Statisztika anyag

7/17/2019 Statisztika anyag

http://slidepdf.com/reader/full/statisztika-anyag 1/35

30

(λ)

λ

8

λ

→

→

Ind( p) p

m, σ

[0, b] b

•

•

•

•

•



(Ω, A, P ) Ω

A σ P

P = P ϑ|ϑ ∈ Θ,

P ϑ

Θ

Θ

Θ

P

X = (X 1, . . . , X n) : Ω → X ⊆ Rn

n

X n X i

n

X

X i P ϑ

X

X i

P ϑ

F n;ϑ|ϑ ∈ Θ

F n;ϑ(x1,...,xn) = P ϑ(X 1 < x1,...,X n < xn)

X n

F n;ϑ(x1,...,xn) =n

i=1

F 1;ϑ(xi),

F 1;ϑ

X i

pn;ϑ(x1, . . . , xn) = P ϑ(X 1 = x1, X 2 = x2,...,X n = xn)

f n;ϑ(x1,...,xn)

X = (X 1, . . . , X 30) : Ω

→ N30

0 X i

i

X X i ∼ Poisson(ϑ)

Θ = (0, ∞) ⊂ R

p30;ϑ(x1, x2, . . . , x30) =30i=1

p1;ϑ(xi) =30i=1

e−ϑ ϑxi

xi! = e−30ϑ ϑ

xi

xi!.

T : X → Rk

T = T (X )

k

X n

T (X ) = X = 1

n

n

i=1

X i

X

T (X ) = S 2X = 1

n

ni=1

(X i − X )2 X

T (X ) = (X (n)1 , X

(n)2 ,...,X

(n)n ) X X

(n)1 ≤ X

(n)2 ≤ ≤ X

(n)n

T (2, 4, 1, 3) = (1, 2, 3, 4)

T (X ) = X (

n)n − X (

n)1

X

T (X ) =

X (n)n+12

n

X (n)n2

+ X (n)n2 +1

2 n

X



(X 1,...,X n)

X i

x1 < x2 < · · · < xm m ≤ n xj

|i|X i = xj|n

= 1

n

ni=1

I (X i = xj),

I (X i = xj) = 1

X i = xj

I (X i = xj) = 0

X i = xj

X

F n(x) = 1

n

ni=1

I (X i < x), x ∈ R.

1

n

2n

3n

4n

nn

X(n)1X(n)2,3

X(n)4 X

(n)n

F n(x) =

0 x ≤ X

(n)1 ,

k

n X (

n)k < x ≤ X (

n)k+1,

1 X (n)n < x.

X 1, . . . , X n F

F n

1

F

P ( limn→∞ sup

x∈R

F n(x) − F (x) = 0) = 1.

x ∈ R

P ( limn→∞

F n(x) − F (x)

= 0) = 1,

I (X i < x)

supx∈R

F n(x) − F (x)

1/√

n

X

X i

i

X = R20+ X i ∼ Exp(ϑ) Θ = (0, ∞)

f 20;ϑ(x1,...,x20) =20i=1

f 1;ϑ(xi) =20i=1

ϑe−ϑxi = ϑ20e−ϑ

xi .

X i

i

X = 0, 110 X i ∼ Ind(ϑ)

Θ = [0, 1]

p10;ϑ(x1, . . . , x10) =10i=1

p1;ϑ(xi) =10i=1

ϑxi(1 − ϑ)1−xi = ϑ

xi(1 − ϑ)10− xi .

X i i

X = R5+

X i ∼ N (ϑ1, ϑ2) Θ = R× R+

f 5;ϑ(x1,...,x5) =5

i=1

f 1;ϑ(xi) =5

i=1

1 2πϑ2

2

e− (xi−ϑ1)

2

2ϑ22 .



X i i

X = R5+

X i ∼ E (0, ϑ) Θ = (0, ∞)

f 5;ϑ(x1,...,x5) =5

i=1

f 1;ϑ(xi) =5

i=1

1

ϑI (0 ≤ xi ≤ ϑ) =

1

ϑ5I (x

(1)1 ≥ 0

x

(5)5 ≤ ϑ).

