# 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 $m\neq 0$ .

I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
what is the stm
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
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What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Mahi
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Rafiq
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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