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athbf 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 transactions of the royal society a 222 594 604 309 368 bibcode 1922rspta 222 309f doi 10 1098 rsta 1922 0009 hdl 2440 15172 jfm 48 1280 02 jstor 91208 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 mr 0326872 casella george berger roger l 2002 statistical inference 2nd ed duxbury press cover thomas m 2006 elements of information theory joy a thomas 2nd ed hoboken n j wiley interscience p 36 isbn 0 471 24195 4 oclc 59879802 halmos p r savage l j 1949 application of the radon nikodym theorem to the theory of sufficient statistics the annals of mathematical statistics 20 2 225 241 doi 10 1214 aoms 1177730032 issn 0003 4851 factorization theorem encyclopedia of mathematics encyclopediaofmath org retrieved 2022 09 07 taraldsen g 2022 the factorization theorem for sufficiency preprint doi 10 13140 rg 2 2 15068 87687 hogg robert v craig allen t 1995 introduction to mathematical statistics prentice hall isbn 978 0 02 355722 4 the fisher neyman factorization theorem webpage at connexions cnx org dodge 2003 entry for minimal sufficient statistics lehmann and casella 1998 theory of point estimation 2nd edition springer p 37 lehmann and casella 1998 theory of point estimation 2nd edition springer page 42 tikochinsky y tishby n z levine r d 1984 11 01 alternative approach to maximum entropy inference physical review a 30 5 2638 2644 bibcode 1984phrva 30 2638t doi 10 1103 physreva 30 2638 issn 0556 2791 andersen erling bernhard september 1970 sufficiency and exponential families for discrete sample spaces journal of the american statistical association 65 331 1248 1255 doi 10 1080 01621459 1970 10481160 issn 0162 1459 bernardo j m smith a f m 1994 section 5 1 4 bayesian theory wiley isbn 0 471 92416 4 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 mr 0663456 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 s0167 7152 99 00089 9 mr 1731351 zbl 0964 62003 goldstein m o hagan a 1996 bayes linear sufficiency and systems of expert posterior assessments journal of the royal statistical society series b 58 2 301 316 doi 10 1111 j 2517 6161 1996 tb02083 x jstor 2345978 godambe v p 1966 a new approach to sampling from finite populations ii distribution free sufficiency journal of the royal statistical society series b 28 2 320 328 doi 10 1111 j 2517 6161 1966 tb00645 x jstor 2984375 witting t 1987 the linear markov property in credibility theory astin bulletin 17 1 71 84 doi 10 2143 ast 17 1 2014984 hdl 20 500 11850 422507 references edit 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 v t e statistics outline index descriptive statistics continuous ...
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