8

X i i X

x1 x2 x3 x4 x5 x6 x7 x8

28

38

28

0 18

19/8 = 2.38

95/64 = 1.48

4

2

1

x(8)1,2 x

(8)3,4,5 x

(8)6,7 x

(8)8

X

F

F n(x)

nF n(x)

∼Bin(n, F (x))

E ( F n(x)) = F (x)

D( F n(x)) = F (x)(1

−F (x))/n

X

ϑ ∈ Θ

P ϑ(X = x)

ϑ ϑ

T (X )

P ϑ(T (X ) = t)

ϑ

P ϑ(X = x) ϑ

T (X )

P ϑ(T (X ) = t)

T (X ) = t X = x x T (x) = t X

P ϑ(X = x|T (X ) = t)

x

T (x) = t P ϑ(X = x

|T (X ) = t) ϑ

P ϑ(X = x) = P (X = x|T (X ) = T (x))P ϑ(T (X ) = T (x))

X = (X 1, . . . , X n)

(Ω, A, P )

T (X ) ϑ x, t P ϑ(X = x|T (X ) = t)

ϑ

X i ∼ Ind( p)

0 ≤ p ≤ 1

i X i

p



p

P p(X = x|n

i=1

X i = t) =

0

ni=1

xi = t

ni=1 pxi(1 − p)1−xi

nt

pt(1 − p)n−t =

p

=t xi (1 − p)

=n−t n − xi

n

t pt(1

− p)n−t

binom;

Ind

= 1

nt

ni=1

xi = t

p

X T (X )

h

gϑ

P ϑ(X = x) = h(x) ·gϑ(T (x))

⇒: T (X )

P ϑ(X = x) = P ϑ(X = x|T (X ) = T (x)) · P ϑ(T (X ) = T (x)) = h(x) · gϑ(T (x)),

ϑ

⇐: P ϑ(X = x) = h(x) · gϑ(T (x)) T (X )

P ϑ(X = x|T (X ) = t) = P ϑ(X = x, T (X ) = t)

P ϑ(T (X ) = t) =

h(x) · gϑ(t)y:T (y)=t

P ϑ(X = y)=

h(x) · gϑ(t)y:T (y)=t

h(y) · gϑ(T (y))=

h(x)y:T (y)=t

h(y),

T (x) = t,

X 1,...,X n ∼ Poisson(λ)

λ

pn;λ(x) = P λ(X = x) =n

i=1

e−λ · λxi

xi! = e−nλ · λ

xin

i=1 xi! =

1ni=1 xi! h(x)

· e−nλ · λ

xi gλ(

xi

:=T (x)

)

,

T T (X ) = t

t

0

X f n;ϑ(x)

T (X )

ϑ

f n;ϑ(x) = h(x) · gϑ(T (x))

X i ∼ E(0, b) b

f n;b(x) =n

i=1

f 1;b(xi) =n

i=1

1

bI (0 ≤ xi ≤ b) = I (x(n)

1 ≥ 0) h(x)

· 1

bn · I (x(n)

n ≤ b) gb(x

(n)n )

,

X (n)n



T S

S

T

X i ∼ Geo( p)

X i ∼ Bin(m, p) m

X i ∼

Exp(λ) X i ∼ N (m, σ) X i ∼ E (−a, a)

X i

pn; p(x) =

ni=1

(1 − p)xi−1 · p = (1 − p)

xi−n pn.

X i

pn; p(x) =n

i=1

m

xi

pxi(1 − p)m−xi = p

xi(1 − p)m·n− xi ·

ni=1

m

xi

.

X i

f n;λ(x) =

ni=1

λe−λxi = λne−λ

xi .

(m, σ) (

X i,

X 2i )

f n;m,σ(x) =

ni=1

1√ 2πσ2

· e−(xi−m)2

2σ2 =

1√ 2πσ2

n

e− 12σ2

(xi−m)2 =

=

1√ 2πσ2

n

· e− 12σ2

(

x2i−2m

xi+nm2).

σ

X i

f n;m

(x) = 1

√ 2πσ2n

·e−

12σ2

x2i

·e−

12σ2

(nm2−2m

xi).

m

(X i − m)2

max |X i|

f n;a(x) =

ni=1

1

2aI (−a < x < +a) =

1

(2a)n · I (|xi| < a i = 1,...,n) =

1

(2a)n · I (|x|(n)

n < a).

X 1,...,X n

F ϑ

ψ(ϑ)

ψ(ϑ)

T (X )

T (X )

ψ(ϑ) T (X )

T (X ) ψ(ϑ)

T (X )

ψ(ϑ)

n T n

T n(X 1,...,X n) → ψ(ϑ)



T (X ) ψ(ϑ)

E ϑ(T (X )) = ψ(ϑ) ∀ϑ ∈ Θ.

T (X ) bT (ϑ) = E ϑ(T (X )) − ψ(ϑ)

X 1,...,X n ∼ F ϑ

ψ(ϑ) = E ϑ(X i) X

E ϑ(X ) = E ϑ(

1

n

n

i=1 X i) =

1

n

n

i=1 E ϑ(X i)

ψ(ϑ)= ψ(ϑ).

