Chapter 6 Principles of Data Reduction


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1 Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a ukow parameter. Notes:. X deotes the radom variables X, X,, X ad x deotes the sample poits x, x,, x.. A statistic, T ( X ), is a form of summary or data reductio. 3. T ( X) ca be thought of as partitioig the sample space,, i terms of the set of values of the values of T. X i that geerate Three Priciples of Data Reductio. Sufficiecy Priciple: promotes a method of data reductio that does ot discard iformatio about parameter.. Likelihood Priciple: describes a fuctio of the parameter, determied by the observed sample, that cotais all the iformatio about that is available from the sample.
2 Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 3. Equivariace Priciple: prescribes a method of data reductio that still preserves some importat features of the model. Sectio 6. Sufficiecy Priciple A sufficiet statistic for a parameter is a statistic that, i a certai sese, captures all the iformatio about cotaied i the sample. Ay additioal iformatio i the sample, besides the value of the sufficiet statistic, does ot cotais ay more iformatio about. Sufficiecy Priciple: If T ( X ) is a sufficiet statistic for, the ay iferece about should deped o the sample X oly through the value of T ( X ). That is, if x ad y are two sample poits such that T( x) T( y ), the the iferece about should be the same whether X=x or X=y is observed. Sectio 6..: Sufficiet Statistics Defiitio 6.. A statistic T ( X ) is a sufficiet statistic for if the coditioal distributio of the sample X give the value of T ( X) does ot deped o. Questio: Is there a simpler way to fid a sufficiet statistic?
3 Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Theorem 6.. If p( x ) is the joit pdf or pmf X ad qt ( ) is the the pdf or pmf of T ( X ), the T ( X) is a sufficiet statistic for if, for every x i the sample space, the ratio p( x ) / qt ( ( x ) ) is a costat fuctio of. Proof: P ( X=x T( X) T( x)) P ( X=x) p( x ). P ( T( X) T( x)) p( T( x) ) P ( X=xad T( X) T( x)) P ( T( X) T( x)) Example 6..3 (Biomial sufficiet statistic) Let X,, X be iid Beroulli with parameter,0. Show that T( X) X X is a sufficiet statistic for. p( x ) Solutio: pt ( ( x) ) x x i i ( ) i =/ ( t x ). i i t t t ( ) t Example 6..4 (Normal sufficiet statistic) Let X,, X be iid (, ), where is kow. Show that T( X ) X is a sufficiet statistic for. Solutio: See the textbook. 3
4 Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Example (Example of a Statistic that Is Not Sufficiet) Cosider the model of Example 6.. agai with 3. The T X X X3 is sufficiet while T XX X3 is ot sufficiet because: PX (, X 0, X3 XX X3 ) PX (, X 0, X3 ) PX (, X 0, X3 ) PX ( 0, X, X3 0) ( ) ( ). ( ) ( ) ( ) ( ) Example 6..5 (Sufficiet order statistic) Let X,, X be iid from a pdf f ad o other iformatio about f is available. The it follows that Where x() x() x( ) are the order statistics. f ( x ) f ( x ) f ( x ), i i i By Theorem 6.., the order statistics are a sufficiet statistic. Without additioal iformatio about f, we caot have further reductio. () i If f is Cauchy, the order statistics. f x ( ) /[ ( x ) ], or logic, f( x) e /[ e ] ( x) ( x), the most reductio we ca get are 4
