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s a minimal sufficient statistic if the parameter space is discrete θ 0 θ k displaystyle left theta _ 0 theta _ k right examples edit bernoulli distribution edit if x 1 x n are independent bernoulli distributed 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 pr x x pr x 1 x 1 x 2 x 2 x n x n displaystyle pr x x pr x_ 1 x_ 1 x_ 2 x_ 2 ldots x_ n x_ n because the observations are independent this can be written as p x 1 1 p 1 x 1 p x 2 1 p 1 x 2 p x n 1 p 1 x n 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 of p and 1 p gives p x i 1 p n x i p t x 1 p n t x displaystyle p sum x_ i 1 p n sum x_ i p t x 1 p n t x 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 as a concrete application this gives a procedure for distinguishing a fair coin from a biased coin uniform distribution edit see also german tank problem if 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 f θ x 1 x n 1 θ 1 0 x 1 θ 1 θ 1 0 x n θ 1 θ n 1 0 min x i 1 max x i θ displaystyle begin aligned f_ theta x_ 1 ldots x_ n frac 1 theta mathbf 1 _ 0 leq x_ 1 leq theta cdots frac 1 theta mathbf 1 _ 0 leq x_ n leq theta 5pt frac 1 theta n mathbf 1 _ 0 leq min x_ i mathbf 1 _ max x_ i leq theta end aligned where 1 is the indicator function thus the density takes form required by the fisher neyman factorization theorem where h x 1 min x i 0 and the rest of the expression is a function of only θ and t x max x i in fact the minimum variance unbiased estimator mvue for θ is n 1 n t x displaystyle frac n 1 n t x 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 edit if x 1 x n displaystyle x_ 1 x_ n are independent and uniformly distributed on the interval α β displaystyle alpha beta where α displaystyle alpha and β displaystyle beta are unknown parameters then t x 1 n min 1 i n x i max 1 i n x i displaystyle t x_ 1 n left min _ 1 leq i leq n x_ i max _ 1 leq i leq n x_ i right is a two dimensional sufficient statistic for α β displaystyle alpha beta to see this consider the joint probability density function of x 1 n x 1 x n displaystyle x_ 1 n x_ 1 ldots x_ n because the observations are independent the pdf can be written as a product of individual densities i e f x 1 n x 1 n i 1 n 1 β α 1 α x i β 1 β α n 1 α x i β i 1 n 1 β α n 1 α min 1 i n x i 1 max 1 i n x i β displaystyle begin aligned f_ x_ 1 n x_ 1 n prod _ i 1 n left 1 over beta alpha right mathbf 1 _ alpha leq x_ i leq beta left 1 over beta alpha right n mathbf 1 _ alpha leq x_ i leq beta forall i 1 ldots n left 1 over beta alpha right n mathbf 1 _ alpha leq min _ 1 leq i leq n x_ i mathbf 1 _ max _ 1 leq i leq n x_ i leq beta end aligned the joint density of the sample takes the form required by the fisher neyman factorization theorem by letting h x 1 n 1 g α β x 1 n 1 β α n 1 α min 1 i n x i 1 max 1 i n x i β displaystyle begin aligned h x_ 1 n 1 quad g_ alpha beta x_ 1 n left 1 over beta alpha right n mathbf 1 _ alpha leq min _ 1 leq i leq n x_ i mathbf 1 _ max _ 1 leq i leq n x_ i leq beta end aligned since h x 1 n displaystyle h x_ 1 n does not depend on the parameter α β displaystyle alpha beta and g α β x 1 n displaystyle g_ alpha beta x_ 1 n depends only on x 1 n displaystyle x_ 1 n through the function t x 1 n min 1 i n x i max 1 i n x i displaystyle t x_ 1 n left min _ 1 leq i leq n x_ i max _ 1 leq i leq n x_ i right the fisher neyman factorization theorem implies t x 1 n min 1 i n x i max 1 i n x i displaystyle t x_ 1 n left min _ 1 leq i leq n x_ i max _ 1 leq i leq n x_ i right is a sufficient statistic for α β displaystyle alpha beta poisson distribution edit 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 pr x x p x 1 x 1 x 2 x 2 x n x n displaystyle pr x x p x_ 1 x_ 1 x_ 2 x_ 2 ldots x_ n x_ n because the observations are independent this can be written as e λ λ x 1 x 1 e λ λ x 2 x 2 e λ λ x n x n displaystyle e lambda lambda x_ 1 over x_ 1 cdot e lambda lambda x_ 2 over x_ 2 cdots e lambda lambda x_ n over x_ n which may be written