Instatistics,sufficiency is a property of astatistic computed on asample dataset in relation to a parametric model of the dataset. A sufficient statistic contains all of the information that the dataset provides about the model parameters. It is closely related to the concepts of anancillary statistic which contains no information about the model parameters, and of acomplete statistic which only contains information about the parameters and no ancillary information.

A related concept is that oflinear sufficiency, which is weaker thansufficiency but can be applied in some cases where there is no sufficient statistic, although it is restricted to linear estimators.^[1] TheKolmogorov structure function deals with individual finite data; the related notion there is the algorithmic sufficient statistic.

The concept is due toSir Ronald Fisher in 1920.^[2]Stephen Stigler noted in 1973 that the concept of sufficiency had fallen out of favor indescriptive statistics because of the strong dependence on an assumption of the distributional form (seePitman–Koopman–Darmois theorem below), but remained very important in theoretical work.^[3]

Roughly, given a set${\displaystyle \mathbf{X}}$ ofindependent identically distributed data conditioned on an unknown parameter${\displaystyle \theta }$, a sufficient statistic is a function${\displaystyle T(\mathbf{X})}$ whose value contains all the information needed to compute any estimate of the parameter (e.g. amaximum likelihood estimate). Due to the factorization theorem (see below), for a sufficient statistic${\displaystyle T(\mathbf{X})}$, the probability density can be written as${\displaystyle f_{\mathbf{X}}(x;\theta )=h(x)g(\theta ,T(x))}$. From this factorization, it can easily be seen that the maximum likelihood estimate of${\displaystyle \theta }$ will interact with${\displaystyle \mathbf{X}}$ only through${\displaystyle T(\mathbf{X})}$. Typically, the sufficient statistic is a simple function of the data, e.g. the sum of all the data points.

More generally, the "unknown parameter" may represent avector of unknown quantities or may represent everything about the model that is unknown or not fully specified. In such a case, the sufficient statistic may be a set of functions, called ajointly sufficient statistic. Typically, there are as many functions as there are parameters. For example, for aGaussian distribution with unknownmean andvariance, the jointly sufficient statistic, from which maximum likelihood estimates of both parameters can be estimated, consists of two functions, the sum of all data points and the sum of all squared data points (or equivalently, thesample mean andsample variance).

In other words,thejoint probability distribution of the data is conditionally independent of the parameter given the value of the sufficient statistic for the parameter. Both the statistic and the underlying parameter can be vectors.

A statistict = T(X) issufficient for underlying parameterθ precisely if theconditional probability distribution of the dataX, given the statistict = T(X), does not depend on the parameterθ.^[4]

Alternatively, one can say the statistic T(X) is sufficient forθ if, for all prior distributions onθ, themutual information betweenθ andT(X) equals the mutual information betweenθ andX.^[5] In other words, thedata processing inequality becomes an equality:

${\displaystyle I(\theta ;T(X))=I(\theta ;X)}$

As an example, the sample mean is sufficient for the (unknown) meanμ of anormal distribution with known variance. Once the sample mean is known, no further information aboutμ can be obtained from the sample itself. On the other hand, for an arbitrary distribution themedian is not sufficient for the mean: even if the median of the sample is known, knowing the sample itself would provide further information about the population mean. For example, if the observations that are less than the median are only slightly less, but observations exceeding the median exceed it by a large amount, then this would have a bearing on one's inference about the population mean.

Fisher's factorization theorem orfactorization criterion provides a convenientcharacterization of a sufficient statistic. If theprobability density function is ƒ_θ(x), thenT is sufficient forθif and only if nonnegative functionsg andh can be found such that

${\displaystyle f(x;\theta )=h(x)g(\theta ,T(x)),}$

i.e., the density ƒ can be factored into a product such that one factor,h, does not depend onθ and the other factor, which does depend onθ, depends onx only throughT(x). A general proof of this was given by Halmos and Savage^[6] and the theorem is sometimes referred to as the Halmos–Savage factorization theorem.^[7] The proofs below handle special cases, but an alternative general proof along the same lines can be given.^[8] In many simple cases the probability density function is fully specified by${\displaystyle \theta }$ and${\displaystyle T(x)}$, and${\displaystyle h(x)=1}$ (seeExamples).

It is easy to see that ifF(t) is a one-to-one function andT is a sufficientstatistic, thenF(T) is a sufficient statistic. In particular we can multiply asufficient statistic by a nonzero constant and get another sufficient statistic.