5 Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Note: Outside the expoetial family of distributios, it is rare to have a sufficiet statistic of smaller dimesio tha the size of the sample ad i may cases order statistics is the best we ca do. Example (Sufficiet Statistic for Poisso Family) Let X,, X be iid Poisso populatio with the parameter 0. The T( X ) X i i is a sufficiet statistic for. Proof: Notice that T has a Poisso distributio with the parameter. Questio: Ca we fid a sufficiet statistic by simple examiatio of the pdf or pmf? Theorem 6..6 (Factorizatio Theorem) Let f ( x ) deote the joit pdf or pmf of a sample X. A statistic T ( X ) is a sufficiet statistic for if ad oly if there exist fuctios gt ( ) ad h( x) such that, for all sample poits x ad all parameter poits, f ( x ) gt ( ( x) ) h( x ). Note: To use the factorizatio Theorem to fid a sufficiet statistic, we factor the joit pdf of the sample ito two parts, with oe part ot depedig o ( h( x )). The other part, the oe that depeds o, usually depeds o the sample X oly through some fuctio T ( X ) ad this fuctio is the sufficiet statistic of. 5
6 Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Example 6..7 Let X,, X be iid (, ), where is kow. Show that T( X ) X is a sufficiet statistic for usig the Factorizatio Theorem. Example 6..8 (Uiform sufficiet statistic) Let X,, X be iid from a discrete uiform o,,. Show that T( X ) X max X is a sufficiet statistic for. ( ) i i Example 6..9 (Normal sufficiet statistic, ad ukow) Let X,, X be iid (, ). Show that T X T X T X X S is a sufficiet statistic for ad ( ) ( ( ), ( )) (, ). Note: For a ormal model (, ), X ad S cotai all iformatio about ad. However, if the model is ot ormal, this may ot ecessarily be true. Example (Sufficiet Statistic for Poisso Family) Let X,, X be iid Poisso populatio with the parameter 0. The use the Factorizatio Theorem to show that both sufficiet statistics for. T( X ) X ad i i T( X ) ( X, X ) i i are Questio: Is there a easy way to fid a sufficiet statistic for a expoetial family of distributios? 6
7 Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Theorem 6..0 Let X,, X be iid from a pdf or pmf f ( x ) that belogs to a expoetial family give by Where θ (,, d ), d k. The is a sufficiet statistic for θ. f ( x θ) h( x) c( θ)exp( w( ) ( )) i i θ ti x, T( X) ( t ( X ), t ( X ),, t ( X )) j j j j j k j k Note: There ca be more tha oe sufficiet statistic for a give model (e.g., X itself is a sufficiet statistic; ay oetooe fuctio of a sufficiet statistic is a sufficiet statistic). Example (Sufficiet Statistic for Poisso Family) Let X,, X be iid Poisso populatio with the parameter x 0. The f( x ) exp( ) exp( )exp( xlog( )), we have hx ( ) / x!, c( ) exp( ), x! x! w( ) log( ), ad tx ( ) x, so T ( X ) t ( X ) X is a sufficiet statistic for. i i i i Sectio 6..: Miimal Sufficiet Statistics 7
8 Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Defiitio 6.. A sufficiet statistic T ( X ) is called miimal sufficiet statistic if, for ay other sufficiet statistic T '( X ), T ( X) is a fuctio of T '( X ). Notes:. The partitio associated with a miimal sufficiet statistic is the coarsest possible partitio for a sufficiet statistic so that it achieves the greatest possible data reductio for a sufficiet statistic.. Miimal sufficiet statistic elimiates all the extraeous iformatio i the sample ad leaves oly that which cotais iformatio about. 3. How to fid the miimal sufficiet statistics? Example 6.. (Two ormal sufficiet statistics) Let X,, X be iid (, ), where is kow. As see i Example 6..9, T '( ) ( X, S ) X is a sufficiet statistic for ( is a kow parameter i this case). However, we ca reduce further T '( X ) by defiig the fuctio rab (, ) aso that if T( ) r( X, S ) X X is a sufficiet statistic for (which we foud from Example 6..7). Note that we have ot show that X is miimal sufficiet for i this case where is kow. 8