as e n λ λ x 1 x 2 x n 1 x 1 x 2 x n displaystyle e n lambda lambda x_ 1 x_ 2 cdots x_ n cdot 1 over x_ 1 x_ 2 cdots x_ n 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 edit if x 1 x n displaystyle x_ 1 ldots x_ n are independent and normally distributed with expected value θ displaystyle theta a parameter and known finite variance σ 2 displaystyle sigma 2 then t x 1 n x 1 n i 1 n x i displaystyle t x_ 1 n overline x frac 1 n sum _ i 1 n x_ i is a sufficient statistic for θ displaystyle theta to see this consider the joint probability density function of x 1 n x 1 x n 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 f x 1 n x 1 n i 1 n 1 2 π σ 2 exp x i θ 2 2 σ 2 2 π σ 2 n 2 exp i 1 n x i θ 2 2 σ 2 2 π σ 2 n 2 exp i 1 n x i x θ x 2 2 σ 2 2 π σ 2 n 2 exp 1 2 σ 2 i 1 n x i x 2 i 1 n θ x 2 2 i 1 n x i x θ x 2 π σ 2 n 2 exp 1 2 σ 2 i 1 n x i x 2 n θ x 2 i 1 n x i x θ x 0 2 π σ 2 n 2 exp 1 2 σ 2 i 1 n x i x 2 exp n 2 σ 2 θ x 2 displaystyle begin aligned f_ x_ 1 n x_ 1 n prod _ i 1 n frac 1 sqrt 2 pi sigma 2 exp left frac x_ i theta 2 2 sigma 2 right 6pt 2 pi sigma 2 frac n 2 exp left sum _ i 1 n frac x_ i theta 2 2 sigma 2 right 6pt 2 pi sigma 2 frac n 2 exp left sum _ i 1 n frac left left x_ i overline x right left theta overline x right right 2 2 sigma 2 right 6pt 2 pi sigma 2 frac n 2 exp left 1 over 2 sigma 2 left sum _ i 1 n x_ i overline x 2 sum _ i 1 n theta overline x 2 2 sum _ i 1 n x_ i overline x theta overline x right right 6pt 2 pi sigma 2 frac n 2 exp left 1 over 2 sigma 2 left sum _ i 1 n x_ i overline x 2 n theta overline x 2 right right sum _ i 1 n x_ i overline x theta overline x 0 6pt 2 pi sigma 2 frac n 2 exp left 1 over 2 sigma 2 sum _ i 1 n x_ i overline x 2 right exp left frac n 2 sigma 2 theta overline x 2 right end aligned the joint density of the sample takes the form required by the fisher neyman factorization theorem by letting h x 1 n 2 π σ 2 n 2 exp 1 2 σ 2 i 1 n x i x 2 g θ x 1 n exp n 2 σ 2 θ x 2 displaystyle begin aligned h x_ 1 n 2 pi sigma 2 frac n 2 exp left 1 over 2 sigma 2 sum _ i 1 n x_ i overline x 2 right 6pt g_ theta x_ 1 n exp left frac n 2 sigma 2 theta overline x 2 right end aligned since h x 1 n displaystyle h x_ 1 n does not depend on the parameter θ displaystyle theta and g θ x 1 n displaystyle g_ theta x_ 1 n depends only on x 1 n displaystyle x_ 1 n through the function t x 1 n x 1 n i 1 n x i displaystyle t x_ 1 n overline x frac 1 n sum _ i 1 n x_ i the fisher neyman factorization theorem implies t x 1 n displaystyle t x_ 1 n is a sufficient statistic for θ displaystyle theta if σ 2 displaystyle sigma 2 is unknown and since s 2 1 n 1 i 1 n x i x 2 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 f x 1 n x 1 n 2 π σ 2 n 2 exp n 1 2 σ 2 s 2 exp n 2 σ 2 θ x 2 displaystyle begin aligned 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 aligned the fisher neyman factorization theorem still holds and implies that x s 2 displaystyle overline x s 2 is a joint sufficient statistic for θ σ 2 displaystyle theta sigma 2 exponential distribution edit if x 1 x n displaystyle x_ 1 dots x_ n are independent and exponentially distributed with expected value θ an unknown real valued positive parameter then t x 1 n i 1 n x i displaystyle t x_ 1 n sum _ i 1 n x_ i is a sufficient statistic for θ to see this consider the joint probability density function of x 1 n x 1 x n 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 f x 1 n x 1 n i 1 n 1 θ e 1 θ x i 1 θ n e 1 θ i 1 n x i displaystyle begin aligned f_ x_ 1 n x_ 1 n prod _ i 1 n 1 over theta e 1 over theta x_ i 1 over theta n e 1 over theta sum _ i 1 n x_ i end aligned the joint density of the sample takes the form required by the fisher neyman factorization theorem by letting h x 1 n 1 g θ x 1 n 1 θ n e 1 θ i 1 n x i displaystyle begin aligned h x_ 1 n 1 g_ theta x_ 1 n 1 over theta n e 1 over theta sum _ i 1 n x_ i end aligned since h x 1 n displaystyle h x_ 1 n does not depend on the parameter θ displaystyle theta and g θ x 1 n displaystyle