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Sufficient statistics arise in nearly every aspect of statistical inference. It is important to understandthem before progressing to areas such as hypothesis testing and parameter estimation.

Suppose we observe an N -dimensional random vector X , characterized by the density or mass function f x , where is a p -dimensional vector of parameters to be estimated. The functional form of f x is assumed known. The parameter completely determines the distribution of X . Conversely, a measurement x of X provides information about through the probability law f x .

Suppose X X 1 X 2 , where X i 1 are IID. Here is a scalar parameter specifying the mean. The distribution of X is determined by through the density f x 1 2 x 1 2 2 1 2 x 2 2 2 On the other hand, if we observe x 100 102 , then we may safely assume 0 is highly unlikely.

The N -dimensional observation X carries information about the p -dimensional parameter vector . If p N , one may ask the following question: Can we compress x into a low-dimensional statistic without any loss of information?Does there exist some function t T x , where the dimension of t is M N , such that t carries all the useful information about ?

If so, for the purpose of studying we could discard the raw measurements x and retain only the low-dimensional statistic t . We call t a sufficient statistic . The following definition captures this notion precisely:

Let X 1 , , X M be a random sample, governed by the density or probability mass function f x . The statistic T x is sufficient for if the conditional distribution of x , given T x t , is independent of . Equivalently, the functional form of f t x does not involve .
How should we interpret this definition? Here are somepossibilities:

1. Let f x t denote the joint density or probability mass function on ( X , T ( X ) ) . If T X is a sufficient statistic for , then

f x f x T x f t x f t f t x f t
Therefore, the parametrization of the probability law for the measurement x is manifested in the parametrization of the probability law for the statistic T x .

2. Given t T x , full knowledge of the measurement x brings no additional information about . Thus, we may discard x and retain on the compressed statistic t .

3. Any inference strategy based on f x may be replaced by a strategy based on f t .

Binary information source

( Scharf, pp.78 ) Suppose a binary information source emitsa sequence of binary (0 or 1) valued, independent variables x 1 , , x N . Each binary symbol may be viewed as a realization of a Bernoulli trial: x n Bernoulli , iid. The parameter 0 1 is to be estimated.

The probability mass function for the random sample x x 1 x N is

f x n 1 N f x n n 1 N f x x n 1 1 x n k 1 N k
where k n 1 N x n is the number of 1's in the sample.

We will show that k is a sufficient statistic for x . This will entail showing that the conditional probability massfunction f k x does not depend on .

The distribution of the number of ones in N independent Bernoulli trials is binomial: f k N k k 1 N k Next, consider the joint distribution of ( x , x n ) . We have f x f x x n Thus, the conditional probability may be written

f k x f x k f k f x f k k 1 N k N k k 1 N k 1 N k
This shows that k is indeed a sufficient statistic for . The N values x 1 , , x N can be replaced by the quantity k without losing information about .

In the previous example , suppose we wish to store in memory the information we possess about . Compare the savings, in terms of bits, we gain by storing the sufficientstatistic k instead of the full sample x 1 , , x N .

Determining sufficient statistics

In the example above , we had to guess the sufficient statistic, and work out theconditional probability by hand. In general, this will be a tedious way to go about finding sufficientstatistics. Fortunately, spotting sufficient statistics can be made easier by the Fisher-Neyman Factorization Theorem .

Uses of sufficient statistics

Sufficient statistics have many uses in statistical inference problems. In hypothesis testing, the Likelihood Ratio Test can often be reduced to a sufficient statistic of the data. In parameter estimation, the Minimum Variance Unbiased Estimator of a parameter can be characterized by sufficient statistics and the Rao-Blackwell Theorem .

Minimality and completeness

Minimal sufficient statistics are, roughly speaking, sufficient statistics that cannot becompressed any more without losing information about the unknown parameter. Completeness is a technical characterization of sufficient statistics that allows one toprove minimality. These topics are covered in detail in this module.

Further examples of sufficient statistics may be found in the module on the Fisher-Neyman Factorization Theorem .

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Signal and information processing for sonar. OpenStax CNX. Dec 04, 2007 Download for free at http://cnx.org/content/col10422/1.5
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