# 2.3 The neyman-pearson criterion  (Page 2/3)

 Page 2 / 3

## Neyman-pearson criterion

$\max\{{P}_{D}\},\text{such that}{P}_{F}\le$

The maximization is over all decision rules (equivalently, over all decision regions ${R}_{0}$ , ${R}_{1}$ ). Using different terminology, the Neyman-Pearson criterionselects the most powerful test of size (not exceeding)  .

Fortunately, the above optimization problem has an explicit solution. This is given by the celebrated Neyman-Pearson lemma , which we now state. To ease the exposition, our initial statement of this result onlyapplies to continuous random variables, and places a technical condition on the densities. A more general statement is givenlater in the module.

## Neyman-pearson lemma: initial statement

Consider the test ${}_{0}:(x, {f}_{0}(x))$ ${}_{1}:(x, {f}_{1}(x))$ where ${f}_{i}(x)$ is a density. Define $(x)=\frac{{f}_{1}(x)}{{f}_{0}(x)}$ , and assume that $(x)$ satisfies the condition that for each $\in \mathbb{R}$ , $(x)$ takes on the value  with probability zero under hypothesis ${}_{0}$ . The solution to the optimization problem in is given by $(x)=\frac{{f}_{1}(x)}{{f}_{0}(x)}\underset{{}_{0}}{\overset{{}_{1}}{}}$ where  is such that ${P}_{F}=\int {f}_{0}(x)\,d x=$ If $=0$ , then  . The optimal test is unique up to a set of probability zero under ${}_{0}$ and ${}_{1}$ .

The optimal decision rule is called the likelihood ratio test . $(x)$ is the likelihood ratio , and  is a threshold . Observe that neither the likelihood ratio nor the threshold depends on the a priori probabilities $({}_{i})$ . they depend only on the conditional densities ${f}_{i}$ and the size constraint  . The threshold can often be solved for as a function of  , as the next example shows.

Continuing with , suppose we wish to design a Neyman-Pearson decision rule withsize constraint  . We have

$(x)=\frac{\frac{1}{\sqrt{2\pi }}e^{-\left(\frac{(x-1)^{2}}{2}\right)}}{\frac{1}{\sqrt{2\pi }}e^{-\left(\frac{x^{2}}{2}\right)}}=e^{x-\frac{1}{2}}$
By taking the natural logarithm of both sides of the LRT and rarranging terms, the decision rule is not changed, and weobtain $x\underset{{}_{0}}{\overset{{}_{1}}{}}\ln +\frac{1}{2}\equiv$ Thus, the optimal rule is in fact a thresholding rule like we considered in . The false-alarm probability was seen to be ${P}_{F}=Q()$ Thus, we may express the value of  required by the Neyman-Pearson lemma in terms of  : $=Q()^{(-1)}$

## Sufficient statistics and monotonic transformations

For hypothesis testing involving multiple or vector-valued data, direct evaluation of the size( ${P}_{F}$ ) and power ( ${P}_{D}$ ) of a Neyman-Pearson decision rule would require integrationover multi-dimensional, and potentially complicated decision regions. In many cases, however, this can be avoided bysimplifying the LRT to a test of the form $t\underset{{}_{0}}{\overset{{}_{1}}{}}$ where the test statistic $t=T(x)$ is a sufficient statistic for the data. Such a simplified form is arrived at by modifying both sides of the LRT withmontonically increasing transformations, and by algebraic simplifications. Since the modifications do not change thedecision rule, we may calculate ${P}_{F}$ and ${P}_{D}$ in terms of the sufficient statistic. For example, the false-alarm probability may be written

${P}_{F}=(\text{declare}{}_{1})=\int {f}_{0}(t)\,d t$
where ${f}_{0}(t)$ denotes the density of $t$ under ${}_{0}$ . Since $t$ is typically of lower dimension than $x$ , evaluation of ${P}_{F}$ and ${P}_{D}$ can be greatly simplified. The key is being able to reduce theLRT to a threshold test involving a sufficient statistic for which we know the distribution .

