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to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 background 2 mathematical definition toggle mathematical definition subsection 2 1 example 3 fisher neyman factorization theorem toggle fisher neyman factorization theorem subsection 3 1 likelihood principle interpretation 3 2 proof 3 3 another proof 4 minimal sufficiency 5 examples toggle examples subsection 5 1 bernoulli distribution 5 2 uniform distribution 5 3 uniform distribution with two parameters 5 4 poisson distribution 5 5 normal distribution 5 6 exponential distribution 5 7 gamma distribution 6 rao blackwell theorem 7 exponential family 8 other types of sufficiency toggle other types of sufficiency subsection 8 1 bayesian sufficiency 8 2 linear sufficiency 9 see also 10 notes 11 references toggle the table of contents sufficient statistic 15 languages deutsch español فارسی français עברית italiano 日本語 한국어 nederlands polski русский українська tiếng việt 粵語 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia statistical principle in statistics sufficiency is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset a sufficient statistic for a model parameter contains all of the information that the dataset provides about that parameter it is closely related to the concepts of an ancillary statistic which contains no information about the model parameters and of a complete statistic which only contains information about the parameters and no ancillary information 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 1 the kolmogorov structure function deals with individual finite data the related notion there is the algorithmic sufficient statistic the concept is due to sir ronald fisher in 1920 2 stephen stigler noted in 1973 that the concept of sufficiency had 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 remained very important in theoretical work 3 background edit roughly given a set x displaystyle mathbf x of independent identically distributed data conditioned on an unknown parameter θ displaystyle theta a sufficient statistic is a function t x displaystyle t mathbf x 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 t x displaystyle t mathbf x the probability density can be written as f x x θ h x g θ t x displaystyle f_ mathbf x x theta h x g theta t mathbf x from this factorization it can easily be seen that the maximum likelihood estimate of θ displaystyle theta will interact with x displaystyle mathbf x only through t x 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 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 in other words given any value of the sufficient statistic for the parameter the joint probability distribution of the data doesn t depend on the parameter both the statistic and the underlying parameter can be vectors mathematical definition edit a statistic t t x is sufficient for underlying parameter θ precisely if the conditional probability distribution of the data x given the statistic t t x does not depend on the parameter θ 4 alternatively one can say the statistic t x is sufficient for θ if for all prior distributions on θ the mutual information between θ and t x equals the mutual information between θ and x 5 in other words the data processing inequality becomes an equality i θ t x i θ x displaystyle i bigl theta t x bigr i theta x example edit as an example the sample mean is sufficient for the unknown 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 for an arbitrary distribution 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 edit fisher s factorization theorem or factorization criterion provides a convenient characterization of a sufficient statistic if the probability density function is ƒ x θ where θ is a parameter then t is sufficient for θ if and only if nonnegative functions g and h can be found such that f x θ h x g θ t x 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 on θ and the other factor which does depend on θ depends on x only through t 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 θ displaystyle theta and t x displaystyle t x and h x 1 displaystyle h x 1 see examples it is easy to see that if f t is a one to one function and t is a sufficient statistic then f t is a sufficient statistic in particular we can multiply a sufficient statistic by a nonzero constant and get another sufficient statistic likelihood principle interpretation edit an implication of the theorem is that when using likelihood based 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 edit due to hogg and craig 9 let x 1 x 2 x n displaystyle x_ 1 x_ 2 ldots x_ n denote a random sample from a distribution having the pdf f x θ for ι θ δ let y 1 u 1 x 1 x 2 x n be a statistic whose pdf is g 1 y 1 θ what we want to prove is that y 1 u 1 x 1 x 2 x n is a sufficient statistic for θ if and only if for some function h i 1 n f x i θ g 1 u 1 x 1 x 2 x n θ h x 1 x 2 x n 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 i 1 n f x i θ g 1 u 1 x 1 x 2 x n θ h x 1 x 2 x n 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 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 j w i y j displaystyle j left w_ i y_ j right thus i 1 n f w i y 1 y 2 y n θ j g 1 y 1 θ h w 1 y 1 y 2 y n w n y 1 y 2 y n 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 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 right hand member g 1 y 1 θ displaystyle g_ 1 y_ 1 theta is the pdf of y 1 displaystyle y_ 1 so that h w 1 w n j displaystyle h w_ 1 dots w_ n j is the quotient of g y 1 y n θ displaystyle g y_ 1 dots y_ n theta and g 1 y 1 θ displaystyle g_ 1 y_ 1 theta that is it is the conditional pdf h y 2 y n y 1 θ displaystyle h y_ 2 dots y_ n mid y_ 1 theta of y 2 y n displaystyle y_ 2 dots y_ n given y 1 y 1 displaystyle y_ 1 y_ 1 but h x 1 x 2 x n displaystyle h x_ 1 x_ 2 dots x_ n and thus h w 1 y 1 y n w n y 1 y n 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 θ displaystyle theta since θ displaystyle theta was not introduced in the transformation and accordingly not in the jacobian j displaystyle j it follows that h y 2 y n y 1 θ displaystyle h y_ 2 dots y_ n mid y_ 1 theta does not depend upon θ displaystyle theta and that y 1 displaystyle y_ 1 is a sufficient statistics for θ displaystyle theta the converse is proven by taking g y 1 y n θ g 1 y 1 θ h y 2 y n y 1 displaystyle g y_ 1 dots y_ n theta g_ 1 y_ 1 theta h y_ 2 dots y_ n mid y_ 1 where h y 2 y n y 1 displaystyle h y_ 2 dots y_ n mid y_ 1 does not depend upon θ displaystyle theta because y 2 y n displaystyle y_ 2 y_ n depend only upon x 1 x n displaystyle x_ 1 x_ n which are independent on θ displaystyle theta when conditioned by y 1 displaystyle y_ 1 a sufficient statistics by hypothesis now divide both members by the absolute value of the non vanishing jacobian j displaystyle j and replace y 1 y n displaystyle y_ 1 dots y_ n by the functions u 1 x 1 x n u n x 1 x n displaystyle u_ 1 x_ 1 dots x_ n dots u_ n x_ 1 dots x_ n in x 1 x n displaystyle x_ 1 dots x_ n this yields g u 1 x 1 x n u n x 1 x n θ j g 1 u 1 x 1 x n θ h u 2 u n u 1 j displaystyle frac g left u_ 1 x_ 1 dots x_ n dots u_ n x_ 1 dots x_ n theta right j g_ 1 left u_ 1 x_ 1 dots x_ n theta right frac h u_ 2 dots u_ n mid u_ 1 j where j displaystyle j is the jacobian with y 1 y n displaystyle y_ 1 dots y_ n replaced by their value in terms x 1 x n displaystyle x_ 1 dots x_ n the left hand member is necessarily the joint pdf f x 1 θ f x n θ displaystyle f x_ 1 theta cdots f x_ n theta of x 1 x n displaystyle x_ 1 dots x_ n since h y 2 y n y 1 displaystyle h y_ 2 dots y_ n mid y_ 1 and thus h u 2 u n u 1 displaystyle h u_ 2 dots u_ n mid u_ 1 does not depend upon θ displaystyle theta then h x 1 x n h u 2 u n u 1 j displaystyle h x_ 1 dots x_ n frac h u_ 2 dots u_ n mid u_ 1 j is a function that does not depend upon θ displaystyle theta another proof edit 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 x t x displaystyle x t x by f θ x t displaystyle f_ theta x t since t displaystyle t is a deterministic function of x displaystyle x we have f θ x t f θ x displaystyle f_ theta x t f_ theta x as long as t t x displaystyle t t x and zero otherwise therefore f θ x f θ x t f θ x t f θ t f x t f θ t displaystyle begin aligned f_ theta x f_ theta x t 5pt f_ theta x mid t f_ theta t 5pt f x mid t f_ theta t end aligned with the last equality being true by the definition of sufficient statistics thus f θ x a x b θ t displaystyle f_ theta x a x b_ theta t with a x f x t x displaystyle a x f_ x mid t x and b θ t f θ t displaystyle b_ theta t f_ theta t conversely if f θ x a x b θ t displaystyle f_ theta x a x b_ theta t we have f θ t x t x t f θ x t x t x t f θ x x t x t a x b θ t x t x t a x b θ t displaystyle begin aligned f_ theta t sum _ x t x t f_ theta x t 5pt sum _ x t x t f_ theta x 5pt sum _ x t x t a x b_ theta t 5pt left sum _ x t x t a x right b_ theta t end aligned 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 displaystyle t let f x t x displaystyle f_ x mid t x denote the conditional probability density of x displaystyle x given t x displaystyle t x then we can derive an explicit expression for this f x t x f θ x t f θ t f θ x f θ t a x b θ t x t x t a x b θ t a x x t x t a x displaystyle begin aligned f_ x mid t x frac f_ theta x t f_ theta t 5pt frac f_ theta x f_ theta t 5pt frac a x b_ theta t left sum _ x t x t a x right b_ theta t 5pt frac a x sum _ x t x t a x end aligned 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 displaystyle theta and thus t displaystyle t is a sufficient statistic 10 minimal sufficiency edit 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 11 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 f θ x f θ y displaystyle frac f_ theta x f_ theta y is independent of θ displaystyle longleftrightarrow s x s y this follows as a consequence from fisher 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 p θ displaystyle p_ theta 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 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 l x θ i l x θ 0 displaystyle left frac l x mid theta _ i l x mid theta _ 0 right for i 1 k displaystyle i 1 k is a minimal sufficient statistic if the parameter space is discrete θ 0 θ k displaystyle left theta _ 0 theta _ k right ex...
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