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If very uncertain about model accuracy, assuming a form for the nominal density may be questionable orquantifying the degree of uncertainty may be unreasonable. In these cases, any formula for the underlying probability densities may be unjustified, but the model evaluationproblem remains. For example, we may want to determine a signal presence or absence in an array output (non-zero mean vs. zeromean) without much knowledge of the contaminating noise. If minimal assumptions can be made about the probability densities, non-parametric model evaluation can be used ( Gibson and Melsa ). In this theoretical framework, no formula for the conditional densities isrequired; instead, we use worst-case densities which conform to the weak problem specification. Because few assumptions about theprobability models are used, non-parametric decision rules are robust: they are insensitive to modeling assumptions because sofew are used. The "robust" test of the previous section are so-named because they explicitly encapsulate model imprecision. Ineither case, one should expect greater performance (smaller error probabilities) in non-parametric decision rules than possible froma "robust" one.

Two hypothesized models are to be tested; 0 is intended to describe the situation "the observed data have zero mean" and the other "a non-zero mean is present."We make the usual assumption that the L observed data values are statistically independent. The only assumption we will make about the probabilistic descriptions underlying these models is that the median of the observationsis zero in the first instance and non-zero in the second. The median of a random variable is the "half-way" value: the probability that the random variable is less than themedian is one-half as is the probability that it is greater. The median and mean of a random variable are not necessarily equal;for the special case of a symmetric probability density they are. In any case, the non-parametric models will be stated interms of the probability that an observation is greater than zero. 0 : r l 0 1 2 1 : r l 0 1 2 The first model is equivalent to a zero-median model for the data; the second implies that the median is greater thanzero. Note that the form of the two underlying probability densities need not be the same to correspond to the two models;they can differ in more general ways than in their means.

To solve this model evaluation problem, we seek (as do all robust techniques) the worst-case density, the densitysatisfying the conditions for one model that is maximally difficult to distinguish from a given density under theother. Several interesting problems arise in this approach. First of all, we seek a non-parametric answer: thesolution must not depend on unstated parameters (we should not have to specify how large the non-zero mean might be). Secondly,the model evaluation rule must not depend on the form for the given density. These seemingly impossible properties are easilysatisfied. To find the worst-case density, first define p + r l 1 r l to be the probability density of the l th observation assuming that 1 is true and that the observation was non-negative. A similar definition for negative values isneeded. p + r l 1 r l p r l 1 r l 0 r l p - r l 1 r l p r l 1 r l 0 r l In terms of these quantities, the conditional density of an observation under 1 is given by p r l 1 r l 1 r l 0 p + r l 1 r l 1 1 r l 0 p - r l 1 r l The worst-case density under 0 would have exactly the same functional form as this one for positive and negative valueswhile having a zero median.

Don't forget that the worst-case density in model evaluation agrees with thegiven one over as large a range as possible.
As depicted in , a density meeting these requirements is p r l 1 r l p + r l 1 r l p - r l 1 r l 2

For each density having a positive-valued median, the worst-case densityhaving zero median would have exactly the same functional form as the given one on the positive and negative real lines, but withthe areas adjusted to be equal. Here, a unit-mean, unit-variance Gaussian density and its corresponding worst-case density isusually discontinuous at the origin; be that as it may, this rather bizarre worst-case density leads to a simple non-parametric decision rule.

The likelihood ratio for a single observation would be 2 1 r l 0 for non-negative values and 2 1 1 r l 0 for negative values. While the likelihood ratio depends on 1 r l 0 , which is not specified in out non-parametric model, the sufficient statistic will not depend on it! To see this, note that the likelihood ratio varies only withthe sign of the observation. Hence, the optimal decision rule amounts to counting how many of the observations are positive;this count can be succinctly expressed with the unit-step function u as l 0 L 1 u r l .

We define the unit-step function as u x 1 x 0 0 x 0 , with the value at the origin undefined. We presume that the densities have no mass at the origin under eithermodel. Although appearing unusual, u r l does indeed yield the number of positively values observations.
Thus, the likelihood ratio for the L statistically independent observations is written 2 L 1 r l 0 l u r l 1 1 r l 0 L l u r l 0 1 Making the usual simplifications, the unknown probability 1 r l 0 can be maneuvered to the right side and merged with the threshold. The optimal non-parametric decision rule thuscompares the sufficient statistic - the count of positive-valued observations - with a threshold determined by the Neyman-Pearsoncriterion.
l 0 L 1 u r l 0 1
This decision rule is called the sign test as it depends only on the signs of the observed data. The sign test isuniformly most powerful and robust.

To find the threshold , we can use the Central Limit Theorem to approximate theprobability distribution of the sum by a Gaussian. Under 0 , the expected value of u r l is 1 2 and the variance is 1 4 . To the degree that the Central Limit Theorem reflects the false-alarm probability (see this problem ), P F is approximately given by P F Q L 2 L 4 and the threshold is found to be L 2 Q P F L 2 As it makes no sense for the threshold to be greater than L (how many positively values observations can there be?), the specified false-alarmprobability must satisfy P F Q L . This restriction means that increasing stringent requirements on the false-alarm probability can only be met ifwe have sufficient data.

Questions & Answers

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where is the latest information on a no technology how can I find it
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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
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Source:  OpenStax, Statistical signal processing. OpenStax CNX. Dec 05, 2011 Download for free at http://cnx.org/content/col11382/1.1
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