## Common variances, uncommon means

Let's design a Neyman-Pearson decision rule of size  for the problem ${}_{0}:(x, (0, ^{2}I))$ ${}_{1}:(x, (1, ^{2}I))$ where $> 0$ , $^{2}> 0$ are known, $0=\left(\begin{array}{c}0\\ \\ 0\end{array}\right)$ , $1=\left(\begin{array}{c}1\\ \\ 1\end{array}\right)$ are $N$ -dimensional vectors, and $I$ is the $N$ $N$ identity matrix. The likelihood ratio is

$(x)=\frac{\prod_{n=1}^{N} \frac{1}{\sqrt{2\pi ^{2}}}e^{-\left(\frac{({x}_{n}-)^{2}}{2^{2}}\right)}}{\prod_{n=1}^{N} \frac{1}{\sqrt{2\pi ^{2}}}e^{-\left(\frac{{x}_{n}^{2}}{2^{2}}\right)}}=\frac{e^{-\sum_{n=1}^{N} \frac{({x}_{n}-)^{2}}{2^{2}}}}{e^{-\sum_{n=1}^{N} \frac{{x}_{n}^{2}}{2^{2}}}}=e^{\frac{1}{2^{2}}\sum_{n=1}^{N} 2{x}_{n}-^{2}}=e^{\frac{1}{^{2}}(-\left(\frac{N^{2}}{2}\right)+\sum_{n=1}^{N} {x}_{n})}$
To simplify the test further we may apply the natural logarithm and rearrange terms to obtain $t\equiv \sum_{n=1}^{N} {x}_{n}\underset{{}_{0}}{\overset{{}_{1}}{}}\frac{^{2}}{}\ln +\frac{N}{2}\equiv$
We have used the assumption $> 0$ . If $< 0$ , then division by  is not a monotonically increasing operation, and the inequalitieswould be reversed.
The test statistic $t$ is sufficient for the unknown mean. To set the threshold  , we write the false-alarm probability (size) as ${P}_{F}=(t> )=\int {f}_{0}(t)\,d t$ To evaluate ${P}_{F}$ , we need to know the density of $t$ under ${}_{0}$ . Fortunately, $t$ is the sum of normal variates, so it is again normally distributed. In particular, we have $t=Ax$ , where $A=1^T$ , so $(t, (A0, A^{2}IA^T)=(0, N^{2}))$ under ${}_{0}$ . Therefore, we may write ${P}_{F}$ in terms of the Q-function as ${P}_{F}=Q(\frac{}{\sqrt{N}})$ The threshold is thus determined by $=\sqrt{N}Q()^{(-1)}$ Under ${}_{1}$ , we have $(t, (A1, A^{2}IA^T)=(N, N^{2}))$ and so the detection probability (power) is ${P}_{D}=(t> )=Q(\frac{-N}{\sqrt{N}})$ Writing ${P}_{D}$ as a function of ${P}_{F}$ , the ROC curve is given by ${P}_{D}=Q(Q({P}_{F})^{(-1)}-\frac{\sqrt{N}}{})$ The quantity $\frac{\sqrt{N}}{}$ is called the signal-to-noise ratio . As its name suggests, a larger SNR corresponds to improved performance of the Neyman-Pearsondecision rule.
In the context of signal processing, the foregoing problem may be viewed as the problem of detecting aconstant (DC) signal in additive white Gaussian noise : ${}_{0}:{x}_{n}={w}_{n},n=1,,N$ ${}_{1}:{x}_{n}=A+{w}_{n},n=1,,N$ where $A$ is a known, fixed amplitude, and $({w}_{n}, (0, ^{2}))$ . Here $A$ corresponds to the mean  in the example.

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
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
Got questions? Join the online conversation and get instant answers!