# 3.6 Asymptotic distribution of maximum likelihood estimators

 Page 1 / 1
This course is a short series of lectures on Introductory Statistics. Topics covered are listed in the Table of Contents. The notes were prepared by EwaPaszek and Marek Kimmel. The development of this course has been supported by NSF 0203396 grant.

## Asymptotic distribution of maximum likelihood estimators

Let consider a distribution with p.d.f. $f\left(x;\theta \right)$ such that the parameter $\theta$ is not involved in the support of the distribution. We want to be able to find the maximum likelihood estimator $\stackrel{^}{\theta }$ by solving $\frac{\partial \left[\mathrm{ln}L\left(\theta \right)\right]}{\partial \theta }=0,$ where here the partial derivative was used because $L\left(\theta \right)$ involves ${x}_{1},{x}_{2},...,{x}_{n}$ .

That is, $\frac{\partial \left[\mathrm{ln}L\left(\stackrel{^}{\theta }\right)\right]}{\partial \theta }=0,$ where now, with $\stackrel{^}{\theta }$ in this expression, $L\left(\stackrel{^}{\theta }\right)=f\left({X}_{1};\stackrel{^}{\theta }\right)f\left({X}_{2};\stackrel{^}{\theta }\right)···f\left({X}_{n};\stackrel{^}{\theta }\right).$

We can approximate the left-hand member of this latter equation by a linear function found from the first two terms of a Taylor’s series expanded about $\theta$ , namely $\frac{\partial \left[\mathrm{ln}L\left(\theta \right)\right]}{\partial \theta }+\left(\stackrel{^}{\theta }-\theta \right)\frac{{\partial }^{2}\left[\mathrm{ln}L\left(\theta \right)\right]}{\partial {\theta }^{2}}\approx 0,$ when $L\left(\theta \right)=f\left({X}_{1};\theta \right)f\left({X}_{2};\theta \right)···f\left({X}_{n};\theta \right).$

Obviously, this approximation is good enough only if $\stackrel{^}{\theta }$ is close to $\theta$ , and an adequate mathematical proof involves those conditions. But a heuristic argument can be made by solving for $\stackrel{^}{\theta }-\theta$ to obtain

Recall that $\mathrm{ln}L\left(\theta \right)=\mathrm{ln}f\left({X}_{1};\theta \right)+\mathrm{ln}f\left({X}_{2};\theta \right)+···+\mathrm{ln}f\left({X}_{n};\theta \right)$ and

$\frac{\partial \mathrm{ln}L\left(\theta \right)}{\partial \theta }=\sum _{i=1}^{n}\frac{\partial \left[\mathrm{ln}f\left({X}_{i};\theta \right)\right]}{\partial \theta };$

The expression (2) is the sum of the n independent and identically distributed random variables ${Y}_{i}=\frac{\partial \left[\mathrm{ln}f\left({X}_{i};\theta \right)\right]}{\partial \theta },i=1,2,...,n.$ and thus the Central Limit Theorem has an approximate normal distribution with mean (in the continuous case) equal to

$\begin{array}{l}\underset{-\infty }{\overset{\infty }{\int }}\frac{\partial \left[\mathrm{ln}f\left({x}_{i};\theta \right)\right]}{\partial \theta }f\left(x;\theta \right)dx=\underset{-\infty }{\overset{\infty }{\int }}\frac{\partial \left[f\left({x}_{i};\theta \right)\right]}{\partial \theta }\frac{f\left({x}_{i};\theta \right)}{f\left({x}_{i};\theta \right)}dx=\underset{-\infty }{\overset{\infty }{\int }}\frac{\partial \left[f\left({x}_{i};\theta \right)\right]}{\partial \theta }dx\\ =\frac{\partial }{d\partial }\left[\underset{-\infty }{\overset{\infty }{\int }}f\left({x}_{i};\theta \right)dx\right]=\frac{\partial }{d\partial }\left[1\right]=0.\end{array}$

Clearly, the mathematical condition is needed that it is permissible to interchange the operations of integration and differentiation in those last steps. Of course, the integral of $f\left({x}_{i};\theta \right)$ is equal to one because it is a p.d.f.

Since we know that the mean of each Y is $\underset{-\infty }{\overset{\infty }{\int }}\frac{\partial \left[\mathrm{ln}f\left({x}_{i};\theta \right)\right]}{\partial \theta }f\left(x;\theta \right)dx=0$ let us take derivatives of each member of this equation with respect to $\theta$ obtaining

$\underset{-\infty }{\overset{\infty }{\int }}\left\{\frac{{\partial }^{2}\left[\mathrm{ln}f\left({x}_{i};\theta \right)\right]}{\partial {\theta }^{2}}f\left(x;\theta \right)+\frac{\partial \left[\mathrm{ln}f\left({x}_{i};\theta \right)\right]}{\partial \theta }\frac{\partial \left[f\left({x}_{i};\theta \right)\right]}{\partial \theta }\right\}dx=0.$

