# 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.

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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