ψ( p) = 1 p

1X

1/p

ψ( p)

n

ψ( p) p

n

T

E p(T (X 1,...,X n)) =

x∈0,1nT (x1,...,xn) · p

xi · (1 − p)n−

xi ,

n p pk I (X 1 = 1) · I (X 2 =1) · · · I (X k = 1) (k ≤ n)

T n(X 1, . . . , X n) ψ(ϑ) ∀ϑ ∈ Θ

E ϑ(T n(X 1, . . . , X n)) → ψ(ϑ) (n → ∞).

T 1, T 2

ψ(ϑ)

T 1

T 2

D2ϑ(T 1) ≤ D2

ϑ(T 2) ϑ ∈ Θ T

ϑ D2ϑ(T 1) < D2

ϑ(T 2) D2ϑ(T 1) > D2

ϑ(T 2)

E ϑ[(T −ψ(ϑ))2]

T 1

T 2 1

P ϑ(T 1 = T 2) = 1

ϑ ∈ Θ

E ϑ(T 1) = E ϑ(T 2) = ψ(ϑ) D2ϑ(T 1) =

D2ϑ(T 2)

θ

T = T 1 + T 2

2 .

T

E ϑ(T ) = ψ(ϑ)

T 1

D2ϑ(T 1) ≤ D2

ϑ(T ) = 1

4

D2

ϑ(T 1) + D2ϑ(T 2) + 2covϑ(T 1, T 2)

=

1

2D2

ϑ(T 1) + 1

2covϑ(T 1, T 2),

D2

ϑ(T 1) ≤ covϑ(T 1, T 2)

1 ≤ covϑ(T 1, T 2)

Dϑ(T 1)Dϑ(T 2) = Rϑ(T 1, T 2),

T 1 = aT 2 + b

1

a = 1 b = 0



X = (X 1, . . . , X n) ψ(ϑ) = E ϑ(X i)

D2

ϑ(X i) < ∞ ϑ

X

ψ(ϑ)

ni=1 ciX i

ni=1 ciX i

ni=1 ci = 1

D2ϑ(X ) =

D2ϑ(X i)

n , D2

ϑ(

ciX i) =

D2

ϑ(ciX i) = (

c2i )D2

ϑ(X i).

1

n ≤

ni=1

c2i .

c2i

n ≥

ci

n =

1

n,

c2i ≥

n

n2 =

1

n.

X i ∼ E (0, b) b

T 1 = n + 1n

X (n)n , T 2 = 2 · X.

T 2 T 1

X i = b ·Y i Y i ∼ E (0, 1)

X (

n)n = b ·Y (

n)n

E (0, 1)

Y

(n)n

P (Y (n)n < t) = tn

Y

(n)n

n · tn−1

E (Y (n)n ) =

1

0

t · ntn−1 dt = n · 1

n + 1 =

n

n + 1.

0 1

↓

nn+1

E b(T 1) = b

D2b (T 2) = 4 · D 2

b (X i)

n =

4

n · b2

12 =

b2

3n.

D2(Y (n)n ) =

1

0

t2 · n · tn−1 dt −

n

n + 1

2

= n

n + 2 −

n

n + 1

2

=

= n(n + 1)2 − n2(n + 2)

(n + 1)2(n + 2) =

n

(n + 1)2(n + 2).

D2

b (X (n)n ) = b2D2

b (Y (n)n )

D2b (T 1) =

n + 1

n

2

· D2b (X (n)

n ) = b2

n(n + 2).

T 1 T 2 n

T n(X 1,...,X n) ψ(ϑ) T n → ψ(ϑ)

(n −→ ∞)

P ϑ(|T n − ψ(ϑ)| > ε) n→∞−→ 0 ∀ϑ ∈ Θ.



X → E ϑ(X i)

E ϑ(T n) = ψ(ϑ) T n

D2ϑ(T n) → 0

n → ∞ T n

P ϑ(|T n − ψ(ϑ)| > ε) ≤ D2

ϑ(T n)

ε2 −→ 0.

X i

∼E (0, ϑ)

T n = n+1

n X (n)n

D2ϑ(

n + 1

n · X (n)

n ) = ϑ2

n(n + 1) −→ 0.

X = (X 1, . . . , X n)

F n;ϑ

Ln(x; ϑ) Ln(x; ϑ) =f n;ϑ(x)

Ln(x; ϑ) = pn;ϑ(x)

n

F n;ϑ

I n(ϑ) = E ϑ([ ∂

∂ϑ ln Ln(X ; ϑ)]2)

X 1, . . . , X n

L1(xi; ϑ)

I 1(ϑ) < ∞

E ϑ( ∂

∂ϑ ln L1(X 1; ϑ)) = 0.

