Statystyka matematyczna i teoria estymacji

X
Rn
X (X , A)
P�
P = {P� : � " &!}
(X , A, P)
X X = (X1, . . . , Xn) X1, . . . , Xn
(X , A, P)n
(X1, . . . , Xn) P� �
Rk T = (T1, . . . , Tk)
X
-1
" T (B) " A.
B"B(Rk)
(X1, . . . , Xn)
n
"
1
X = Xi.
n
i=1
(X1, . . . , Xn)
n
"
1
S2 = (Xi - X)2.
n - 1
i=1
(X1, . . . , Xn)
n
"
1
Fn(t) = 1 (Xi), t " R.
(-",t]
n
i=1
Fn
(X1, . . . , Xn)
F t " R
EF [Fn(t)] = F (t)
( )
PF lim Fn(t) = F (t) = 1
n"
"
Fn(t) - F (t)
n
F (t)[1 - F (t)]
N (0, 1) n "
Dn = sup |Fn(x) - F (x)|
x"R
(X1, . . . , Xn) F Fn
Dn 0 n "
1
(X1, . . . , Xn)
P " P F Fn
Dn = sup |Fn(x) - F (x)| x " R
n"N
"
lim P ( nDn x) = K(x),
n"
"
"
2k2x2
K(x) = (-1)k 1 (x).
(0,")
k=-"
(X1, . . . , Xn) P " P
F
n
"
1
Ak(X) = Xik.
n
i=1
x Ak
Fn(�; x) Fn(�; X)
+"
n
"
1
Ak(x) = xk = tkdFn(t, x).
i
n
i=1
R
k
EXk < " Ak(x) EX1
(X1, . . . , Xn)
n
"
1
Mk(X) = (Xi - X)k,
n
i=1
X
X1, . . . , Xn
�
X1(�), . . . , Xn(�)
X(1)(�) X(2)(�) . . . X(n)(�).
� X(1) . . . X(n)
X(1) = min{X1, . . . , Xn} X(n) = max{X1, . . . , Xn}
X1, . . . , Xn P " P
F X(k)
F
+"(x)
n!
FX (x) = tk-1(1 - t)n-kdt.
(k)
(k - 1)!(n - k)!
0
FX
(k)
B(k, n - k + 1) F
X1, . . . , Xn
f k
Xk
n!
fX (x) = f(x)[F (x)]k-1[1 - F (x)]n-k.
(k)
(k - 1)!(n - k)!
p " (0, 1) (X1, . . . , Xn)
{
X(np), np " Z,
Zn,p =
X([np]+1), np " Z,
/
(X1, . . . , Xn) P " P
F p " (0, 1) śp
F (x - 0) p F (x) F (x - 0) = limx0- F (x) śp
p F śp Zn,p śp
n "
1
2
Med X = Zn, 1
2
{
n+1
X( ), n ,
2
Med X =
1
n n
(X( ) + X( +1)), n ,
2
2 2
T X
P = {P� : � " &!}
X � t
X T = t �
(X , A) X A
� X � (X , A)
+"
fd� = 0.
A
f = 0 A �(A) = 0
f = 0 A
f A 0
f A
+"
fd� = 0 �! f = 0 A,
A
f A
+"
fd� = 0 �! �(A) = 0.
A
f
+"
�(A) = fd�,
A
A � � �
(X , A)
(X , A) � �
A " A
+"
�(A) = f�.
A
�(A) = 0 �! �(A) = 0.
�
�
f
+"
" fd� = �(A).
A"A
A
f � �
� � f
�
�
+"
g �(A) = gd� f = g
A
+"
� fd� = 0
A
A " A f = 0 � �
� � << �
P
� " P << � P
P "P
�
" X
card X 5!0 A X
A = 2X P
" X
A X P
(X , A)
A P
" P (A) = 0.
P "P
P
P P
� " P (A) = 0
P "P
�(A) = 0 �
P
T
P = {P� : � " &!} �
� g� h p�
P�
p�(x) = g�[T (x)]h(x) �.
(X1, . . . , Xn)
� Xi <" Poiss(�) i = 1, . . . , n
n -�
1 -�
�x �x
p�(x1, . . . , xn) = p�(x1) . . . p�(xn) = � . . . � =
x1! xn!
"n
-n� n
"n "
i=1
� 1
-n�
i=1
= � .
x1! � . . . � xn! xi!
i=1
n
"
1
T (x) = xi g�[T (x)] = �nT (x) -n�
n
i=1
n
"
1
h(x) =
xi!
i=1
p�(x1, . . . , xn) = g�[T (x)]h(x).
n
"
1
T (X) = Xi
n
i=1
T U P
g h T = g(U) P U = h(T ) P
T
U g T = G(U) P
V (X)
�
E�[V (X)] �
�
T
T
E�[f(T )] = c � " &!
f(t) = c P
" E�[f(T )] = 0 �! f(t) = 0 P.
�"&!
T
" E�[f(T )] = 0 �! f(t) = 0 P
�"&!
(X1, . . . , Xn) Xi <" U(0, �) � > 0
T = X(n) �
" E�[f(T )] = 0.
�>0
t
Xn : fX (t) = n(� )n-1 11 (t)
(0,�)
(n)
�
+"� ( )n-1
t
" E�[f(T )] = f(t)n dt = 0.
�>0
�
0
f(t) = 0
P
f(t) = 0 P T
{P� : � " &!}
