Useful Statistical Identities, Inequalities and Manipulations

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Leslie Marsh
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1 Useful Statstcal Iettes, Iequaltes a Mapulatos Cotets Itroucto Probablty Os, Log-Os PA ( ) π Let π PA ( ) We efe the Os Rato θ If θ we say that the patet s three PA ( ) π tmes more lely to have the sease We also symmetrze ths to evece va the log os efto: π L Logt( θ) log( θ) log, L (, ) π he uts are ether Bels (base 0) or Napers (base e) We ca retur to probabltes by tag the verse L e π PA ( ) + e L a π PA ( c ) + L e Momets Calculatos volvg populato/sample meas a varaces, a er/outer proucts: Populato [Sample sze ] NOE: θ σ ( E( X ), Var( X )) (, ) (" "," ") E X ( X, S! Var( X ) Cov( X, X ) E( X ) ( ) ( x ) ( x x) X R ( ) uless X s a screte rv a f ( x X ) x Supt( f ) Var( X) Cov( XX, ) EX ( )( X ) EXX X R ( ) X X X ) Cov( X, ) E( X X)( ) EX X X (x), X R R etc N J Dobelma - StatHayoc, Rev 6//5 of 7 Prte: 6/8/5

2 Useful Statstcal Iettes, Iequaltes a Mapulatos Sample [sze ] ˆ m X ; m X ˆ ; m m S ( X X ) ( x ) ( x x) + x ( ) ( ) ( X X) X XX + X X X S ( ) S ( S ubase) NOE: x ( x x) + x S X X S S S ( )( ) X X X X XX XX Σx ( x ) X R, s s s s s s s s x Σ Σ j Sj Σxx j xx j, j,, s s s s, For X x, x Σ x Σx x Σxx xx Σx x Σxx xx xx xx x x Σ Σ ` Σxx x x x x Σ `, ( )( ) X X X X, X R, R, ( x ) X X Calculatos volvg populato/sample hgher-orer momets ( r a th), a tesor otato: Populato [] NOE: Sew[ess] Pearsoa otato: Sew S where / me mo σ x x, the stace from mea to moe He also κ use β Later, Moer otato for sewess becameγ β β / / κ σ Recall that E( X )( X )( X ) ( X ) f( x θ) x For vector X, tesor otato s requre to geeralze J Dobelma - StatHayoc, Rev 6//5 of 7 Prte: 6/8/5

3 Useful Statstcal Iettes, Iequaltes a Mapulatos Kurtoss β Sce β for the ormal strbuto, Pearso also use a excess urtoss β β hs κ ow has a moer otato ofγ β Note that terms of the cumulats, γ κ Sample [sze ] he sewess a urtoss are ormally estmate usg metho of momets Staar errors for these statstcs for both the ormal a the Pearso system of strbutos are prove Pearso, Flo, etc [895, 900, 90, 907] Sample versos of requre moe estmato of etermato of a Pearso curve sce may moe estmators are hghly sestve to samplg varato More later No-epeece of hgher momets ˆβ a ˆβ are ecessarly ot epeet of themselves a the usual estmators of a Covarace Correlato hese ettes requre creatg a xagoal matrx wth the varaces as elemets hese ca be geerate by frst creatg a varace vector, a by usg the ut matrces E : E 0,{ E } a e : e 0, { e } Populato V σ ag( ),, or σ Eve where v ( σ ) σ x Ρ V V V V σ j ρj σσ j V Ρ V σ ρσσ Note: I Matlab a R, V j j j chol( V ) J Dobelma - StatHayoc, Rev 6//5 of 7 Prte: 6/8/5