I n(ϑ) = n

·I 1(ϑ)

I 1(ϑ) = D2ϑ( ∂ ∂ϑ ln L1(X 1; ϑ))

E ϑ( ∂

∂ϑ ln Ln(X ; ϑ)) = E ϑ(

∂

∂ϑ ln

ni=1

L1(X i; ϑ)) =

ni=1

E ϑ( ∂

∂ϑ ln L1(X i; ϑ))

0

= 0.

I n(ϑ) = D2ϑ(

∂

∂ϑ ln Ln(X ; ϑ)) = D2

ϑ(n

i=1

∂

∂ϑ ln L1(X i; ϑ)) =

ni=1

D2ϑ(

∂

∂ϑ ln L1(X i; ϑ)

I 1(ϑ)

) = n · I 1(ϑ).

+∞

−∞f 1;ϑ(x) dx = 1 ⇒ ∂

∂ϑ

+∞

−∞f 1;ϑ(x) dx = 0.

0 =

+∞

−∞

∂

∂ϑf 1;ϑ(x) dx =

+∞

−∞

∂ ∂ϑ f 1;ϑ(x)

f 1;ϑ(x) ·f 1;ϑ(x) dx =

+∞

−∞

∂

∂ϑ ln f 1;ϑ(x)·f 1;ϑ(x) dx = E ϑ(

∂

∂ϑ ln f 1;ϑ(X 1)).



∂

∂ϑ

x1

p1;ϑ(x1) =x1

∂

∂ϑ p1;ϑ(x1).

X i ∼

Ind( p) L1(x, p) = px(1

− p)1−x

x

∈ 0, 1

∂

∂p ln L1(x, p) =

x

p − 1 − x

1 − p.

E p

∂

∂p ln L1(X i, p)

=

p

p − 1 − p

1 − p = 0.

I 1( p) = D2 p

∂

∂p ln L1(X i, p)

= D2

p

X i

p(1 − p)

=

1

p(1 − p),

I n( p) = n · I 1( p) = n

p(1 − p)

X i ∼ N (ϑ, σ0) σ0

L1(x, ϑ) = 1

2πσ20

· e− (x−ϑ)2

2σ20 .

ln L1(x, ϑ) = ln 1

2πσ2

0

− (x − ϑ)2

2σ20

.

E ϑ

(

∂

∂ϑ ln L1(X i, ϑ)

= E ϑ

2(X i − ϑ)

2σ20

= 0.

I 1(ϑ) = D2ϑ

X i − ϑ

σ20

=

D2ϑ(X i)

σ40

= 1

σ20

,

I n(ϑ) =

n

σ20

ψ(ϑ)

X

I X(ϑ)

T = T (X ) I T (ϑ)

I T (ϑ) ≤ I X(ϑ)

T

L1(x, ϑ)

ϑ

∞ > I 1(ϑ) > 0

I 1(ϑ)

ϑ



X = (X 1,...,X n)

T (X )

ψ(ϑ)

D2

ϑ(T ) < ∞ ϑ

ψ(ϑ) = E ϑ

T (X )

∂

∂ϑ ln Ln(X ; ϑ)

.

D2ϑ(T ) ≥ (ψ(ϑ))2

I n(ϑ) ϑ

S

covϑ(T, S )2 ≤ D2ϑ(T )D2

ϑ(S ) ⇒ D2ϑ(T ) ≥ covϑ(T, S )2

D2ϑ(S )

.

S

covϑ(T, S ) = E ϑ(T S ) − E ϑ(T )E ϑ(S ) = E ϑ(T S ) − ψ(ϑ)E ϑ(S )

T S = ∂

∂ϑ ln Ln(X ; ϑ) E ϑ(S ) = 0

covϑ(T, S ) = E ϑ(T · S ) = ψ (ϑ),

D2

ϑ(S ) = I n(ϑ)

ψ(ϑ) = E ϑ(T ) =

T (x)f n;ϑ(x) dx .

ψ(ϑ) =

T (x)

∂

∂ϑf n;ϑ(x) dx = E ϑ

T (X )

∂

∂ϑ ln Ln(X ; ϑ)

.

(ψ(ϑ))2

I n(ϑ) D2

ϑ(T ) = (ψ(ϑ))2

I n(ϑ)

ϑ

T

T

ψ(ϑ) = ϑ

1/I n(ϑ)

X i ∼ N (ϑ, σ0)

ϑ

1/I n(ϑ) =

σ20/n

D2

ϑ(X ) = σ20/n

X

ϑ

X i ∼ Ind( p)

p

1/I n( p) =

p(1 − p)/n

D2 p(X ) = p(1 − p)/n

X

p

X i ∼ Exp(λ) ψ(λ) = 1λ X

D2λ(X ) =

1λ2

n =

1

λ2

·n

.

E λ

∂

∂λ ln f 1;λ(X i)

= E λ

∂

∂λ(ln λ − λX i)

= E λ

1

λ − X i

= 0.

