 Sufficient statistic

In statistics, a sufficient statistic is a statistic which has the property of sufficiency with respect to a statistical model and its associated unknown parameter, meaning that "no other statistic which can be calculated from the same sample provides any additional information as to the value of the parameter".^{[1]} A statistic is sufficient for a family of probability distributions if the sample from which it is calculated gives no additional information than does the statistic, as to which of those probability distributions is that of the population from which the sample was taken.
In practical terms, given a set of independent identically distributed data conditioned on an unknown parameter θ, a sufficient statistic is a function whose value contains all the information needed to compute any estimate of the parameter (e.g. a maximum likelihood estimate). Due to the factorization theorem (see below), for a sufficient statistic , the joint distribution can be written as . From this factorization, it can easily be seen that the maximum likelihood estimate of θ will interact with only through . 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 a vector 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 a jointly sufficient statistic. Typically, there are as many functions as there are parameters. For example, for a Gaussian distribution with unknown mean and variance, 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, the sample mean and sample variance).
The concept, due to Sir Ronald Fisher, is equivalent to the statement that, conditional on the value of a sufficient statistic for a parameter, the joint probability distribution of the data does not depend on that parameter. Both the statistic and the underlying parameter can be vectors.
A related concept is that of linear sufficiency, which is weaker than sufficiency but can be applied in some cases where there is no sufficient statistic, although it is restricted to linear estimators.^{[2]}
The concept of sufficiency has fallen out of favor in descriptive statistics because of the strong dependence on an assumption of the distributional form (see Pitman–Koopman–Darmois theorem below), but remains very important in theoretical work.^{[3]}
Contents
Mathematical definition
A statistic T(X) is sufficient for underlying parameter θ precisely if the conditional probability distribution of the data X, given the statistic T(X), does not depend on the parameter θ,^{[4]} i.e.
or in shorthand
Example
As an example, the sample mean is sufficient for the mean (μ) of a normal distribution with known variance. Once the sample mean is known, no further information about μ can be obtained from the sample itself. On the other hand, the median 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–Neyman factorization theorem
Fisher's factorization theorem or factorization criterion provides a convenient characterization of a sufficient statistic. If the probability density function is ƒ_{θ}(x), then T is sufficient for θ if and only if functions g and h can be found such that
i.e. the density ƒ can be factored into a product such that one factor, h, does not depend on θ and the other factor, which does depend on θ, depends on x only through T(x).
Likelihood principle interpretation
An implication of the theorem is that when using likelihoodbased inference, two sets of data yielding the same value for the sufficient statistic T(X) will always yield the same inferences about θ. By the factorization criterion, the likelihood's dependence on θ is only in conjunction with T(X). As this is the same in both cases, the dependence on θ will be the same as well, leading to identical inferences.
Proof
Due to Hogg and Craig.^{[5]} Let , denote a random sample from a distribution having the pdf f(x, θ) for γ < θ < δ. Let Y = u(X_{1}, X_{2}, ..., X_{n}) be a statistic whose pdf is g(y; θ). Then Y = u(X_{1}, X_{2}, ..., X_{n}) is a sufficient statistic for θ if and only if, for some function H,
First, suppose that
We shall make the transformation y_{i} = u_{i}(x_{1}, x_{2}, ..., x_{n}), for i = 1, ..., n, having inverse functions x_{i} = w_{i}(y_{1}, y_{2}, ..., y_{n}), for i = 1, ..., n, and Jacobian . Thus,
The lefthand member is the joint pdf g(y_{1}, y_{2}, ..., y_{n}; θ) of Y_{1} = u_{1}(X_{1}, ..., X_{n}), ..., Y_{n} = u_{n}(X_{1}, ..., X_{n}). In the righthand member, is the pdf of Y_{1}, so that is the quotient of and g_{1}(y_{1};θ); that is, it is the conditional pdf of given Y_{1} = y_{1}.
But , and thus , was given not to depend upon θ. Since θ was not introduced in the transformation and accordingly not in the Jacobian J, it follows that does not depend upon θ and that Y_{1} is a sufficient statistics for θ.
The converse is proven by taking:
where does not depend upon θ because Y_{2}...Y_{n} depend only upon X_{1}...X_{n} which are independent on Θ when conditioned by Y_{1}, a sufficient statistics by hypothesis. Now divide both members by the absolute value of the nonvanishing Jacobian J, and replace by the functions in . This yields
where J * is the Jacobian with replaced by their value in terms . The lefthand member is necessarily the joint pdf of . Since , and thus , does not depend upon θ, then
is a function that does not depend upon θ.
Another Proof
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 of (X,T(X)) by f_{θ}(x,t). Since T is a function of X, we have f_{θ}(x,t) = f_{θ}(x) and thus:
 f_{θ}(x) = f_{θ}(x,t) = f_{θ  t}(x)f_{θ}(t)
with the last equality being true by the definition of conditional probability distributions. Thus f_{θ}(x) = a(x)b_{θ}(t) with a(x) = f_{θ  t}(x) and b(x) = f_{θ}(t).