However, $\frac{\partial \left[f\left({x}_{i};\theta \right)\right]}{\partial \theta }=\frac{\partial \left[\mathrm{ln}f\left({x}_{i};\theta \right)\right]}{\partial \theta }f\left(x;\theta \right)$ so ${\underset{-\infty }{\overset{\infty }{\int }}\left\{\frac{\partial \left[\mathrm{ln}f\left({x}_{i};\theta \right)\right]}{\partial \theta }\right\}}^{2}f\left(x;\theta \right)dx=-\underset{-\infty }{\overset{\infty }{\int }}\frac{{\partial }^{2}\left[\mathrm{ln}f\left({x}_{i};\theta \right)\right]}{\partial {\theta }^{2}}f\left({x}_{i};\theta \right)dx.$

Since $E\left(Y\right)=0$ , this last expression provides the variance of $Y=\partial \left[\mathrm{ln}f\left(X;\theta \right)\right]/d\partial .$ Then the variance of expression (2) is n times this value, namely

$-nE\left\{\frac{{\partial }^{2}\left[\mathrm{ln}f\left({x}_{i};\theta \right)\right]}{\partial {\theta }^{2}}\right\}.$

Let us rewrite (1) as

$\frac{\sqrt{n}\left(\stackrel{^}{\theta }-\theta \right)}{1-\sqrt{-E\left\{{\partial }^{2}\left[\mathrm{ln}f\left(X;\theta \right)\right]/\partial {\theta }^{2}\right\}}}=\frac{\frac{\partial \left[\mathrm{ln}L\left(\theta \right)\right]/\partial \theta }{\sqrt{-E\left\{{\partial }^{2}\left[\mathrm{ln}f\left(X;\theta \right)\right]/\partial {\theta }^{2}\right\}}}}{\frac{-\frac{1}{n}\frac{{\partial }^{2}\left[\mathrm{ln}L\left(\theta \right)\right]}{\partial {\theta }^{2}}}{E\left\{-{\partial }^{2}\left[\mathrm{ln}f\left(X;\theta \right)\right]/\partial {\theta }^{2}\right\}}}$

The numerator of (4) has an approximate $N\left(0,1\right)$ distribution; and those unstated mathematical condition require, in some sense for $-\frac{1}{n}\frac{{\partial }^{2}\left[\mathrm{ln}L\left(\theta \right)\right]}{\partial {\theta }^{2}}$ to converge to $E\left[-{\partial }^{2}\left[\mathrm{ln}f\left(X;\theta \right)\right]/\partial {\theta }^{2}\right]$ . Accordingly, the ratios given in equation (4) must be approximately $N\left(0,1\right)$ . That is, $\stackrel{^}{\theta }$ has an approximate normal distribution with mean $\theta$ and standard deviation $\frac{1}{\sqrt{-nE\left\{{\partial }^{2}\left[\mathrm{ln}f\left(X;\theta \right)\right]/\partial {\theta }^{2}\right\}}}$ .

With the underlying exponential p.d.f. $f\left(x;\theta \right)=\frac{1}{\theta }{e}^{-x/\theta },0 $\overline{X}$ is the maximum likelihood estimator. Since $\mathrm{ln}f\left(x;\theta \right)=-\mathrm{ln}\theta -\frac{x}{\theta }$ and $\frac{\partial \left[\mathrm{ln}f\left(x;\theta \right)\right]}{\partial \theta }=-\frac{1}{\theta }+\frac{x}{{\theta }^{2}}$ and $\frac{{\partial }^{2}\left[\mathrm{ln}f\left(x;\theta \right)\right]}{\partial {\theta }^{}}=\frac{1}{{\theta }^{2}}-\frac{2x}{{\theta }^{3}}$ , we have $-E\left[\frac{1}{{\theta }^{2}}-\frac{2X}{{\theta }^{3}}\right]=-\frac{1}{\theta }+\frac{2\theta }{{\theta }^{3}}=\frac{1}{{\theta }^{2}}$ because $E\left(X\right)=\theta$ . That is, $\overline{X}$ has an approximate distribution with mean $\theta$ and standard deviation $\theta /\sqrt{n}$ . Thus the random interval $\overline{X}±1.96\left(\theta /\sqrt{n}\right)$ has an approximate probability of 0.95 for covering $\theta$ . Substituting the observed $\overline{x}$ for $\theta$ , as well as for $\overline{X}$ , we say that $\overline{x}±1.96\overline{x}/\sqrt{n}$ is an approximate 95% confidence interval for $\theta$ .