I 1(λ) = D2

λ

1λ − X i

= D2

λ(X i) = 1

λ2 I n(λ) =

n

λ2

ψ(λ) = 1/λ ψ(λ) = −1/λ2

− 1λ2

2

nλ2

= 1

n · λ2.

X 1/λ



X i ∼ (λ) X λ

X i ∼ E (0, ϑ) x

f 1;ϑ(x) ϑ

f 1;ϑ(x) =

1ϑ

0 ≤ x ≤ ϑ0

I 1(ϑ) = E ϑ ∂

∂ϑ

ln L1(X 1; ϑ)2

= E ϑ ∂

∂ϑ

ln 1

ϑ2

= −

1

ϑ2

= 1

ϑ2

,

1

X 1 < ϑ

1

L1(X 1; ϑ)

I n(ϑ) = E ϑ

∂ ∂ϑ ln

1ϑ

n2

=−n

ϑ

2=

n2

ϑ2 = n2 · I 1(ϑ)

x

1ϑ

↓

f 1;ϑ(x)

ϑ

ϑ

f 1;ϑ(x)

ϑ

X, Y

X

Y = y E (X |Y = y) =

x x · P (X = x|Y = y) = V (y) X

Y

E (X |Y ) = V (Y )

E (c|Y ) = c

E (X 1 + X 2|Y ) = E (X 1|Y ) + E (X 2|Y ) E (cX |Y ) = cE (X |Y )

X 1 ≤ X 2 ⇒ E (X 1|Y ) ≤ E (X 2|Y )

E (E (X |Y )) = E (V (Y )) =y

V (y) · P (Y = y) =y

x

x · P (X = x|Y = y) · P (Y = y) =

x

xy

P (X = x|Y = y) · P (Y = y)

P (X=x)

=x

x · P (X = x) = E (X ).

E (XW (Y )|Y ) = W (Y )E (X |Y ) Y W (Y )

D2(X ) = E

D2(X |Y )

+ D2 (E (X |Y )) D2(X |Y ) =

E

(X − E (X |Y ))2|Y

D2(X |Y ) = E

X 2 − 2XE (X |Y ) + E (X |Y )2|Y

= E (X 2|Y ) − E (2XE (X |Y )|Y ) + E

E (X |Y )2|Y

=

E (X 2|Y ) − 2E (X |Y )E (X |Y ) + E (X |Y )2 = E (X 2|Y ) − E (X |Y )2.



E

D2(X |Y )

= E

E (X 2|Y )

E (X2)

−E

E (X |Y )2

= E (X 2) − E (X )2 D2(X)

− E

E (X |Y )2− E (X )2

D2(E (X|Y ))

.

D2 (E (X |Y )) ≤ D2(X )

n

X

Y

Y = y

X Bin(n

−y, 1/3) E (X

|Y ) = (n

−Y )/3

E (E (X |Y )) = E ((n − Y )/3) = n − E (Y )

3 =

n

6, E (X ) =

n

6.

D2(X |Y ) = (n − Y ) · 1

3 · 2

3 =

2(n − Y )

9 , E (D2(X |Y )) =

n

9.

D2(X ) = 5n

36, D2(E (X |Y )) =

n

36.

X

S

ψ(ϑ)

T

ϑ

U = E (S |T )

ψ(ϑ)

S

E ϑ(U ) = E ϑ (E ϑ(S |T )) = E ϑ(S ) = ψ(ϑ), D2ϑ(U ) = D2

ϑ (E ϑ(S |T )) ≤ D2ϑ(S ).

T

E (S |T )

ϑ

X i ∼ Geo( p) p T =

ni=1

X i

S = I (X 1 = 1) U = E (S |T ) = V (T )

V () = E p(S |T = ) = 0 · P p(S = 0|T = ) + 1 · P p(S = 1|T = ) = P p(X 1 = 1|n

i=1

X i = ) =

P p(X 1 = 1 , n

i=1 X i = )

P (n

i=1 X i = ) .

P p(X 1 = 1,

ni=1

X 1 = ) = P p(X 1 = 1,

ni=2

X i = − 1) = P p(X 1 = 1)P p(

ni=2

X i = − 1).

ni=1 X i ∼ NegBin(n, p)

V () = P p(X 1 = 1)P p(

ni=2 X i = − 1)

P (n

i=1 X i = ) =

p−2n−2

pn−1(1 − p)−n

−1n−1

pn(1 − p)−n

= n − 1

− 1.

U = V (n

i=1 X i) = n − 1ni=1 X i − 1

S



X i ∼ Poisson(λ) ψ(λ) = e−λ

e−X

e−λ = P λ(X 1 = 0) S = I (X 1 = 0)

T = n

i=1 X i

E λ(S |T = ) = P λ(X 1 = 0|n

i=1

X i = ) = e−λ · e−(n−1)λ · ((n−1)λ)

!

e−nλ · (nλ)

!