P�
�
�ł �ł
s
"
�ł
p� = exp �i(�)Ti(x) - B(�)łł � h(x),
i=1
�i B � Ti
X
s
�ł �ł
s
"
�ł
p�(x) = exp �i(�)Ti(x) - B(�)łł � h(x),
i=1
T = (T1, . . . , Ts)
T
P = {P� : � " &!} V
T
(X , A, P) P = {P� : � " &!}
X
g(�) &!
g(�) �(X)
R
�(x) �(X) X
x g(�)
g(�)
g(�)
d L(�, d)
L
L(�, d) 0, �, d
L[�, g(�)] = 0 �
L(�, d) = (d - g(�))2
�
R(�, �) := E�[L(�, �(X))],
� �
�
L[�, g(�)] = 0 � �0
�(X) x g(�0)
g(�)
�
�(X) g(�)
E�[�(X)] - g(�) �(X)
�(X) g(�)
E�[�(X)] = g(�)
�
Ć
I X
P (X " I) = 1
Ć(EX) E[Ć(X)].
Ć
X X
X X
2
P� " P = {P� : �2 " &!} T
P � g(�) L(�, d)
d d
R(�, �) = EL[�, �(X)] < "
�(t) = E[�(X)|T = t].
�(T )
R(�, �) < R(�, �),
�(X) = �(T )
�
�2
" R(�, �2 ) R(�, �)
�"&!
R(�0, �2 ) < R(�0, �) �0.
�
L �
g(�) �2
R(�, �) = R(�, �2 ) � � = �2
ń : R R X
Ć(a) = E[ń(X - a)] a
Ć Ć(a)
ń
�(X) g(�)
" E�[�(X)] = g(�).
�"&!
X
1
Bin(�, n) � " (0, 1) g(�) =
�
( )
n
"
n 1
" �(x) �x(1 - �)x = .
�"(0,1)
x �
x=0
�
�(0) +" � 0
�0 g(�)
� =
�0 - U U
" E�U = 0.
�"&!
�
� �0 U �0
Var� � = Var�(�0 - U) = E�(�0 - U)2 - (E��)2 = E�(�0 - U)2 - [g(�)]2.
� E�(�0 -
U)2
" � " "
E��2 < " � " &!
X P (X = -1) = p
P (X = k) = q2pk k = 0, 1, 2, . . . p " (0, 1) q = 1 - p
p q2
� " "
" " E�(�U) = 0,
�"&! U"U
U "
E�U = 0
(1) �! " " Cov�(�, U) = 0.
�"&! U"U
g
g(�)
NJMW
NJMW
U
NJMW
U
NJMW
U NJMW
T P = {P� : � " &!}
g(�) U
� g(�)
T
T �(T ) �(T )
E�[�(T )] = E�[�(T )] = E[E[�(X)|T ]] = E�[�(X)] = g(�)
� " &!
g(�) T �(Y )
NJMW g(�)
X P = {P� :
� " &!} T P
g(�)
T
P
X
P = {P� : � " &!} T
P L(�, d)
g(�)
(NJMW )
NJMW
T
NJMW g(�)
P s
� = (�1, . . . , �2) T = (T1, . . . , Ts)
X1, . . . , Xn N (�, �2)
�2 �2
n
"
1
�(X) = (Xi - X)2,
n - 1
i=1
n n n
" " "
1
X = Xi T = ( Xi, (Xi-
n
i=1 i=1 i=1
X)2) � NJMW
T
NJMW U g(�)
E�[�(T )] = g(�)
� " &!
�(X)
g(�) NJMW
� T
�
�2 (T ) = E[�(X)|T ]
(X , A, P) P << � � �
P = {P� : � " &!}
p� {x : p�(x) > 0} X R
&! R
p�(x) &! P
"
I(�) = E�["� ln p�(X)]2 " (0, ") I(�)
+"
"
p�(x)d�(x) = 0
"�
X
(X1, . . . , Xn)
�
U g(�)
+" +"
" "
g2 (�) = �(x) p�(x)d�n(x) = �(x)p�(x)d�n(x),
"� "�
n n
X X
[g2 (�)]2
Var� �
nI(�)
"
ln p�(x) = a(�)[�(x) - g(�)] �,
"�
a(�) �
� g(�)
� " &!
R(�, �)
+"
R(�, �)d�(�).
�
+"
d�(�) = 1.
�
+"
� R(�, �)d�(�)
�
�
Ś �
�
�
n
�
�
�
X = x �
� X = x
d E[L(�, d)]
X = x �(X)
E[L(Ś, �(X))|X = x]
Ś � Ś = �
X P� g(�)
L(�, d)
�0
x ��(x) E[L(Ś, �(X))|X =
x]
��(X)
L(�, d) = (d - g(�))2 ��(x) = E[g(Ś)|X = x]
E[w(Ś)-g(Ś)|X=x]
L(�, d) = w(�)(d - g(�))2 ��(x) =
E[w(Ś)|X=x]
L(�, d) = |d - g(�)| ��
g(Ś)|X = x
� sup R(�, d)
�"&!
sup R(�, �)
�"&!
��
+"
r� = R(�, ��)d�(�).
�
2
�2 r� r�
��
� ��
(�n) �n
�n
+"
rn := R(�, �n)d�n(�) r.
(�n)
� r� < r
(�n) �
�n
+"
rn := R(�, �n)d� r
�
sup R(�, �) = r.
�"&!
�
(�n)
(X1, . . . , Xn) p�(x)
� � " &! �" Rd L(�; x1, . . . , xn) : &!
R
n
"
L(�; x1, . . . , xn) = p�(xi)
i=1