g_ theta x_ 1 n depends only on x 1 n displaystyle x_ 1 n through the function t x 1 n i 1 n x i displaystyle t x_ 1 n sum _ i 1 n x_ i the fisher neyman factorization theorem implies t x 1 n i 1 n x i displaystyle t x_ 1 n sum _ i 1 n x_ i is a sufficient statistic for θ displaystyle theta gamma distribution edit if x 1 x n displaystyle x_ 1 dots x_ n are independent and distributed as a γ α β displaystyle gamma alpha beta where α displaystyle alpha and β displaystyle beta are unknown parameters of a gamma distribution then t x 1 n i 1 n x i i 1 n x i 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 α β displaystyle alpha beta to see this consider the joint probability density function of x 1 n x 1 x n 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 f x 1 n x 1 n i 1 n 1 γ α β α x i α 1 e 1 β x i 1 γ α β α n i 1 n x i α 1 e 1 β i 1 n x i displaystyle begin aligned f_ x_ 1 n x_ 1 n prod _ i 1 n left 1 over gamma alpha beta alpha right x_ i alpha 1 e 1 beta x_ i 5pt left 1 over gamma alpha beta alpha right n left prod _ i 1 n x_ i right alpha 1 e 1 over beta sum _ i 1 n x_ i end aligned the joint density of the sample takes the form required by the fisher neyman factorization theorem by letting h x 1 n 1 g α β x 1 n 1 γ α β α n i 1 n x i α 1 e 1 β i 1 n x i displaystyle begin aligned h x_ 1 n 1 g_ alpha beta x_ 1 n left 1 over gamma alpha beta alpha right n left prod _ i 1 n x_ i right alpha 1 e 1 over beta sum _ i 1 n x_ i end aligned since h x 1 n displaystyle h x_ 1 n does not depend on the parameter α β displaystyle alpha beta and g α β x 1 n displaystyle g_ alpha beta x_ 1 n depends only on x 1 n displaystyle x_ 1 n through the function t x 1 n i 1 n x i i 1 n x i 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 t x 1 n i 1 n x i i 1 n x i displaystyle t x_ 1 n left prod _ i 1 n x_ i sum _ i 1 n x_ i right is a sufficient statistic for α β displaystyle alpha beta rao blackwell theorem edit sufficiency finds a useful application in the rao blackwell theorem which 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 in the sense of having lower variance 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 edit main article exponential family according 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 intuitively this states that nonexponential families of distributions on the real line require nonparametric statistics to fully capture the information in the data less tersely suppose x n n 1 2 3 displaystyle x_ n n 1 2 3 dots are independent identically distributed real random variables whose distribution is known to be in some family of probability distributions parametrized by θ displaystyle theta satisfying certain technical regularity conditions then that family is an exponential family if and only if there is a r m displaystyle mathbb r m valued sufficient statistic t x 1 x n displaystyle t x_ 1 dots x_ n whose number of scalar components m displaystyle m does not increase as the sample size n 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 the real line when the parameters or the random variables are no longer real valued the situation is more complex 15 other types of sufficiency edit bayesian sufficiency edit 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 every x pr θ x x pr θ t x t x 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 statistics t is predictive sufficient if pr x x x x pr x x t x t x displaystyle pr x x mid x x pr x x 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 linear sufficiency edit 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 vector y based on x as e y x displaystyle hat e y mid x then a linear statistic t x is linear sufficient 21 if e θ x e θ t x displaystyle hat e theta mid x hat e theta mid t x see also edit 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 notes edit dodge y 2003 entry for linear sufficiency fisher r a 1922 on the mathematical foundations of theoretical statistics philosophical 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