Reciprocally, if f_{θ}(x) = a(x)b_{θ}(t), we have
With the first equality by the definition of pdf for multiple variables, the second by the remark above, the third by hypothesis, and the fourth because the summation is not over t.
Thus, the conditional probability distribution is:
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 θ and thus T is a sufficient statistic.^{[6]}
Minimal sufficiency
A sufficient statistic is minimal sufficient if it can be represented as a function of any other sufficient statistic. In other words, S(X) is minimal sufficient if and only if
 S(X) is sufficient, and
 if T(X) is sufficient, then there exists a function f such that S(X) = f(T(X)).
Intuitively, a minimal sufficient statistic most efficiently captures all possible information about the parameter θ.
A useful characterization of minimal sufficiency is that when the density f_{θ} exists, S(X) is minimal sufficient if and only if
 is independent of θ : S(x) = S(y)
This follows as a direct consequence from Fisher's factorization theorem stated above.
A case in which there is no minimal sufficient statistic was shown by Bahadur, 1957.^{[7]} 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 P_{θ} ) are all discrete or are all continuous.
If there exists a minimal sufficient statistic, and this is usually the case, then every complete sufficient statistic is necessarily minimal sufficient^{[8]}(note that this statement does not exclude the option of 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 statistic.
The collection of likelihood ratios is a minimal sufficient statistic if P(X  θ) is discrete or has a density function.
Examples
Bernoulli distribution
If X_{1}, ...., X_{n} are independent Bernoullidistributed random variables with expected value p, then the sum T(X) = X_{1} + ... + X_{n} is a sufficient statistic for p (here 'success' corresponds to X_{i} = 1 and 'failure' to X_{i} = 0; so T is the total number of successes)
This is seen by considering the joint probability distribution:
Because the observations are independent, this can be written as
and, collecting powers of p and 1 − p, gives
which satisfies the factorization criterion, with h(x) = 1 being just a constant.
Note the crucial feature: the unknown parameter p interacts with the data x only via the statistic T(x) = Σ x_{i}.
Uniform distribution
See also: German tank problemIf X_{1}, ...., X_{n} are independent and uniformly distributed on the interval [0,θ], then T(X) = max(X_{1}, …, X_{n}) is sufficient for θ — the sample maximum is a sufficient statistic for the population maximum.
To see this, consider the joint probability density function of X=(X_{1},…,X_{n}). Because the observations are independent, the pdf can be written as a product of individual densities
where 1_{{...}} is the indicator function. Thus the density takes form required by the Fisher–Neyman factorization theorem, where h(x) = 1_{{min{xi}≥0}}, and the rest of the expression is a function of only θ and T(x) = max{x_{i}}.
In fact, the minimumvariance unbiased estimator (MVUE) for θ is
This is the sample maximum, scaled to correct for the bias, and is MVUE by the Lehmann–Scheffé theorem. Unscaled sample maximum T(X) is the maximum likelihood estimator for θ.
Uniform distribution (with two parameters)
If are independent and uniformly distributed on the interval (where and are unknown parameters), then is a twodimensional sufficient statistic for .
To see this, consider the joint probability density function of . 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 does not depend on the parameter (α,β) and depends only on through the function
the Fisher–Neyman factorization theorem implies is a sufficient statistic for .
Poisson distribution
If X_{1}, ...., X_{n} are independent and have a Poisson distribution with parameter λ, then the sum T(X) = X_{1} + ... + X_{n} is a sufficient statistic for λ.
To see this, consider the joint probability distribution:
Because the observations are independent, this can be written as
which may be written as
which shows that the factorization criterion is satisfied, where h(x) is the reciprocal of the product of the factorials. Note the parameter λ interacts with the data only through its sum T(X).
Normal distribution
If are independent and normally distributed with expected value θ (a parameter) and known finite variance σ^{2}, then is a sufficient statistic for θ.
To see this, consider the joint probability density function of . Because the observations are independent, the pdf can be written as a product of individual densities, i.e. 
Then, since , which can be shown simply by expanding this term,
The joint density of the sample takes the form required by the Fisher–Neyman factorization theorem, by letting
Since does not depend on the parameter θ and depends only on through the function
the Fisher–Neyman factorization theorem implies is a sufficient statistic for θ.
Exponential distribution
If are independent and exponentially distributed with expected value θ (an unknown realvalued positive parameter), then is a sufficient statistic for θ.
To see this, consider the joint probability density function of . 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 does not depend on the parameter θ and depends only on through the function
the Fisher–Neyman factorization theorem implies is a sufficient statistic for θ.