An implication of the theorem is that when using likelihood-based inference, two sets of data yielding the same value for the sufficient statisticT(X) will always yield the same inferences aboutθ. By the factorization criterion, the likelihood's dependence onθ is only in conjunction withT(X). As this is the same in both cases, the dependence onθ will be the same as well, leading to identical inferences.

Due to Hogg and Craig.^[9] Let${\displaystyle X_1,X_2,\dots ,X_n}$, denote a random sample from a distribution having thepdff(x, θ) forι < θ < δ. LetY₁ = u₁(X₁, X₂, ..., X_n) be a statistic whose pdf isg₁(y₁; θ). What we want to prove is thatY₁ = u₁(X₁,X₂, ..., X_n) is a sufficient statistic forθ if and only if, for some functionH,

${\displaystyle \prod _{i=1}^{n}f(x_i;\theta )=g_1\left[u_1(x_1,x_2,\dots ,x_n);\theta \right]H(x_1,x_2,\dots ,x_n).}$

First, suppose that

${\displaystyle \prod _{i=1}^{n}f(x_i;\theta )=g_1\left[u_1(x_1,x_2,\dots ,x_n);\theta \right]H(x_1,x_2,\dots ,x_n).}$

We shall make the transformationy_i = u_i(x₁, x₂, ..., x_n), fori = 1, ..., n, having inverse functionsx_i = w_i(y₁, y₂, ..., y_n), fori = 1, ..., n, andJacobian${\displaystyle J=\left[w_i/y_j\right]}$. Thus,

${\displaystyle \prod _{i=1}^{n}f\left[w_i(y_1,y_2,\dots ,y_n);\theta \right]=|J|g_1(y_1;\theta )H\left[w_1(y_1,y_2,\dots ,y_n),\dots ,w_n(y_1,y_2,\dots ,y_n)\right].}$

The left-hand member is the joint pdfg(y₁,y₂, ...,y_n; θ) ofY₁ =u₁(X₁, ...,X_n), ...,Y_n =u_n(X₁, ...,X_n). In the right-hand member,${\displaystyle g_1(y_1;\theta )}$ is the pdf of${\displaystyle Y_1}$, so that${\displaystyle H[w_1,\dots ,w_n]|J|}$ is the quotient of${\displaystyle g(y_1,\dots ,y_n;\theta )}$ and${\displaystyle g_1(y_1;\theta )}$; that is, it is the conditional pdf${\displaystyle h(y_2,\dots ,y_n\mid y_1;\theta )}$ of${\displaystyle Y_2,\dots ,Y_n}$ given${\displaystyle Y_1=y_1}$.

But${\displaystyle H(x_1,x_2,\dots ,x_n)}$, and thus${\displaystyle H\left[w_1(y_1,\dots ,y_n),\dots ,w_n(y_1,\dots ,y_n))\right]}$, was given not to depend upon${\displaystyle \theta }$. Since${\displaystyle \theta }$ was not introduced in the transformation and accordingly not in the Jacobian${\displaystyle J}$, it follows that${\displaystyle h(y_2,\dots ,y_n\mid y_1;\theta )}$ does not depend upon${\displaystyle \theta }$ and that${\displaystyle Y_1}$ is a sufficient statistics for${\displaystyle \theta }$.

The converse is proven by taking:

${\displaystyle g(y_1,\dots ,y_n;\theta )=g_1(y_1;\theta )h(y_2,\dots ,y_n\mid y_1),}$

where${\displaystyle h(y_2,\dots ,y_n\mid y_1)}$ does not depend upon${\displaystyle \theta }$ because${\displaystyle Y_2...Y_n}$ depend only upon${\displaystyle X_1...X_n}$, which are independent on${\displaystyle \mathrm{\Theta }}$ when conditioned by${\displaystyle Y_1}$, a sufficient statistics by hypothesis. Now divide both members by the absolute value of the non-vanishing Jacobian${\displaystyle J}$, and replace${\displaystyle y_1,\dots ,y_n}$ by the functions${\displaystyle u_1(x_1,\dots ,x_n),\dots ,u_n(x_1,\dots ,x_n)}$ in${\displaystyle x_1,\dots ,x_n}$. This yields

