統計學 : 應用與進階 第 7 章 : 抽樣與抽樣分配

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• : 7 :

• , , , , , , (),

• , , , , , (sample)(statistical inferences)

• (random samples)

n

• :{X1, X2, . . . , Xn} f (),{X1, X2, . . . , Xn} , I.I.D. (I.I.D. samples): , 2 , , I.I.D. , i.i.d. (, )

• : i.i.d. (, ) E(X1) = E(X2) = = E(Xn) = ,Var (X1) = Var (X2) = = Var (Xn) = ,i j ,E(XiXj ) = E(Xi )E(Xj).

! (ex ante)

• (statistics): T = t(X1, X2, . . . , Xn), , (sample mean)

(sample variance)

• (statistics)(sample covariance)

(sample correlation coefficient)

• (statistics)r (sample r -th moments)

• , , , {X1, X2, . . . , Xn} ,t(X1, X2, . . . , Xn)

• , (exact distribution), , , (asymptotic theory) ,(asymptotic distribution)

• : Z1, Z2 . . . , Zk k N(0, 1),W (Chi-square distribution)

k (degree of freedom)

• : , (support) supp(W) = {w|0 w < }

:

, Gamma

• : r

,

,

• : k N(0, 1) , (k) , W (k) , W k N(0, 1)

• : : X (kX ), Y (kY ), X, Y, X + Y (kX + kY ), , , n > 2, n ,

• (k = 3, 7, 9)

• : t t William Sealy Gosset (18761937) Gosset (Guinness Brewing Co.) , , , Gosset Student , t Students t

• : t : Z N(0, 1),W (k),

U t (t distribution) :

k t

• : t t :

supp(U) = {u| u < }

• t , t , t(k) N(0, 1) as k .

k = 1, t(1) , : E[t(1)] = , t(1) (standard Cauchy distribution)

• t : k = 1(), k = 300 ()

• : F X1 X2 :

(n1, n2) F

• : F :

(support) supp(F) = {f |0 f < }

• : F X F(n1, n2) Y = 1/X , Y F(n2, n1)t t(k), t F(1, k)

• F (n1 = 10, n2 = 10)

• Critical Values of F-Test 00F0Note!

• , (exact distribution) N(,) n

• ,

66The lower F is not the reciprocal of the upper F.What do you do if equal sample sizes?

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