=

1 − 1

n

.

n

i=1 X i ∼

(nλ) n

i=2 X i ∼

((n − 1)λ)

V =

1 − 1

n

Xi

=

1 − 1

n

n

−→e−1

X

.

X i ∼ ( p) ψ( p) = p2

X

2

S = I (X 1 = 1, X 2 = 1) T = ni=1 X i

E p(S |T = ) = P p(X 1 = 1, X 2 = 1 |n

i=1

X i = ) = p · p · n−2

−2

p−2(1 − p)n

n

p(1 − p)n−

= ( − 1)

n(n − 1),

V =

X i

n ·

X i − 1

n − 1

E (X ) 1

n

ni=1 X i = X

D2(X ) 1

n

ni=1(X i − X )2 = S 2n

sup x | F (x) < 1 − inf x | F (x) > 0 X (

n)n − X (

n)1

F (x) = P (X < x) 1n

ni=1 I (X i < x) = F n(x)

X 1,...,X n F ϑ ϑ ∈ Θ

ϑ

ϑ Ln(X ; ϑ) = maxLn(X ; ϑ) : ϑ ∈ Θ

Ln(X ; ϑ)

∂ ∂ϑ ln Ln(X ; ϑ) = 0

X i ∼ Exp(λ)

Ln(X ; λ) = n

i=1 λe−λXi = λne−λ

Xi

ln Ln(X ; λ) = n · ln λ − λ

X i

∂ ∂λ ln Ln(X ; λ) = n

λ −

X i 0

λ =

nX i

= 1

X

λ = 1

X

X i ∼ (ϑ, ϑ + 1) Ln(X ; ϑ) = I (ϑ ≤ X i ≤ ϑ + 1, ∀i)

11

n= I (ϑ ≤ X

(n)1 , X

(n)n ≤

ϑ + 1)

ϑ ∈

X (n)

n − 1, X (n)

1



T

Ln(X ; ϑ) = h(X ) · gϑ(T (X )) h(X )

ϑ

T (X )

X 1,...,X n F ϑ ϑ

ϑ ψ(ϑ)

ψ(ϑ)

Ln(x; ψ) = sup Ln(x; ϑ) | ψ(ϑ) = ψ

ψ(ϑ)

Ln(x; ψ)

ψ

Ln(x; ψ) ≤ Ln(x; ϑ)

Ln(x; ψ(ϑ)) = Ln(x; ϑ)

Ln(x; ψ(ϑ)) = maxψ

Ln(x; ψ)

n

ϑn

√ n · (ϑn − ϑ) −→ N (m(ϑ), σ(ϑ))

(n → ∞)

m(ϑ) = 0

σ2(ϑ) = 1I 1(ϑ)

X i ∼ E (a, b)

Ln(X ; a, b) =

1

b − a

n

· I (a ≤ X i ≤ b ∀i) [a, b]

a = X (n)1 b = X

(n)n

X i ∼ (m, σ)

Ln(X ; m, σ) =n

i=1

1√ 2πσ

e−(Xi−m)2

2σ2 =

1√

2π

n

· 1

σn · e−

(Xi−m)2

2σ2

ln Ln(X ; m, σ) = ln

1√

2π

n

− n ln σ −

(X i − m)2

2σ2

∂ ∂m ln Ln(X ; m, σ) = (X i−

m)2

σ2 = 0

∂ ∂σ ln Ln(X ; m, σ) = −n

σ − 2

(X i − m)2

2σ3 = 0

X i = n · m ⇒ m =

Xi

n = X

(X i − m)2 = n · σ2 ⇒ σ2 =

(Xi−m)2

n

σ2 = 1n

(X i − X )2 = S 2n

. µ = E ϑ(X i )

E ϑ(X i )

1n

ni=1

X i

X i ∼ E (a, b)

µ1 = E a,b(X i) = a + b

2 µ2 = E a,b(X 2i ) = D2

a,b(X i) + E a,b(X i)2 = (b − a)2

12 +

a + b

2

2



(b − a)2

12 = µ2 − µ2

1 ⇒ b − a =

12(µ2 − µ21)

b = µ1 +

3(µ2 − µ21)

a = µ1 −

3(µ2 − µ21)

a = X −

3( 1n

X 2i − X

2) = X − √

3S n

b = X +

3( 1

n

X 2i − X

2)X +

√ 3S n

X i ∼ E (−a, a)

µ1 = E a(X i) = 0

µ2 = E a(X 2i ) = a2

3 ⇒ a = √ 3µ2

a =

3 · 1n ·

X 2i .