${\displaystyle \frac{g\left[u_1(x_1,\dots ,x_n),\dots ,u_n(x_1,\dots ,x_n);\theta \right]}{|J^{\ast }|}=g_1\left[u_1(x_1,\dots ,x_n);\theta \right]\frac{h(u_2,\dots ,u_n\mid u_1)}{|J^{\ast }|}}$

where${\displaystyle J^{\ast }}$ is the Jacobian with${\displaystyle y_1,\dots ,y_n}$ replaced by their value in terms${\displaystyle x_1,\dots ,x_n}$. The left-hand member is necessarily the joint pdf${\displaystyle f(x_1;\theta )\cdots f(x_n;\theta )}$ of${\displaystyle X_1,\dots ,X_n}$. Since${\displaystyle h(y_2,\dots ,y_n\mid y_1)}$, and thus${\displaystyle h(u_2,\dots ,u_n\mid u_1)}$, does not depend upon${\displaystyle \theta }$, then

${\displaystyle H(x_1,\dots ,x_n)=\frac{h(u_2,\dots ,u_n\mid u_1)}{|J^{\ast }|}}$

is a function that does not depend upon${\displaystyle \theta }$.

A simpler more illustrative proof is as follows, although it applies only in the discrete case.

We use the shorthand notation to denote the joint probability density of${\displaystyle (X,T(X))}$ by${\displaystyle f_{\theta }(x,t)}$. Since${\displaystyle T}$ is a deterministic function of${\displaystyle X}$, we have${\displaystyle f_{\theta }(x,t)=f_{\theta }(x)}$, as long as${\displaystyle t=T(x)}$ and zero otherwise. Therefore:

with the last equality being true by the definition of sufficient statistics. Thus${\displaystyle f_{\theta }(x)=a(x)b_{\theta }(t)}$ with${\displaystyle a(x)=f_{X\mid t}(x)}$ and${\displaystyle b_{\theta }(t)=f_{\theta }(t)}$.

Conversely, if${\displaystyle f_{\theta }(x)=a(x)b_{\theta }(t)}$, we have

With the first equality by thedefinition of pdf for multiple variables, the second by the remark above, the third by hypothesis, and the fourth because the summation is not over${\displaystyle t}$.

Let${\displaystyle f_{X\mid t}(x)}$ denote the conditional probability density of${\displaystyle X}$ given${\displaystyle T(X)}$. Then we can derive an explicit expression for this:

With the first equality by definition of conditional probability density, the second by the remark above, the third by the equality proven above, and the fourth by simplification. This expression does not depend on${\displaystyle \theta }$ and thus${\displaystyle T}$ is a sufficient statistic.^[10]

A sufficient statistic isminimal sufficient if it can be represented as a function of any other sufficient statistic. In other words,S(X) isminimal sufficient if and only if^[11]

1. S(X) is sufficient, and
2. ifT(X) is sufficient, then there exists a functionf such thatS(X) =f(T(X)).

Intuitively, a minimal sufficient statisticmost efficiently captures all possible information about the parameterθ.

A useful characterization of minimal sufficiency is that when the densityf_θ exists,S(X) isminimal sufficient if

${\displaystyle \frac{f_{\theta }(x)}{f_{\theta }(y)}}$ is independent ofθ :${\displaystyle ⟺}$S(x) =S(y)

This follows as a consequence fromFisher's factorization theorem stated above.

A case in which there is no minimal sufficient statistic was shown by Bahadur, 1954.^[12] However, under mild conditions, a minimal sufficient statistic does always exist. In particular, in Euclidean space, these conditions always hold if the random variables (associated with${\displaystyle P_{\theta }}$ ) are all discrete or are all continuous.

If there exists a minimal sufficient statistic, and this is usually the case, then everycomplete sufficient statistic is necessarily minimal sufficient^[13] (note that this statement does not exclude a pathological case in which a complete sufficient exists while there is no minimal sufficient statistic). While it is hard to find cases in which a minimal sufficient statistic does not exist, it is not so hard to find cases in which there is no complete sufficient statistic.

The collection of likelihood ratios${\displaystyle \left\{\frac{L(X\mid {\theta }_i)}{L(X\mid {\theta }_0)}\right\}}$ for${\displaystyle i=1,...,k}$, is a minimal sufficient statistic if the parameter space is discrete${\displaystyle \left\{{\theta }_0,...,{\theta }_k\right\}}$.