� �(X1, . . . , Xn)

L(�(x1, . . . , xn); x1, . . . , xn) = sup L(�; x1, . . . , xn)
�"&!
(x1, . . . , xn)

g(�) g(�) �
�
P P� � " &!
p� � &!
�0
� = �0 (X1, . . . , Xn)
8
P
P� (p� (X1) � . . . � p� (Xn) > p�(X1) � . . . � p�(Xn)) 1,
0 0 0
n "
X P
T
[�] [�] T
(X1, . . . , Xn)
k
�ł �ł
k
"
�ł
p�(x) = exp �i � Ti(x) - B(�)łł � h(x),
i=1
� " &! �" Rd
�ł łł
"2B(�)
�ł �ł
C =
"�i"�j
1 i,j k

�(X1, . . . , Xn)

(x1, . . . , xn) �(x1, . . . , xn)
n
"
1 "
Ti(xj) = B(�), i = 1, . . . , k.
n "�i
j=1
(X1, . . . , Xn)
T �
T [�]
(X , A, P) P =
{P� : � " Ś}
(X1, . . . , Xn)
� " Ś
� " Ś 1 - ą (0 < ą < 1)
(�1, �2)
�1 = �1(X1, . . . , Xn) �2 = �2(X1, . . . , Xn)
(X1, . . . , Xn) �
� " Ś
P (�1(X1, . . . , Xn) < � < �2(X1, . . . , Xn)) = 1 - ą.
(�1, �2)
�
(�1, �2) 1 - ą
(X1, . . . , Xn)
EXi = � Var Xi = �2 < "
�2 �2
P (|Xn - �| �) �! 1 - P (|Xn - �| < �) �!
n�2 n�2
�2 �2
P (|Xn - �| < �) 1 - �! P (Xn - � < � < Xn + �) 1 - .
n�2 n�2
�2 = 1 0, 99
10
"
n�2 100 � =
n
" "
P (Xn - 10/ n < � < Xn + 10/ n) 0, 99.
(X1, . . . , Xn) N (�, �2)
" � �2 Xn N (�, �/
n)
Xn - �
U = "
�/ n
N (0, 1) ą u1 u2
P (u1 < U < u2) = Ś(u2) - Ś(u1) = 1 - ą.
ą ą = ą1 + ą2 ą1, ą2 > 0
u1 = u(ą1) u2 = u(1 - ą2) u(ą1) u(1 - ą2)
ą1 1 - ą2 U
( )
"
Xn - �
P u(ą1) < n < u(1 - ą2) = 1 - ą.
�
ą
ą1 = ą2 =
2
�ł �ł
( (
ą) � ą) �
�ł łł
Xn - u 1 - " , Xn + u 1 - " .
2 n 2 n
(X1, . . . , Xn) N (�, �2)
� �2
Xn - �
"
t = ,
Sn/ n - 1
n
"
2 1
Sn = (Xi-Xn)2 t n-1
n
i=1
t(ą, n - 1) ą
�ł �ł
( ) ( )
ą Xn - � ą
�ł łł
"
P t , n - 1 < < t 1 - , n - 1 = 1 - ą.
2 Sn/ n - 1 2
ą
t(ą, n - 1) = -t(1 - , n - 1)
2 2
�ł �ł
( ) ( )
ą Sn ą Sn
�ł łł
" "
Xn - t 1 - , n - 1 , Xn + t 1 - , n - 1
