Chapter 5 Properties of a Random Sample

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1 Lecture 3 o BST 63: Statstcal Theory I Ku Zhag, /6/006 Revew for the revous lecture Cocets: radom samle, samle mea, samle varace Theorems: roertes of a radom samle, samle mea, samle varace Examles: how to calculate the df (mf) of the samle mea Chater 5 Proertes of a Radom Samle Secto 5 Sums of Radom Varables from a Radom Samle Two mortat famles of dstrbutos: x µ locato-scale famly - f( x µσ, ) = fz( ): σ σ Cosder a radom samle,, ad Z,, Z where = µ + σ Z( =,, ), the = σ Z + µ Therefore, f g ( z ) s the df of Z, the has the df (a locato-scale famly dstrbuto): Z x µ f ( x) = g ( ) Z σ σ Examle: If Z s Cauchy(0,) (from Examle 50), the t follows that s Cauchy( µ, σ ) f,, s the radom samle from the oulato wth Cauchy( µ, σ )

2 Lecture 3 o BST 63: Statstcal Theory I Ku Zhag, /6/006 Exoetal famly - f ( x θ) = h( x) c( θ)ex( w( θ) t ( x)) : = Theorem 5: Suose,, s a radom varable from a df or mf f ( x θ ) where s a member of a exoetal famly Defe T, T by f ( x θ) = h( x) c( θ)ex( w( θ) t( x)), j= j = T(,, ) = t ( ), =,, If the set {( w ( θ ),, w ( θ)), θ Θ} cotas a oe subset of exoetal famly of the form R, the the dstrbuto of s T θ = θ = θ f ( u,, u ) H( u,, u )[ c( )] ex( w( ) u ) ( T,, T ) a Notes: c( θ ) ad w ( θ ) are the same for f ad f T Ths s ot true for the curved exoetal famly Examle 5 (sum of Beroull radom varables): Suose,, s a radom varable from a x x Beroll( ) oulato: f ( x ) = ( ) = ( )ex( xlog( /( ))), thus =, c( ) =, w ( ) = log( /( )), ad t ( x) = x Thus T(,, ) = + + From the theorem 5, we have f ( u) = H( u)( ) ex( ulog( /( ))) T

3 Lecture 3 o BST 63: Statstcal Theory I Ku Zhag, /6/006 Secto 53 Samlg from the Normal Dstrbuto Theorem 53: Let,, be a radom varable from a = [/( )] ( ) = S a ad b has a S are deedet The ( µσ, / ) dstrbuto ( µ, σ ) dstrbuto, ad let = (/ ) ad = c ( ) S / σ has a ch squared dstrbuto wth degrees of freedom Lemma 53 (Facts about ch squared dstrbuto): We use χ deote a ch squared radom varable wth degrees of freedom a If Z s (0,) radom varable, the b If,, are deedet ad + + Z ~ χ ~ χ, the + + s ch squared dstrbuted wth degrees of freedom Lemma 533: Let j ~ ( µ j, σ j), j =,,, deedet For costats a j ad b rj ( j =,, ; =,, ; r =,, m), where + m, defe U = a,,,, j j j = ad = Vr = b,,, j rj j r = m = 3

4 Lecture 3 o BST 63: Statstcal Theory I Ku Zhag, /6/006 a U ad V r are deedet f ad oly f Cov( U, V r) = 0 Furthermore, r a jb j rjσ = j Cov( U, V ) = b The vectors ( U,, U ) ad ( V,, V m ) are deedet f ad oly f U s deedet of V r for all ars (, r)( =,, r ; =,, m) Note: Ths result (a) of ths Lemma mles that for two ormal radom varables to be deedet, we oly eed to show that ther covarace s 0 Suose,, be a radom varable from a ( µ, σ ) oulato We wat to ow the dstrbuto of where U ~ (0,) ad V ~ µ ( µ )/ σ / U = = S S / σ V / /( ) χ ad U ad V are deedet, Defto 534 (Studet s T dstrbuto) Let,, be a radom samle from a ( µ, σ ) oulato The µ quatty has a Studet s t dstrbuto wth degrees of freedom Equvaletly, a radom varable S/ T t f t has a df gve by ~ + Γ( ) f () T t =, < t < / ( + )/ Γ( ) ( π ) ( + t / ) 4

5 Lecture 3 o BST 63: Statstcal Theory I Ku Zhag, /6/006 Notes: If =, we get a Cauchy(0,) dstrbuto I the radom samle settg, ths haes whe = If T ~ t the oly momets exst I artcular, E( T ) = 0, > ad Var( T ) = 3 Mgf for Studet s t dstrbuto does ot exst 4 I geeral, f U ~ (0,), ~ f > V χ ad U ad V are deedet, the T = U / V / ~ t Next cosder two deedet radom samles: We wat to ow the dstrbuto of µ σ ad,, ~ (, ) S / S σ σ Y / Y Y Y µ σ,, ~ (, ) Y Y Defto 535: (Sedecor s F dstrbuto amed hoor of Sr Roald Fsher) Let,, be a radom samle from a ( µ, σ ) oulato ad,, Y Ym be a radom samle from a deedet ( µ Y, σ Y) oulato S / SY The radom varable F = has a F dstrbuto wth (umerator degrees of freedom) ad σ / σ Y (deomator degrees of freedom) m Equvaletly, the radom varable F has the F dstrbuto wth ad q degrees of freedom f t has df 5

6 Lecture 3 o BST 63: Statstcal Theory I Ku Zhag, /6/006 Notes: + q Γ( ) x f x x q Γ( ) Γ( ) q [ + / q) x] ( /) / F ( ) = ( ),0 < < ( + q)/ Keler (970) showed that as log as the aret oulatos have a certa tye of symmetry (shercal symmetry), the the rato the defto wll have a F dstrbuto I geeral, f U χ ad V ~ χ q ad U ad V are deedet, the F, ~ ad q degrees of freedom 3 F dstrbuto s commoly used Aalyss of Varace methods 4 If ~ F q, the / ~ F q, 5 If ~ t q, the ~ F, q q U / = has a F dstrbuto wth V / q 6