# 4.16 Robust hypothesis testing

"Robust" is a technical word that implies insensitivity to modeling assumptions. As we have seen, some algorithms arerobust while others are not. The intent of robust signal processing is to derive algorithms that are explicitly insensitive to the underlying signal and/or noise models. The way in which modelingincertainties are described is typified by the approach we shall use in the following discussion of robust model evaluation.

We assume that two nominal models of the generation of the statistically independent observations areknown; the "actual" conditional probability density that describes the data under the assumptions of each model is notknown exactly, but is "close" to the nominal. Letting $p()$ be the actual probability density for each observation and ${p}^{o}()$ the nominal, we say that ( Huber; 1981 ) $p(x)=1{p}^{o}(x)+{p}^{d}(x)$ where ${p}^{d}$ is the unknown disturbance density and  is the uncertainty variable ( $0\le < 1$ ). The uncertainty variable specifies how accurate the nominal model is through to be: the smaller  , the smaller the contribution of the disturbance. It is assumed that some valuefor  can be rationally assigned. The disturbance density is entirely unknown and isassumed to be any value probability density function. The expression given above is normalized so that $p()$ has unit density ranging about it. An example of densities described this way are shown in .

The robust model evaluation problem is formally stated as ${}_{0}:p(r, {}_{0}, r)=\prod_{l=0}^{L-1} 1{p}^{o}({r}_{l}, {}_{0}, {r}_{l})+{p}^{d}({r}_{l}, {}_{0}, {r}_{l})$ ${}_{1}:p(r, {}_{1}, r)=\prod_{l=0}^{L-1} 1{p}^{o}({r}_{l}, {}_{1}, {r}_{l})+{p}^{d}({r}_{l}, {}_{1}, {r}_{l})$ The nominal densities under each model correspond to the conditional densities that we have been using until now. Thedisturbance densities are intended to model imprecision of both descriptions; hence, they are assumed to be different in thecontext of each model. Note that the measure of imprecision  is assumed to be the same under either model.

To solve this problem, we take what is known as a minimax approach : find the worst-case combinations of a priori densities (max), then minimize the consequences of this situation (mini)according to some criterion. In this way, bad situations are handles as well as can be expected while the more tolerable onesare (hopefully) processed well also. The "mini" phase of the minimax solution corresponds to the likelihood ratio for manycriteria. Thus, the "max" phase amounts to finding the worst-case probability distributions for the likelihood ratiotest as described in the previous section: find the disturbance densities that can result in a constant value for the ratio overlarge domains of functions. When the two nominal distributions scaled by $1-$ can be brought together so that they are equal for some disturbance, then the likelihood ratio will be constant inthat domain. Of most interest here is the case where the models differ only in the value of the mean, as shown in . "Bringing the distributions together" means, in this case, scaling the distribution for ${}_{0}$ by $1-$ while adding the constant  to the scaled distribution for ${}_{1}$ . One can shown in general that if the ratio of the nominal densities is monotonic, this procedure finds theworst-case distribution ( Huber; 1965 ). The distributions overlap for small and for large values of the data with no overlap in a central region. As weshall see, the size of this central region depends greatly on the choice of  . The tails of the worst-case distributions under each model are equal; conceptually, we consider that theworst-case densities have exponential tails in the model evaluation problem.

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
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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
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