sufficient statistic

Let {fθ} be a statistical model with parameter θ. Let 𝑿=(X1,,Xn) be a random vector of random variablesMathworldPlanetmath representing n observations. A statisticMathworldMathworldPlanetmath T=T(𝑿) of 𝑿 for the parameter θ is called a sufficient statistic, or a sufficient estimator, if the conditional probability distribution of 𝑿 given T(𝑿)=t is not a function of θ (equivalently, does not depend on θ).

In other words, all the information about the unknown parameter θ is captured in the sufficient statistic T. If, say, we are interested in finding out the percentage of defective light bulbs in a shipment of new ones, it is enough, or sufficient, to count the number of defective ones (sum of the Xi’s), rather than worrying about which individual light bulbs are the defective ones (the vector (X1,,Xn)). By taking the sum, a certain “reduction” of data has been achieved.


  1. 1.

    Let X1,,Xn be n independentPlanetmathPlanetmath observations from a uniform distributionMathworldPlanetmath on integers 1,,θ. Let T=max{X1,,Xn} be a statistic for θ. Then the conditional probability distribution of 𝑿=(X1,,Xn) given T=t is


    The numerator is 0 if max{xn}t. So in this case, P(𝑿t)=0 and is not a function of θ. Otherwise, the numerator is θ-n and P(𝑿t) becomes


    where X(i)’s are the rearrangements of the Xi’s in a non-decreasing order from i=1 to n. For the denominator, we first note that

    P(X(1)X(n)=t) = P(X(1)X(n)t)-P(X(1)X(n)<t)
    = P(X(1)X(n)t)-P(X(1)X(n)t-1).

    From the above equation, we find that there are tn-(t-1)n ways to form non-decreasing finite sequencesPlanetmathPlanetmath of n positive integers such that the maximum of the sequence is t. So


    again is not a function of θ. Therefore, T=max{Xi} is a sufficient statistic for θ. Here, we see that a reduction of data has been achieved by taking only the largest member of set of observations, not the entire set.

  2. 2.

    If we set T(X1,,Xn)=(X1,,Xn), then we see that T is trivially a sufficient statistic for any parameter θ. The conditional probability distribution of (X1,,Xn) given T is 1. Even though this is a sufficient statistic by definition (of course, the individual observations provide as much information there is to know about θ as possible), and there is no loss of data in T (which is simply a list of all observations), there is really no reduction of data to speak of here.

  3. 3.

    The sample mean


    of n independent observations from a normal distributionMathworldPlanetmath N(μ,σ2) (both μ and σ2 unknown) is a sufficient statistic for μ. This is the result of the factorization criterion. Similarly, one sees that any partitionMathworldPlanetmathPlanetmath of the sum of n observations Xi into m subtotals is a sufficient statistic for μ. For instance,


    is a sufficient statistic for μ.

  4. 4.

    Again, assume there are n independent observations Xi from a normal distribution N(μ,σ2) with unknown mean and varianceMathworldPlanetmath. The sample variance


    is not a sufficient statistic for σ2. However, if μ is a known constant, then


    is a sufficient statistic for σ2.

A sufficient statistic for a parameter θ is called a minimal sufficient statistic if it can be expressed as a function of any sufficient statistic for θ.

Example. In example 3 above, both the sample mean X¯ and the finite sum S=X1++Xn are minimal sufficient statistics for the mean μ. Since, by the factorization criterion, any sufficient statistic T for μ is a vector whose coordinates form a partition of the finite sum, taking the sum of these coordinates is just the finite sum S. So, we have just expressed S as a function of T. Therefore, S is minimalPlanetmathPlanetmath. Similarly, X¯ is minimal.

Two sufficient statistics T1,T2 for a parameter θ are said to be equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath provided that there is a bijectionMathworldPlanetmath g such that gT1=T2. X¯ and S from the above example are two equivalent sufficient statistics. Two minimal sufficient statistics for the same parameter are equivalent.

Title sufficient statistic
Canonical name SufficientStatistic
Date of creation 2013-03-22 15:02:42
Last modified on 2013-03-22 15:02:42
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Definition
Classification msc 62B05
Synonym sufficient estimator
Synonym minimally sufficient statistic
Synonym minimal sufficient
Synonym minimally sufficient
Defines minimal sufficient statistic
Defines equivalent statistic