IfX₁, ...., X_n are independentBernoulli-distributed random variables with expected valuep, then the sumT(X) = X₁ + ... + X_n is a sufficient statistic forp (here 'success' corresponds toX_i = 1 and 'failure' toX_i = 0; soT is the total number of successes)

This is seen by considering the joint probability distribution:

${\displaystyle Pr\{X=x\}=Pr\{X_1=x_1,X_2=x_2,\dots ,X_n=x_n\}.}$

Because the observations are independent, this can be written as

${\displaystyle p^{x_1}(1-p{)}^{1-x_1}{p}^{x_2}(1-p{)}^{1-x_2}\cdots p^{x_n}(1-p{)}^{1-x_n}}$

and, collecting powers ofp and 1 − p, gives

${\displaystyle p^{\sum x_i}(1-p{)}^{n-\sum x_i}=p^{T(x)}(1-p{)}^{n-T(x)}}$

which satisfies the factorization criterion, withh(x) = 1 being just a constant.

Note the crucial feature: the unknown parameterp interacts with the datax only via the statisticT(x) = Σ x_i.

As a concrete application, this gives a procedure for distinguishing afair coin from a biased coin.

IfX₁, ....,X_n are independent anduniformly distributed on the interval [0,θ], thenT(X) = max(X₁, ...,X_n) is sufficient for θ — thesample maximum is a sufficient statistic for the population maximum.

To see this, consider the jointprobability density function ofX  (X₁,...,X_n). Because the observations are independent, the pdf can be written as a product of individual densities

where1_{...} is theindicator function. Thus the density takes form required by the Fisher–Neyman factorization theorem, whereh(x) = 1_{{min{xi}≥0}}, and the rest of the expression is a function of onlyθ andT(x) = max{x_i}.

In fact, theminimum-variance unbiased estimator (MVUE) forθ is

${\displaystyle \frac{n+1}{n}T(X).}$

This is the sample maximum, scaled to correct for thebias, and is MVUE by theLehmann–Scheffé theorem. Unscaled sample maximumT(X) is themaximum likelihood estimator forθ.

If${\displaystyle X_1,...,X_n}$ are independent anduniformly distributed on the interval${\displaystyle [\alpha ,\beta ]}$ (where${\displaystyle \alpha }$ and${\displaystyle \beta }$ are unknown parameters), then${\displaystyle T(X_1^n)=\left(\underset{1\le i\le n}{min}{X}_i,\underset{1\le i\le n}{max}{X}_i\right)}$ is a two-dimensional sufficient statistic for${\displaystyle (\alpha ,\beta )}$.

To see this, consider the jointprobability density function of${\displaystyle X_1^n=(X_1,\dots ,X_n)}$. Because the observations are independent, the pdf can be written as a product of individual densities, i.e.

The joint density of the sample takes the form required by the Fisher–Neyman factorization theorem, by letting

${\displaystyle \begin{array}{r}h(x_1^n)=1,g_{(\alpha ,\beta )}(x_1^n)={\left(\frac{1}{\beta -\alpha }\right)}^n{\mathbf{1}}_{\{\alpha \le \underset{1\le i\le n}{min}{X}_i\}}{\mathbf{1}}_{\{\underset{1\le i\le n}{max}{X}_i\le \beta \}}.\end{array}}$

Since${\displaystyle h(x_1^n)}$ does not depend on the parameter${\displaystyle (\alpha ,\beta )}$ and${\displaystyle g_{(\alpha ,\beta )}(x_1^n)}$ depends only on${\displaystyle x_1^n}$ through the function${\displaystyle T(X_1^n)=\left(\underset{1\le i\le n}{min}{X}_i,\underset{1\le i\le n}{max}{X}_i\right),}$

the Fisher–Neyman factorization theorem implies${\displaystyle T(X_1^n)=\left(\underset{1\le i\le n}{min}{X}_i,\underset{1\le i\le n}{max}{X}_i\right)}$ is a sufficient statistic for${\displaystyle (\alpha ,\beta )}$.

IfX₁, ...., X_n are independent and have aPoisson distribution with parameterλ, then the sumT(X) = X₁ + ... + X_n is a sufficient statistic for λ.

To see this, consider the joint probability distribution:

${\displaystyle Pr(X=x)=P(X_1=x_1,X_2=x_2,\dots ,X_n=x_n).}$

Because the observations are independent, this can be written as

${\displaystyle \frac{e^{-\lambda }{\lambda }^{x_1}}{x_1!}\cdot \frac{e^{-\lambda }{\lambda }^{x_2}}{x_2!}\cdots \frac{e^{-\lambda }{\lambda }^{x_n}}{x_n!}}$

which may be written as

${\displaystyle e^{-n\lambda }{\lambda }^{(x_1+x_2+\cdots +x_n)}\cdot \frac{1}{x_1!x_2!\cdots x_n!}}$

which shows that the factorization criterion is satisfied, whereh(x) is the reciprocal of the product of the factorials. Note the parameter λ interacts with the data only through its sumT(X).

