2.10 Non-random parameters

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 $m\sum_{l=0}^{L-1} {r}_{l}\underset{{}_{0}}{\overset{{}_{1}}{}}(^{2}\ln +\frac{Lm^{2}}{2})$ 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 $=\sqrt{L}Q^{(-1)}({P}_{F})$ 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.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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