: 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?