The maximum likelihood estimator for $\lambda$ in $f\left(x;\lambda \right)=\frac{{\lambda }^{x}{e}^{-\lambda }}{x!},x=0,1,2,...;\theta \in \Omega =\left\{\theta :0<\theta <\infty \right\}$ is $\stackrel{^}{\lambda }=\overline{X}$ Now $\mathrm{ln}f\left(x;\lambda \right)=x\mathrm{ln}\lambda -\lambda -\mathrm{ln}x!$ and $\frac{\partial \left[\mathrm{ln}f\left(x;\lambda \right)\right]}{\partial \lambda }=\frac{x}{\lambda }-1$ and $\frac{{\partial }^{2}\left[\mathrm{ln}f\left(x;\lambda \right)\right]}{\partial {\lambda }^{2}}=\frac{x}{{\lambda }^{2}}$ . Thus $-E\left(-\frac{X}{{\lambda }^{2}}\right)=\frac{\lambda }{{\lambda }^{2}}=\frac{1}{\lambda }$ and $\stackrel{^}{\lambda }=\overline{X}$ has an approximate normal distribution with mean $\lambda$ and standard deviation $\sqrt{\lambda /n}$ . Finally $\overline{x}±1.645\sqrt{\overline{x}/n}$ serves as an approximate 90% confidence interval for $\lambda$ . With the data from example(…) $\overline{x}=2.225$ and hence this interval is from 1.887 to 2.563.

It is interesting that there is another theorem which is somewhat related to the preceding result in that the variance of $\stackrel{^}{\theta }$ serves as a lower bound for the variance of every unbiased estimator of $\theta$ . Thus we know that if a certain unbiased estimator has a variance equal to that lower bound, we cannot find a better one and hence it is the best in the sense of being the unbiased minimum variance estimator . This is called the Rao-Cramer Inequality .

Let ${X}_{1},{X}_{2},...,{X}_{n}$ be a random sample from a distribution with p.d.f. $f\left(x;\theta \right),\theta \in \Omega =\left\{\theta :c<\theta where the support X does not depend upon $\theta$ so that we can differentiate, with respect to $\theta$ , under integral signs like that in the following integral:

$\underset{-\infty }{\overset{\infty }{\int }}f\left(x;\theta \right)dx=1.$

If $Y=u\left({X}_{1},{X}_{2},...,{X}_{n}\right)$ is an unbiased estimator of $\theta$ , then

$Var\left(Y\right)\ge \frac{1}{n\underset{-\infty }{\overset{\infty }{\int }}{\left\{\left[\partial \mathrm{ln}f\left(x;\theta \right)/\partial \theta \right]\right\}}^{2}f\left(x;\theta \right)dx}=\frac{-1}{n\underset{-\infty }{\overset{\infty }{\int }}\left[{\partial }^{2}\mathrm{ln}f\left(x;\theta \right)/\partial {\theta }^{2}\right]f\left(x;\theta \right)dx}.$

Note that the two integrals in the respective denominators are the expectations $E\left\{{\left[\frac{\partial \mathrm{ln}f\left(X;\theta \right)}{\partial \theta }\right]}^{2}\right\}$ and $E\left[\frac{{\partial }^{2}\mathrm{ln}f\left(X;\theta \right)}{\partial {\theta }^{2}}\right]$ sometimes one is easier to compute that the other.

Note that above the lower bound of two distributions: exponential and Poisson was computed. Those respective lower bounds were ${\theta }^{2}}{n}$ and $\lambda }{n}$ . Since in each case, the variance of $\overline{X}$ equals the lower bound, then $\overline{X}$ is the unbiased minimum variance estimator.

The sample arises from a distribution with p.d.f. $f\left(x;\theta \right)=\theta {x}^{\theta -1},0

We have $\mathrm{ln}f\left(x;\theta \right)=\mathrm{ln}\theta +\left(\theta -1\right)\mathrm{ln}x,\frac{\partial \mathrm{ln}f\left(x;\theta \right)}{\partial \theta }=\frac{1}{\theta }+\mathrm{ln}x,$ and $\frac{{\partial }^{2}\mathrm{ln}f\left(x;\theta \right)}{\partial {\theta }^{2}}=-\frac{1}{{\theta }^{2}}.$

Since $E\left(-1/{\theta }^{2}\right)=-1/{\theta }^{2}$ , the lower bound of the variance of every unbiased estimator of $\theta$ is ${\theta }^{2}/n$ . Moreover, the maximum likelihood estimator $\stackrel{^}{\theta }=-n/\mathrm{ln}\prod _{i=1}^{n}{X}_{i}$ has an approximate normal distribution with mean $\theta$ and variance ${\theta }^{2}/n$ . Thus, in a limiting sense, $\stackrel{^}{\theta }$ is the unbiased minimum variance estimator of $\theta$ .

To measure the value of estimators; their variances are compared to the Rao-Cramer lower bound. The ratio of the Rao-Cramer lower bound to the actual variance of any unbiased estimator is called the efficiency of that estimator. As estimator with efficiency of 50% requires that 1/0.5=2 times as many sample observations are needed to do as well in estimation as can be done with the unbiased minimum variance estimator (then 100% efficient estimator).

what is variations in raman spectra for nanomaterials
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
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
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
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
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
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 did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers! By By    By By By  