If${\displaystyle X_1,\dots ,X_n}$ are independent andnormally distributed with expected value${\displaystyle \theta }$ (a parameter) and known finite variance${\displaystyle {\sigma }^2,}$ then

${\displaystyle T(X_1^n)=\overline{x}=\frac{1}{n}\sum _{i=1}^n{X}_i}$

is a sufficient statistic for${\displaystyle \theta .}$

To see this, consider the jointprobability density function of${\displaystyle X_1^n=(X_1,\dots ,X_n)}$. Because the observations are independent, the pdf can be written as a product of individual densities, i.e.

The joint density of the sample takes the form required by the Fisher–Neyman factorization theorem, by letting

Since${\displaystyle h(x_1^n)}$ does not depend on the parameter${\displaystyle \theta }$ and${\displaystyle g_{\theta }(x_1^n)}$ depends only on${\displaystyle x_1^n}$ through the function

${\displaystyle T(X_1^n)=\overline{x}=\frac{1}{n}\sum _{i=1}^n{X}_i,}$

the Fisher–Neyman factorization theorem implies${\displaystyle T(X_1^n)}$ is a sufficient statistic for${\displaystyle \theta }$.

If${\displaystyle {\sigma }^2}$ is unknown and since${\displaystyle s^2=\frac{1}{n-1}\sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2}$, the above likelihood can be rewritten as

${\displaystyle \begin{array}{r}{f}_{X_1^n}(x_1^n)=(2\pi {\sigma }^2{)}^{-n/2}\exp \left(-\frac{n-1}{2{\sigma }^2}{s}^2\right)\exp \left(-\frac{n}{2{\sigma }^2}(\theta -\overline{x}{)}^2\right).\end{array}}$

The Fisher–Neyman factorization theorem still holds and implies that${\displaystyle (\overline{x},s^2)}$ is a joint sufficient statistic for${\displaystyle (\theta ,{\sigma }^2)}$.

If${\displaystyle X_1,\dots ,X_n}$ are independent andexponentially distributed with expected valueθ (an unknown real-valued positive parameter), then${\displaystyle T(X_1^n)=\sum _{i=1}^n{X}_i}$ is a sufficient statistic for θ.

To see this, consider the jointprobability density function of${\displaystyle X_1^n=(X_1,\dots ,X_n)}$. Because the observations are independent, the pdf can be written as a product of individual densities, i.e.

The joint density of the sample takes the form required by the Fisher–Neyman factorization theorem, by letting

${\displaystyle \begin{array}{r}h(x_1^n)=1,g_{\theta }(x_1^n)=\frac{1}{{\theta }^n}{e}^{\frac{-1}{\theta }\sum _{i=1}^n{x}_i}.\end{array}}$

Since${\displaystyle h(x_1^n)}$ does not depend on the parameter${\displaystyle \theta }$ and${\displaystyle g_{\theta }(x_1^n)}$ depends only on${\displaystyle x_1^n}$ through the function${\displaystyle T(X_1^n)=\sum _{i=1}^n{X}_i}$

the Fisher–Neyman factorization theorem implies${\displaystyle T(X_1^n)=\sum _{i=1}^n{X}_i}$ is a sufficient statistic for${\displaystyle \theta }$.

If${\displaystyle X_1,\dots ,X_n}$ are independent and distributed as a${\displaystyle \mathrm{\Gamma }(\alpha ,\beta )}$, where${\displaystyle \alpha }$ and${\displaystyle \beta }$ are unknown parameters of aGamma distribution, then${\displaystyle T(X_1^n)=\left(\prod _{i=1}^n{X}_i,\sum _{i=1}^n{X}_i\right)}$ is a two-dimensional sufficient statistic for${\displaystyle (\alpha ,\beta )}$.

To see this, consider the jointprobability density function of${\displaystyle X_1^n=(X_1,\dots ,X_n)}$. Because the observations are independent, the pdf can be written as a product of individual densities, i.e.

