R2_STAT_2006

R2_STAT_2006 - 1 R. 2 James B. McDonal d Br i gham Young...

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Unformatted text preview: 1 R. 2 James B. McDonal d Br i gham Young Uni ver si t y 8/ 01 R. 2 An Out l i ne of St at i st i cal Concept s A. Basi c Concept s and Def i ni t i ons 1. Sampl e space 2. Random var i abl es di scr et e cont i nuous 3. Densi t y f unct i ons pr oper t i es uses exampl es B. Ext ensi ons of Basi c Concept s 1. Joi nt densi t y of sever al var i abl es 2. Rel at i onshi p bet ween t he j oi nt densi t y and mar gi nal densi t i es 3. St ochast i c i ndependence C. Mat hemat i cal Expect at i on 1. Def i ni t i ons a. Expect ed val ue of r andom var i abl e b. Expect ed val ue of a f unct i on of r andom var i abl es c. Popul at i on var i ance as an expect ed val ue d. Popul at i on covar i ance and cor r el at i on 2. Basi c pr oper t i es of expect at i on a. Expect at i on and var i ance of a l i near combi nat i on of r andom var i abl b. Var i ance of a l i near combi nat i on of r andom var i abl es 3. Exampl es 2 R. 2 D. Some Random Var i abl es and Thei r Pr oper t i es and Rel at i onshi ps densi t y par amet er s mean 1. Nor mal , 2 2. Chi - squar e v=d. f v 3. t - st at i st i c v=d. f 0 4. F- st at i st i c v 1 =d. f . 1 v 2 v 2 =d. f . 2 v 2- 2 E. Met hods of Est i mat i on Random sampl e: Y 1 , Y 2 , . . . , Y n Unknown Par amet er s: 1. Met hod of moment s Pr i nci pl e: Sel ect par amet er est i mat or s whi ch equat e t he sampl e moment s t o t he cor r espondi ng t heor et i cal moment s. Exampl e: Pr oper t i es: Consi st ent and asympt ot i cal l y nor mal under f ai r l y gener al condi t i ons. 2. Least Squar es Pr i nci pl e: Sel ect par amet er est i mat or s whi ch mi ni mi ze t he sum of squar ed er r or s ( ver t i cal devi at i ons) Exampl e: Pr oper t i es: Case- by- case 3. Maxi mum l i kel i hood Pr i nci pl e: Sel ect par amet er s whi ch maxi m i ze t he l i kel i hood f unct i on Exampl e: Pr oper t i es: Consi st ent , asympt ot i cal l y ef f i ci ent and asympt ot i cal l y nor mal l y di st r i but ed under qui t e gener al condi t i ons 4. Best Li near Unbi ased Est i mat or s ( BLUE) Pr i nci pl e: Const r uct l i near unbi ased est i mat or s whi ch have mi ni mum var i ance Exampl e: Pr oper t i es: Unbi ased, mi ni mum var i ance of l i near unbi ased est i mat or s 3 R. 2 F. Pr oper t i es of Est i mat or s 1. Smal l sampl e pr oper t i es unbi ased est i mat or mi ni mum var i ance est i mat or MSE 2. Lar ge Sampl e pr oper t i es asympt ot i cal l y unbi ased consi st ency asympt ot i cal l y ef f i ci ent 3. Exampl es G. I nt er val Est i mat i on and Hypot hesi s Test i ng Test St at i st i cs Hypot hesi s ________ N( 0, 1) _____ _ t ____ ___ 2 ________ F H o : = o unknown H o : = o known H o : 2 = 2 H o : 2 1 = 2 2 4 R. 2 R. 2 Revi ew of St at i st i cal Concept s A. Basi c Concept s and Def i ni t i ons 1. The sampl e space associ at ed wi t h an exper i ment i s a set S of el ement s such t hat any out come of t he exper i ment cor r esponds t o one and onl y one el ement of t he set . one and onl y one el ement of t he set ....
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R2_STAT_2006 - 1 R. 2 James B. McDonal d Br i gham Young...

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