2·

X = (X 1,...,X n) F ϑ ϑ

(T 1(X ), T 2(X )) 1 − α ϑ

1 − α

P ϑ (T 1(X ) < ϑ < T 2(X )) ≥ 1 − α ∀ϑ.

inf ϑ∈Θ

P ϑ(ϑ ∈ (T 1, T 2)) .

(T 1, T 2) 1 − α ϑ (S 1, S 2) 1 − α ψ(ϑ)

S 1 = inf ψ(ϑ) | ϑ ∈ (T 1, T 2) S 2 = sup ψ(ϑ) | ϑ ∈ (T 1, T 2) .

X i ∼ N (µ, σ)

σ

µ

µ

KI (1 − α)

X ∼ N (µ, σ√

n)

√ n · X − µ

σ ∼ N (0, 1)

P

√ n · X − µ

σ

< uα2

= 1 − α,

uα

Φ(uα) = 1 − α

−uα2

uα2

α2

α2

1 − α

√ n

σ · X − µ

< uα2

⇐⇒ X − µ < σ · uα

2√ n

⇐⇒ X − σ · uα2√

n < µ < X +

σ · uα2√

n .

T 1 = X − σ · uα2√

n , T 2 = X +

σ · uα2√

n .



σ

χ2

t

X i ∼ N (0, 1)

Y =

ni=1

X 2i Y

n

Y ∼ χ2n

√ Y

n

√ Y ∼ χn

Y ∼

χ2n E (Y ) = n D2(Y ) = 2n

n = 1

1√ 2πx

e−x/2 n = 2 Exp(1/2) n ≥ 3

X ∼ N (0, 1)

Y ∼ χn

Z =√

n · X

Y

Z

n

t

n

Z ∼ tn

tn

E (Z ) = 0 n > 1 D2(Z ) = nn−2

n > 2

tn

n → ∞

x cnx−(n+1)

α

tn(α)

X i ∼ N (µ, σ)

X

S 2n

n · S 2nσ2

∼ χ2n−1

X i ∼ N (µ, σ) Y i = (X i − µ)/σ

n · S 2nσ2

=

ni=1

(Y i − Y )2 (Y i − Y )

X i ∼ N (µ, σ)

µ, σ

µ

KI (1 − α)

S ∗n =

1n−1

(X i − X )2

√ n · X − µ

S ∗n=

√ n − 1 ·

√ n · X − µ

σ ∼

(0, 1)√

n − 1 · S ∗nσ

∼ χn−1

∼ tn−1.

P

√

n · X − µ

S ∗n

< tn−1(α

2)

= 1 − α,

T 1 = X − S ∗n · tn−1(α2 )√ n

T 2 = X + S ∗n · tn−1(α2 )√

n .

X i ∼ Ind( p) p KI (1 − α)

X

p

X ≈ N

p,

p(1− p)n

n

√ n · X − p

p(1 − p)≈

(0, 1) ⇒ P

√ n

p(1 − p)· X − p

< uα2

≈ 1 − α.



p p

X √

n X (1 − X )

· X − p < uα

2

X i ∼ Exp(λ) KI (1 − α) λ

16

10.10 8.94 9.61 10.00 10.42 10.33 10.68 9.2510.32 10.26 10.32 10.74 9.98 10.23 9.88 9.89

10.06

0.46

m KI (0.95) σ = 0.5

KI

m

KI (0.9) m3

p

p ≤ 0.05

P p(

) = P p(≥ 4

) =25k=4

25

k

pk · (1 − p)25−k.

p = 0.05

α = sup p≤0.05

P p(≥ 4

) = P 0.05(≥ 4

) = 0.034.

p > 0.05

P p(≤ 3

) =3

k=0

25

k

pk · (1 − p)25−k,

p = 0.1 0.763



(Ω, A, P ) P = P ϑ : ϑ ∈ Θ Θ

H 0 : ϑ ∈ Θ0

H 1 : ϑ ∈ Θ1

Θ = Θ0 ∪ Θ1 Θ0 ∩ Θ1 = ∅

X = (X 1,...,X n)

X

X e

X k

X = X e ∪ X k X e ∩ X k = ∅

X ∈ X e

H 0 X ∈ X k

H 0

Θ = [0, 1]

Θ0 = [0, 0.05]

Θ1 = (0.05, 1]

X = (X 1, . . . , X 25)

X i = 1 i X i = 0 i

X = 0, 125

X e = x ∈ X :25

i=1

xi ≤ 3, X k = x ∈ X :25

i=1

xi ≥ 4.

H 0

H 0 H 0

H 0

H 1

H 0

P ϑ(X

∈ X k) ϑ

∈ Θ0

α = supϑ∈Θ0

P ϑ(X ∈ X k).