Gamma distribution
If are independent and distributed as a , where and are unknown parameters of a Gamma distribution, then is a twodimensional sufficient statistic for (α,β).
To see this, consider the joint probability density function of . 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 does not depend on the parameter and depends only on through the function
the Fisher–Neyman factorization theorem implies is a sufficient statistic for
Rao–Blackwell theorem
Sufficiency finds a useful application in the Rao–Blackwell theorem. It states that if g(X) is any kind of estimator of θ, then typically the conditional expectation of g(X) given sufficient statistic T(X) is a better estimator of θ, and is never worse. Sometimes one can very easily construct a very crude estimator g(X), and then evaluate that conditional expected value to get an estimator that is in various senses optimal.
Exponential family
Main article: Exponential familyAccording to the Pitman–Koopman–Darmois theorem, among families of probability distributions whose domain does not vary with the parameter being estimated, only in exponential families is there a sufficient statistic whose dimension remains bounded as sample size increases. Less tersely, suppose are independent identically distributed random variables whose distribution is known to be in some family of probability distributions. Only if that family is an exponential family is there a (possibly vectorvalued) sufficient statistic whose number of scalar components does not increase as the sample size n increases.
This theorem shows that sufficiency (or rather, the existence of a scalar or vectorvalued of bounded dimension sufficient statistic) sharply restricts the possible forms of the distribution.
Other types of sufficiency
Bayesian sufficiency
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 dataset and by using only a statistic. Thus the requirement is that, for almost every x,
It turns out that this "Bayesian sufficiency" is a consequence of the formulation above,^{[9]} however they are not directly equivalent in the infinitedimensional case.^{[10]} A range of theoretical results for sufficiency in a Bayesian context is available.^{[11]}
Linear sufficiency
A concept called "linear sufficiency" can be formulated in a Bayesian context,^{[12]} and more generally.^{[13]} First define the best linear predictor of a vector Y based on X as . Then a linear statistic T(x) is linear sufficient^{[14]} if
See also
 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
 Sufficient dimension reduction
 Ancillary statistic
Notes
 ^ Fisher, R.A. (1922). "On the mathematical foundations of theoretical statistics". Philosophical Transactions of the Royal Society of London. Series A 222: 309–368. doi:10.1098/rsta.1922.0009. JFM 48.1280.02. JSTOR 91208. http://digital.library.adelaide.edu.au/dspace/handle/2440/15172.
 ^ Dodge, Y. (2003) — entry for linear sufficiency
 ^ Stigler, Stephen (December 1973). "Studies in the History of Probability and Statistics. XXXII: Laplace, Fisher and the Discovery of the Concept of Sufficiency". Biometrika 60 (3): 439–445. doi:10.1093/biomet/60.3.439. JSTOR 2334992. MR0326872.
 ^ Casella, George; Berger, Roger L. (2002). Statistical Inference, 2nd ed. Duxbury Press.
 ^ Hogg, Robert V.; Craig, Allen T. (1995). Introduction to Mathematical Statistics. Prentice Hall. ISBN 9780023557224.
 ^ "The Fisher–Neyman Factorization Theorem". http://cnx.org/content/m11480/1.6/.
 ^ Lehmann and Casella
 ^ Lehmann and Casella , page 72
 ^ Bernardo, J.M.; Smith, A.F.M. (1994). "Section 5.1.4". Bayesian Theory. Wiley. ISBN 0471924164.
 ^ Blackwell, D.; Ramamoorthi, R. V. (1982). "A Bayes but not classically sufficient statistic.". Annals of Statistics 10 (3): 1025–1026. doi:10.1214/aos/1176345895. MR663456. Zbl 0485.62004.
 ^ Nogales, A.G.; Oyola, J.A.; Perez, P. (2000). "On conditional independence and the relationship between sufficiency and invariance under the Bayesian point of view". Statistics & Probability Letters 46 (1): 75–84. doi:10.1016/S01677152(99)000899. MR1731351. Zbl 0964.62003.
 ^ M. Goldstein and A. O'Hagan (1996) "Bayes Linear Sufficiency and Systems of Expert Posterior Assessments", Journal of the Royal Statistical Society, Series B), 58(2), 301–316 jstor
 ^ Godambe, V.P. (1966) "A New Approach to Sampling from Finite Populations. II DistributionFree Sufficiency", Journal of the Royal Statistical Society, Series B, 28 (2), 320–328. jstor
 ^ Witting, T. (1987) "The linear Markov property in credibility theory", ASTIN Bulletin, 17 (1), 71–84.
References
 Kholevo, A.S. (2001), "Sufficient statistic", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 9781556080104, http://eom.springer.de/S/s091070.htm
 Lehmann, E. L.; Casella, G. (1998). Theory of Point Estimation (2nd ed.). Springer. Chapter 4. ISBN 0387985026.
 Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0199206139
Categories: Statistical theory
 Statistical principles
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