# 3.10 Estimation theory: problems  (Page 2/3)

In this example , we derived the maximum likelihood estimate of the mean andvariance of a Gaussian random vector. You might wonder why we chose to estimate the variance $^{2}$ rather than the standard deviation  . Using the same assumptions provided in the example, let's explore theconsequences of estimating a function of a parameter ( van Trees: Probs 2.4.9, 2.4.10 ).

Assuming that the mean is known, find the maximum likelihood estimates of first the variance, then thestandard deviation.

Are these estimates biased?

Describe how these two estimates are related. Assuming that $f()$ is a monotonic function, how are $({}_{\mathrm{ML}})$ and $({f\left(\right)}_{\mathrm{ML}})$ related in general? These results suggest a general question. Consider the problem of estimating somefunction of a parameter  , say ${f}_{1}()$ . The observed quantity is $r$ and the conditional density $p(r, , r)$ is known. Assume that  is a nonrandom parameter.

What are the conditions for an efficient estimate $({f}_{1}())$ to exist?

What is the lower bound on the variance of the error of any unbiased estimate of ${f}_{1}()$ ?

Assume an efficient estimate of ${f}_{1}()$ exists; when can an efficient estimate of some other function ${f}_{2}()$ exist?

Let the observations $r(l)$ consist of statistically independent, identically distributed Gaussian random variables having zero mean butunknown variance. We wish to estimate $^{2}$ , their variance.

Find the maximum likelihood estimate $({}_{\mathrm{ML}}^{2})$ and compute the resulting mean-squared error.

Show that this estimate is efficient.

Consider a new estimate $({}_{\mathrm{NEW}}^{2})$ given by $({}_{\mathrm{NEW}}^{2})=({}_{\mathrm{ML}}^{2})$ , where  is a constant. Find the value of  that minimizes the mean-squared error for $({}_{\mathrm{NEW}}^{2})$ . Show that the mean-squared error of $({}_{\mathrm{NEW}}^{2})$ is less than that of $({}_{\mathrm{ML}}^{2})$ . Is this result compatible with this previous part ?

Let the observations be of the form $r=H+n$ where  and $n$ are statistically independent Gaussian random vectors. $(, (0, {K}_{}()))$ $(n, (0, {K}_{n}()))$ The vector  has dimension $M$ ; the vectors $r$ and $n$ have dimension $N$ .

Derive the minimum mean-squared error estimate of  , $({}_{\mathrm{MMSE}})$ , from the relationship $({}_{\mathrm{MMSE}})=(r, )$

Show that this estimate and the optimum linear estimate $({}_{\mathrm{LIN}})$ derived by the Orthogonality Principle are equal.

Find an expression for the mean-squared error when these estimates are used.

To illustrate the power of importance sampling, let's consider a somewhat nave example. Let $r$ have a zero-mean Laplacian distribution; we want to employ importance sampling techniques to estimate $(r> )$ (despite the fact that we can calculate it easily). Let the density for $\stackrel{}{r}$ be Laplacian having mean  .

Find the weight ${c}_{l}$ that must be applied to each decision based on the variable $\stackrel{}{r}$ .

Find the importance sampling gain. Show that this gain means that a fixed number of simulations are needed to achieve a given percentageestimation error (as defined by the coefficient of variation). Express this number as a function of thecriterion value for the coefficient of variation.

Now assume that the density for $\stackrel{}{r}$ is Laplacian, but with mean $m$ . Optimize $m$ by finding the value that maximizes the importance sampling gain.

Suppose we consider an estimate of the parameter  having the form $()=(r)+C$ , where $r$ denotes the vector of the observables and $()$ is a linear operator. The quantity $C$ is a constant. This estimate is not a linear function of the observables unless $C=0$ . We are interested in finding applications for which it is advantageous to allow $C\neq 0$ . Estimates of this form we term "quasi-linear" .

Show that the optimum (minimum mean-squared error) quasi-linear estimate satisfies $({}_{}(r)+{C}_{}-\dot (r)+C)=0$ for all $()$ and $C$ where $({}_{\mathrm{QLIN}})={}_{}(r)+{C}_{}$ .

Find a general expression for the mean-squared error incurred by the optimum quasi-linear estimate.

Such estimates yield a smaller mean-squared error when the parameter  has a nonzero mean. Let  be a scalar parameter with mean $m$ . The observables comprise a vector $r$ having components given by ${r}_{l}=+{n}_{l}$ , $l\in \{1, , N\}$ where ${n}_{l}$ are statistically independent Gaussian random variables [ $({n}_{l}, (0, {}_{n}^{2}()))$ ] independent of  . Compute expressions for $({}_{\mathrm{QLIN}})$ and $({}_{\mathrm{LIN}})$ . Verify that $({}_{\mathrm{QLIN}})$ yields a smaller mean-squared error when $m\neq 0$ .

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
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
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
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
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