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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

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.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
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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