2 n - 1 2 n - 1
� 1 - ą
X
EX = � Var X = �2 > 0
n 100
Xn - �
U = "
�/ n
n-1
"
"2 1
N (0, 1) Sn = (Xi - Xn)2
n
i=1
�2 n
"2
�2 Sn
�ł �ł
( (
ą) S" ą) S"
�ł łł
Xn - u 1 - "n , Xn + u 1 - "n
2 n 2 n
1 - ą
X1, . . . , Xn N (�, �2)
� �
2
nSn
�2 =
�2
(n-1)
�2(p, n - 1) p
�ł �ł
( ) )
2
ą nSn ( ą
�ł łł
P �2 , n - 1 < < �2 1 - , n - 1 = 1 - ą.
2 �2 2
�2
�ł �ł
2 2
nSn nSn
�ł łł
, ,
ą
�2(1 - , n - 1) �2(ą, n - 1)
2 2
�
�ł" �ł
"
n n
�ł
Sn, Snłł.
ą
�2(1 - , n - 1) �2(ą, n - 1)
2 2
X1, . . . , Xn N (�, �2)
� � n > 50
"
" "
2
n
2�2 a" 2nS N ( 2n - 3, 1)
�2
�ł �ł
( (
"
ą) Sn " " ą)łł
�ł
P 2n - 3 - u 1 - < 2n < 2n - 3 + u 1 - H" 1 - ą.
2 � 2
�ł �ł
" "
Sn 2n Sn 2n
�ł łł
" "
,
ą ą
2n - 3 + u(1 - ) 2n - 3 - u(1 - )
2 2
� 1 - ą
�2
(X , A, P) P = {P� : � " Ś}
X �
� Ś0
H : � " Ś0
H0
X
T (X) K
R
H0 T (X) " K H0
H0
Ć(X) [0, 1] Ć(X)
Ć H0 Ć
H0
ąĆ(�) = E�[Ć(X)], � " Ś0.
sup ąĆ(�) ą
�"Ś0
ą ą
H0
H1 : � " Ś1 Ś1
Ś0 )" Ś1 = "
�Ć(�) = 1 - E�[Ć(X)], � " Ś1.
MĆ(�) = E�[Ć(X)],
� " Ś0 )" Ś1 Ć
MĆ(�) H0
� � " Ś0 MĆ(�) = ąĆ(�)
� " Ś1 MĆ(�) = 1 - �Ć(�)
(X1, . . . , Xn)
N (�, �2) � �2
H0 : � = �0 H1 : � = �0,
8
�0 ą
"
Xn - �0
T = n,
�
n
"
1
Xn = Xi T N (0, 1) H0
n
i=1
ą ą
H0 T < -u1- T > u1- up
2 2
p N (0, 1) H0
H1 : � > �0 H0 T > u1-ą
H1 : � < �0 H0 T < -u1-ą
(X1, . . . , Xn) N (�, �2) �
�2
H0 : � = �0 H1 : � = �0,
8
�0 ą
"
n(Xn - �0)
T = ,
Sn
n n
" "
1 2 1
Xn = Xi Sn = (Xi - Xn)2 t
n n-1
i=1 i=1
n - 1 H0 H0
ą ą
T < -t1- (n - 1) T > t1- (n - 1) tp(n - 1)
2 2
p t n - 1
(X1, . . . , Xn) N (�, �2) � �2
2 2
H0 : �2 = �0 H1 : �2 = �0,
8
2
�0
2
(n - 1)Sn
�2 = ,
2
�0
n - 1 �2(n - 1)
ą
H0 H0 �2 < �2 (n - 1) �2 >
2
�2 ą (n-1) �2(n-1) p
1- p
2
n - 1
n 50