4 Useful Statstcal Iettes, Iequaltes a Mapulatos Sample S E se where s ag( ˆ ) R S ˆ S r j sj sj s s ss jj j ˆ SRS s r ss, or ˆ σ r ˆˆ σσ j j j j j j Dstrbutoal Fouatos Bra Name Dsrbutos See for example my Rce webste Beta( α, β ): Dstrbutoal Relatoshps a Iettes beta(x, theta[], theta[]) Beta Dstrbutos \equatos\strbutos\hayrelatoshpswmf x p FX( ) bom( x, p) Beta( y, + ) y F( p) x 0 0 If z φ ( z) N(0, ), z ~ χ, or For z zz χ N(0, ), σz ε σ σ ε ε χ ; so f ~ gamma (, ) gamma(,) χ, or gamma(,) χ ~ gamma(, ) a X y ~N (, y ) X β, ( β) X~ gamma(,) χ Σ, the X~ t (, Σ, ) X ~ gamma( α, β), ~ Pos( x β ), the PX ( < x) P ( α) X~Pos( ) E ( ) E ( ), δ 0 θ, ( S X δ δ+ ) ( X ) X ~ expo( β ) the X / γ ~ Webull( γ, β) X ~ expo( β ) the X β ~ Raylegh X ~ expo( β ) a β the α γl X ~ Gumbel( αγ, ), α R, γ > 0 X ~ expo( λ) a ~ Erlag(, λ ) the X/ ~ Pareto(, ) J Dobelma - StatHayoc, Rev 6//5 of 7 Prte: 6/8/5

5 Useful Statstcal Iettes, Iequaltes a Mapulatos X ~ gamma(, β) expo( β) X ~ gamma( α, β ) the / X ~ IG( α, β ) X ~ gamma(5, β ) the X/ β ~ Maxwell X X ~ gamma( αθ, ), ~ gamma( βθ, ) the ~ beta( α, β) X+ X X ~ χ ( α), ~ χ ( β) the ~ beta(, ) X+ X ~ uf(0, ), α > 0 the X / γ ~ beta( α, ) X ~ uf(0, ) beta(, ) X ~ beta(, ) the l X ~ expo() a f X ~ beta( α, ) the l X ~ expo( α) X ~ beta( α, β ) the X ~ beta( βα, ) Samplg a Certa Statstcs Quatle fucto he Quatle fucto s the verse of the f: ξ p FX ( p ) f{ xfx : ( ) p } ξ ξ p q ξr ξ s a b c Some Samplg Facts XS, X ~ Normal E( S ) σ ; moreover, { } X must be a SRS J Dobelma - StatHayoc, Rev 6//5 5 of 7 Prte: 6/8/5

6 Useful Statstcal Iettes, Iequaltes a Mapulatos pf: E( S) E ( X X) E X X pf: σ ( E( ) E( )) X X σ σ + ( ) σ Var( ) E( ) E( ) 0 S E( ) E( ) Let σ E( S ) E( S), or E( S) σ ( ) ( ) I geeral, ( + ) X N σ ~ (, ): σ, σ σ ( ) X ~ Uf( ab, ) : ( b a) 9, 5 96( ) 5( )( b a) 70 ( ) X ~ t :, ( )( ) ( 5) ( ) ( ) X ~ χ :, ( + ) ( )( + ) ( ) ( ) X ~ expo( θ ) : θ, 9θ J Dobelma - StatHayoc, Rev 6//5 6 of 7 Prte: 6/8/5

7 Useful Statstcal Iettes, Iequaltes a Mapulatos ( ) θ ( ) X ~ Posso( λ ) : λ, λ(λ + ) λ λ λ λ + + Orer Statstcs Dahya a Gurla, Bometrcs v5, No (060) pp7-7 Cho, E & M J Cho, 008 Varace of sample varace Proceegs of the 008 Jot Statstcal Meetgs, Secto o Survey Research Methos, Amerca Statstcal Assocato, Washgto DC,9-9 J Dobelma - StatHayoc, Rev 6//5 7 of 7 Prte: 6/8/5

M2S1 - EXERCISES 8: SOLUTIONS

M2S1 - EXERCISES 8: SOLUTIONS MS - EXERCISES 8: SOLUTIONS. As X,..., X P ossoλ, a gve that T ˉX, the usg elemetary propertes of expectatos, we have E ft [T E fx [X λ λ, so that T s a ubase estmator of λ. T X X X Furthermore X X X From