<< Chapter < Page Chapter >> Page >

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 do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
Abdullahi
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Statistical signal processing. OpenStax CNX. Dec 05, 2011 Download for free at http://cnx.org/content/col11382/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Statistical signal processing' conversation and receive update notifications?

Ask