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Introduction

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

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
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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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