9 Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Theorem 6..3 Let f ( x ) be the pmf or pdf of a sample X. Suppose there exists a fuctio T ( X ) such that, for every two sample poits x ad y, the ratio of f ( x ) / f ( y ) is costat as a fuctio of if ad oly if T( x) T( y ). The T ( X ) is a miimal sufficiet statistic for. Example 6..4 (Normal miimal sufficiet statistic) Let X,, X be iid (, ), where both ad are ukow. Let x ad y be two sample poits with correspodig sample meas ad variaces Show that ( X, S ) is a miimal sufficiet statistic for (, ). (, S x ) x ad (, S y ) y. Note: If the set of x values o which the pdf or pmf is positive depeds o the parameter, the for the ratio i Theorem 6..3 to be a costat fuctio of, the umerator ad deomiator must be positive for exactly the same values of. Example 6..5 (Uiform miimal sufficiet statistic) Suppose X,, X are iid uiform observatios o the iterval (, ),. Show that T( X) ( X(), X( ) ) is a miimal sufficiet statistic. (I this example, the dimesio of the miimal sufficiet statistic does ot match the dimesio of the parameter.) Note: A miimal sufficiet statistic is ot uique! Ay oetooe fuctio of a miimal sufficiet statistic is also a miimal sufficiet statistic. 9
10 Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Illustratios:. Let X,, X be iid uiform observatios o the iterval (, ),. T( X) ( X(), X( ) ) is a miimal sufficiet for. Hece ( X( ) X(),( X( ) X() )/) is also a miimal sufficiet statistic.. Let X,, X be iid (, ), ad are ukow. ( X, S ) is a miimal sufficiet statistic for (, ). Hece, is also miimal sufficiet statistic. ( X, ) i X i i i Questio: Let X,, X be iid uiform observatios o the iterval (, ),. T( X) ( X(), X( ) ) is a miimal sufficiet for, is X( ) X() also a miimal sufficiet statistic? Sectio 6..3 Acillary Statistics Defiitio 6..6 A statistic S( X ) whose distributio does ot deped o the parameter is called a acillary statistic. Example 6..7 (Uiform acillary statistic) Let X,, X be iid uiform observatios o the iterval (, ),. Show that the rage statistic, R X( ) X(), is a acillary statistic. 0
11 Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Example 6..8 (Locatio family acillary statistic) Suppose X,, X are iid observatios from a locatio parameter family with cdf F( x ),. Show that the rage statistic, R X( ) X(), is a acillary statistic. Example 6..9 (Scale family acillary statistic) Suppose X,, X are iid observatios from a locatio parameter family with cdf F( x/ ), 0. The ay statistic that depeds o the sample oly through the values X / X,, X / X is a acillary statistic. Note: From Chapter 4 (Example 4.3.6), it was show that if X ad X are iid is Cauchy(0, ). I fact, this also holds for ay 0. (0, ),where, the X / X Sectio 6..4 Sufficiet, Acillary ad Complete Statistics Questio: Is a acillary statistic ot related at all to miimal sufficiet statistic? Recall that if X,, X are iid uiform observatios o the iterval (, ),, the ( X(), X ( ) ) ad ( X( ) X(),( X( ) X() )/) are miimal sufficiet statistics for. But we also kow that R X( ) X() is a acillary statistic. Hece, i this case the miimal sufficiet ad acillary statistics are related.
12 Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Questio: Whe is a miimal sufficiet statistic idepedet of every acillary statistic? Defiitio 6.. Let f ( t ) be a family of pdfs ad pmfs for a statistic T ( X ). The family of probability distributios is called complete if EgT ( ) 0 for all implies P ( g( T) 0). Equivaletly, T ( X ) is called a complete statistic. Illustratio: Cosider the family of distributios (,),. If g( X) X, the EgX ( ) EX 0 whe 0 but PgX ( ( ) 0) PX ( 0) 0 sice X is a cotiuous radom variable. So this family of distributios is complete for. Example 6.. (Biomial complete sufficiet statistic) Suppose that T has a biomial( pdistributio, ) with 0 p. Show that T is a complete statistic. Example 6..3 (Uiform complete sufficiet statistic) Let X,, X be iid uiform (0, ),0, observatios. Show that T X( ) is a complete statistic.
