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In those cases where a probability density for the parameters cannot be assigned, the model evaluation problem can be solvedin several ways; the methods used depend on the form of the likelihood ratio and the way in which the parameter(s) enter theproblem. In the Gaussian problem we have discussed so often, the threshold used in the likelihood ratio test may be unity. In this case, examination of the resulting computations required reveals that implementing the test does not require knowledge of the variance of the observations (see this problem ). Thus, if the common variance of the underlying Gaussian distributions is not known,this lack of knowledge has no effect on the optimum decision rule. This happy situation - knowledge of thevalue of a parameter is not required by the optimum decision rule - occurs rarely, but should be checked before using morecomplicated procedures.

A second fortuitous situation occurs when the sufficient statistic as well as its probability density under one of themodels do not depend on the unknown parameter(s). Although the sufficient statistic's threshold expressed in terms of the likelihood ratio's threshold depends on the unknown parameters, may be computed as a single value using the Neyman-Pearson criterion if the computation of the false-alarm probability does not involve the unknown parameters .

Continuing the example of the previous section , let's consider the situation where the value of the mean of each observationunder model 1 is not known. The sufficient statistic is the sum of the observations (that quantity doesn't depend on m ) and the distribution of the observation vector under model 0 does not depend on m (allowing computation of the false-alarm probability). However, a subtlety emerges; in the derivation of thesufficient statistic, we had to divide by the value of the mean. The critical step occurs once the logarithm of thelikelihood ratio is manipulated to obtain m l 0 L 1 r l 0 1 2 L m 2 2 Recall that only positively monotonic transformations can be applied; if a negatively monotonicoperation is applied to this inequality (such as multiplying both sides by -1), the inequality reverses . If the sign of m is known, it can be taken into account explicitly and a sufficient statistic results. If, however, the sign is notknown, the above expression cannot be manipulated further and the left side constitutes the sufficient statistic for thisproblem. The sufficient statistic then depends on the unknown parameter and we cannot develop a decision rule in this case.If the sign is known, we can proceed. Assuming the sign of m is positive, the sufficient statistic is the sum of the observations and the threshold is found by L Q P F Note that if the variance 2 instead of the mean were unknown, we could not compute the threshold. The difficulty lies not with the sufficientstatistic (it doesn't depend on the variance), but with the false alarm probability as the expression indicates. Anotherapproach is required to deal with the unknown-variance problem.

Questions & Answers

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.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Signal and information processing for sonar. OpenStax CNX. Dec 04, 2007 Download for free at http://cnx.org/content/col10422/1.5
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