2

2(n - 1)Sn "

= 2�2
2
�0
"
N ( - 3, 1) H0
"2n "
ą
) + 2n - 3
" ą "H0 2�2 < -u(1 - 2
2�2 > u(1 - ) + 2n - 3 u(p) p
2
N (0, 1)
(X1, . . . , Xk) N (�X, �2)
(Y1, . . . , Yn) N (�Y , �2)
2 2
�2 Xk Y SX SY
n
(Xi) (Yj)
H0 : �X = �Y H1 : �X = �Y .
8
"
Xk - Y kn
n
"
T = (k + n - 2),
2 2
(k - 1)SX + (k - 1)SY k + n
t t(k + n - 2) H0
ą ą
H0 T < -t1- (k + n - 2) T > t1- (k + n - 2)
2 2
2 2 2 2
H0 : �X = �Y H1 : �X = �Y .
8
2
SX
T = ,
2
SY
(k - 1, n - 1) H0
ą
H0 T < F (k - 1, n - 1) T >
2
ą
F1- (k - 1, n - 1) Fp(k - 1, n - 1) p
2
(k - 1, n - 1)
X xi
P (X = xi) i = 1, . . . , k
H0 : P (X = xi) = pi 1 i k,
p1, . . . , pk
(X1, . . . , Xn) X ni
xi n1 + . . . +
nk = n
k
"
(ni - npi)2
�2 = .
npi
i=1
" �2
k - 1 H0
�1-ą(k - 1) H0 �2 > �2 (k - 1)
1-ą
n 100 npi 5 i
X d
� �
� pi �2
k - d - 1
n " H0 �2 > �2 (k - d - 1)
1-ą
X
(a, b) -" a < b "
k
(a, x1], (x1, x2], . . . , (xk-1, b),
(xn) i
pi X i
H0
npi 5 i
ni X1, . . . , Xn
i
X F0
F0
X1, . . . , Xn
H0 X1, . . . , Xn
F
H0 : F = F0,
F0 H1 :
F = F0
8
Dn = sup |Fn(x) - F0(x)|,
-" Fn X1, . . . , Xn
Dn
H0
ą
Dn(ą)
PF (Dn > Dn(ą)) < ą.
0
Dn(ą) H0
ą
F0
X1, . . . , Xn
F F
Dn X1, . . . , Xn
sup |Fn(x) - F0(x)|
-" Fn
Dn
x
( )
i
+
Dn = max - zi ,
1 i n
n
(
i - 1)
-
Dn = max zi - ,
1 i n
n
+ -
Dn = max{Dn , Dn },
zi = F (Xi:n)
n 100 Dn
"
nDn
1
ą [(1 - ą)"n, ") (p)
p

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