling scheme is known as phase-shift keying (PSK). After transmitting a signal across the noisy channel, the receiver faces an M -ary hypothesis testing problem: H 0 : x s 1 w H M : x s M w where w 0 2 I .

In many binary hypothesis tests, one hypothesis represents the absence of a ceratinfeature. In such cases, the hypothesis is usually labelled H 0 and called the null hypothesis. The other hypothesis is labelled H 1 and called the alternative hypothesis.

Waveform detection

Consider the problem of detecting a known signal s s 1 s N in additive white Gaussian noise (AWGN). This scenario is common in sonar and radar systems. Denotingthe data as x x 1 x N , our hypothesis testing problem is H 0 : x w H 1 : x s w where w 0 2 I . H 0 is the null hypothesis, corresponding to the absence of a signal.

Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Statistical signal processing. OpenStax CNX. Jun 14, 2004 Download for free at http://cnx.org/content/col10232/1.1
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