13 Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Theorem 6..4 (Basu s Theorem) If T ( X ) is a complete ad miimal sufficiet statistic, the T ( X ) is idepedet of every acillary statistic. Note: Basu s Theorem allows us to deduce the idepedece of two statistics without ever fidig the joit distributio of the two statistics. Questio: Is there a easier way to fid a complete statistic? Theorem 6..5 (Complete statistics i the expoetial family) Let X,, X be iid observatios from a expoetial family with pdf or pmf of the form Where (,, k ). The the statistic f ( x θ) h( x) c( θ)exp( wi( θ ) ti( x)), k i j j j j j k j T( X) ( t ( X ), t ( X ),, t ( X )) is complete as log as the parameter space cotais a ope set i k R. Note: The distributio (, ) (recall from Example that this distributio is a member of the curved expoetial family of distributios) does ot cotai a twodimesioal ope set because it cotais oly poits o the parabola. Hece, this distributio would ot satisfy the coditios of Theorem
14 Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 I fact, oe ca fid a fuctio, gt ( ( X)) 0 such that EgT ( ( X )) 0. (Exercise 6.5) A Reparameterizatio of Expoetial Families (Caoical Form): f ( x ) h ( x ) c ( *)exp( t ( x )), X where hx ( ) ad t ( x) are the same as i the origial parameterizatio. The set k i i k { (,, ): hx ( )exp( t( x)) dx }, which is called the atural parameter space for the family. i i k i i i Example (Reparameterizatio of the Normal Distributio) Solutio: f ( x, ) exp( )exp( x x ), where / ad /. Defiitio 3.4.7: A Curved expoetial family is a family of desities of the form (3.4.) for which the dimesio of the vector is equal to d k. If d k, the family is a full expoetial family. Example 3.4.8: Normal with mea ad variace. 4
15 Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Example 6..6 (Usig Basu s Theorem I) Let X,, X be iid expoetial( ) observatios. Compute Eg( X ) where X g( X) X X. Example 6..7 (Usig Basu s Theorem II) Let X,, X be iid observatios from Basu s Theorem, show that X ad S are idepedet. (, ) populatio. Usig Theorem 6..8 If a miimal sufficiet statistic exists, the ay complete statistic is also miimal sufficiet. Note: A miimal sufficiet statistics may ot be a complete statistic. Example (A miimal sufficiet statistics that is complete) Let X,, X be iid uiform observatios o the iterval (, ),. From example 6..5, we kow that T( X ) ( X(), X( ) ) is a miimal sufficiet statistic. From example 6..7, we kow that R( X ) X( ) X(), is a acillary statistic. I additio, we kow that R( X ) has a beta pdf with ad. Thus we have E ( X( ) X() ) 0, but PX ( ( ) X() 0) 0. So T( ) ( X, X ) X is ot complete. () ( ) 5
16 Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Sectio 6.3 Likelihood Priciple I his sectio, we study a specific, importat statistic called the likelihood fuctio that also ca be used to summarize data. Defiitio 6.3. Let f ( x ) deote the joit pdf or pmf of the sample X ( X,, X ). The give that X=x is observed, the fuctio of defied by L( x) f ( x ) is called the likelihood fuctio. Questio: What is the differece betwee the likelihood fuctio ad the pdf (or pmf)? Aswer: They have the same formula. The distictio betwee them is which variable is fixed ad which is varyig. Likelihood Priciple If x ad y are two sample poits such that L( x ) is proportioal to L( y ), i.e., there exists a costat C( x,y ) such that L( x) C( x,y) L( y ), the the coclusios draw from x ad y for should be idetical. Notes: 6
17 Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0. Likelihood priciple states that eve if two sample poits have proportioal likelihoods, the they will cotai equivalet iformatio about.. Give two parameter values ad, the likelihood fuctio tells us if is a more plausible (ot probable) parameter value tha i light of the data gathered. 3. Fiducial iferece (Fisher, 930) iterprets likelihood as probabilities for very much like Bayesia iferece. Skip Sectio 6.3. Formal likelihood priciple Example 6.3. (Negative biomial likelihood) Let X have a egative biomial distributio with r 3 ad success probability p. If x is observed, the the likelihood fuctio is the fifthdegree polyomial o 0 p defied by 4 3 L( p ) Pp ( X ) p ( p). I geeral, if X x is observed, the the likelihood fuctio is the polyomial of degree 3 x, 3 x L p x p p x 3 ( ) ( ) x. 7
18 Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Example (Normal fiducial distributio) Let X,, X be iid L (, ), kow. The / ( x) ( ) exp( ( x i ) / ( )) i / ( ) exp{ ( ( x ) ( ) ) / ( )} i i x x / / i i We first ote that C( x,y ) exists if ad oly if x exp( ( x x) / ( ))( / ) exp( ( x) / ( / )). y, i which case i i i i C ( x,y ) exp( ( x x ) /( ) ( y y ) /( )). Thus, the Likelihood Priciple states that the same coclusio about should be draw for ay sample poits satisfyig x y. Secod, up to a costat, L( x ) has a ormal distributio (, / ) x, thus we have x 0.95 P(.96.96) P( x.96 / x.96 / ). / Example (Likelihood Fuctio for Uiform Distributio) Let X,, X be iid uiform (0, ), the the likelihood fuctio is: L( x) I[0 x( ) ]( x,, x ). 8
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