The joint density of the sample takes the form required by the Fisher–Neyman factorization theorem, by letting

${\displaystyle \begin{array}{r}h(x_1^n)=1,g_{(\alpha ,\beta )}(x_1^n)={\left(\frac{1}{\mathrm{\Gamma }(\alpha ){\beta }^{\alpha }}\right)}^n{\left(\prod _{i=1}^n{x}_i\right)}^{\alpha -1}{e}^{\frac{-1}{\beta }\sum _{i=1}^n{x}_i}.\end{array}}$

Since${\displaystyle h(x_1^n)}$ does not depend on the parameter${\displaystyle (\alpha ,\beta )}$ and${\displaystyle g_{(\alpha ,\beta )}(x_1^n)}$ depends only on${\displaystyle x_1^n}$ through the function${\displaystyle T(x_1^n)=\left(\prod _{i=1}^n{x}_i,\sum _{i=1}^n{x}_i\right),}$

the Fisher–Neyman factorization theorem implies${\displaystyle T(X_1^n)=\left(\prod _{i=1}^n{X}_i,\sum _{i=1}^n{X}_i\right)}$ is a sufficient statistic for${\displaystyle (\alpha ,\beta ).}$

Sufficiency finds a useful application in theRao–Blackwell theorem, which states that ifg(X) is any kind of estimator ofθ, then typically theconditional expectation ofg(X) given sufficient statisticT(X) is a better (in the sense of having lowervariance) estimator ofθ, and is never worse. Sometimes one can very easily construct a very crude estimatorg(X), and then evaluate that conditional expected value to get an estimator that is in various senses optimal.

According to thePitman–Koopman–Darmois theorem, among families of probability distributions whose domain does not vary with the parameter being estimated, only inexponential families is there a sufficient statistic whose dimension remains bounded as sample size increases. Intuitively, this states that nonexponential families of distributions on the real line requirenonparametric statistics to fully capture the information in the data.

Less tersely, suppose${\displaystyle X_n,n=1,2,3,\dots }$ areindependent identically distributedreal random variables whose distribution is known to be in some family of probability distributions, parametrized by${\displaystyle \theta }$, satisfying certain technical regularity conditions, then that family is anexponential family if and only if there is a${\displaystyle {\mathbb{R}}^m}$-valued sufficient statistic${\displaystyle T(X_1,\dots ,X_n)}$ whose number of scalar components${\displaystyle m}$ does not increase as the sample sizen increases.^[14]

This theorem shows that the existence of a finite-dimensional, real-vector-valued sufficient statistics sharply restricts the possible forms of a family of distributions on thereal line.

When the parameters or the random variables are no longer real-valued, the situation is more complex.^[15]

An alternative formulation of the condition that a statistic be sufficient, set in a Bayesian context, involves the posterior distributions obtained by using the full data-set and by using only a statistic. Thus the requirement is that, for almost everyx,

${\displaystyle Pr(\theta \mid X=x)=Pr(\theta \mid T(X)=t(x)).}$

More generally, without assuming a parametric model, we can say that the statisticsT ispredictive sufficient if

${\displaystyle Pr(X^{\prime }=x^{\prime }\mid X=x)=Pr(X^{\prime }=x^{\prime }\mid T(X)=t(x)).}$

It turns out that this "Bayesian sufficiency" is a consequence of the formulation above,^[16] however they are not directly equivalent in the infinite-dimensional case.^[17] A range of theoretical results for sufficiency in a Bayesian context is available.^[18]

A concept called "linear sufficiency" can be formulated in a Bayesian context,^[19] and more generally.^[20] First define the best linear predictor of a vectorY based onX as${\displaystyle \hat{E}[Y\mid X]}$. Then a linear statisticT(x) is linear sufficient^[21] if

${\displaystyle \hat{E}[\theta \mid X]=\hat{E}[\theta \mid T(X)].}$

• Completeness of a statistic
• Basu's theorem on independence of complete sufficient and ancillary statistics
• Lehmann–Scheffé theorem: a complete sufficient estimator is the best estimator of its expectation
• Rao–Blackwell theorem
• Chentsov's theorem
• Sufficient dimension reduction
• Ancillary statistic

• Kholevo, A.S. (2001) [1994],"Sufficient statistic",Encyclopedia of Mathematics,EMS Press
• Lehmann, E. L.; Casella, G. (1998).Theory of Point Estimation (2nd ed.). Springer. Chapter 4.ISBN 0-387-98502-6.
• Dodge, Y. (2003)The Oxford Dictionary of Statistical Terms, OUP.ISBN 0-19-920613-9

• Statistical theory
• Statistical principles
• Factorization

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