H 0

P ϑ(X ∈ X e) ϑ ∈ Θ1

β (ϑ) = 1 − P ϑ(X ∈ X e) = P ϑ(X ∈ X k) ϑ ∈ Θ1.

β (ϑ1)

H 1 : ϑ = ϑ1

n

(X ne , X nk )

α αn β

β n

X ∼ E (−t, 1 + 2t) t > 0

H 0 : t = 0 Θ0

H 1 : t > 0

Θ1

X e = (0.1, 0.85)

X = R

α = P 0(X ∈ X k) = 1 − P 0(X ∈ X e) = 1 − P 0(0.1 < X < 0.85) = 1 − 0.75 = 0.25

.

β (t) = P t(X ∈ X k) = 1 − P t(0.1 < X < 0.85) = 1 − 0.751+3t



12

14

β (t)

β (ϑ) ≥ α ∀ϑ ∈ Θ1

(X e, X k) (X e, X k)

β (ϑ) = P ϑ(X ∈ X k) ≥ β (ϑ) = P ϑ(X ∈ X k) ∀ϑ ∈ Θ1

(X ne , X nk )

α

αn ≤ α ∀n

β n(ϑ) n→∞−→ 1 ∀ϑ ∈ Θ1

(

X e,

X k)

Ψ : X → 0, 1

x ∈ X Ψ(x) H 0

X e = x | Ψ(x) = 0 X k = x | Ψ(x) = 1

Ψ : X → [0, 1]

x

∈ X

Ψ(x)

H 0

P ϑ(H 0 ) =

x

Ψ(x)P ϑ(X = x) = E ϑ(Ψ(X )).

α = supϑ∈Θ0

E ϑ(Ψ(X ))

β (ϑ) = E ϑ(Ψ(X )) (ϑ ∈ Θ1)

H 0 H 1

H 0 : ϑ = ϑ0 Ln(0; x)

H 1 : ϑ = ϑ1 Ln(1; x)

X

n

Ψ(x) =

1

Ln(1,x)Ln(0;x) > c

γ

Ln(1,x)Ln(0;x) = c

0

Ln(1,x)Ln(0;x)

< c

.

0 < α ≤ 1 c γ Ψ α

Ψ ≤ α

Y = Ln(1;X)Ln(0;X)

α = P 0(Y > c) + γ · P 0(Y = c) = 1 − P 0(Y ≤ c) + γ · P 0(Y = c),



1−α = P 0(Y ≤ c)−γ · P 0(Y = c) c0 P 0(Y ≤ c0) = 1 −α c = c0, γ = 0

c0 c0

P 0(Y < c0) ≤ 1 − α < P 0(Y ≤ c0)

c = c0, γ = P 0(Y ≤ c0) − (1 − α)

P 0(Y = c0) (0 < γ ≤ 1).

Ψ

E 0(Ψ(X )) ≤ α = E 0(Ψ(X ))

E 1(Ψ(X )) ≤ E 1(Ψ(X ))

Ln(1; x) = f n;1(x)

f n;1(x)

H 1

Ln(0; x) = f n;0(x)

f n;0(x)

H 0

Rn

(Ψ(x) − Ψ(x)) · (f n;1(x) − c · f n;0(x)) dx ≥ 0.

f n;1(x) − c · f n;0(x) = 0 f n;1(x) − c · f n;0(x) > 0

Ψ(x) = 1

Ψ(x) ≤ 1

Ψ(x) − Ψ(x) ≥ 0

≥

f n;1(x) − c · f n;0(x) < 0

Ψ(x) = 0 Ψ(x) − Ψ(x) ≤ 0 ≥

0 ≤

Ψf n;1 −

Ψf n;1 −c ·

Ψf n;0 +c·

Ψf n;0 = E 1(Ψ(X ))−E 1(Ψ(X ))−c·E 0(Ψ(X ))+ c·E 0(Ψ(X )).

E 1(Ψ(X )) − E 1(Ψ(X )) ≥ c · [E 0(Ψ(X )) − E 0(Ψ(X ))] ≥ 0.

β n ≥ 1 − cn

c

H 0 : P (

) = 1

2 H 1 : P (

) = 1

6 X =

F F , F I , I F , I I L1 L0

F F F I IF II L1

136

536

536

2536

L014

14

14

14

L1

L0

19

59

59

259

α = 0.25

γ, c

Ψ(x) =

1

L1

L0> c

γ

L1

L0= c

0

L1

L0< c

α 0.25 = α = 1 · P 0(L1

L0> c) + γ · P 0(L1

L0= c)

25/9 c ∈ ( 59

, 259

)

γ = 0

II

H 0

H 0

α = 0.3

5

9 < c < 25

9

0.25 < 0.3

19

< c < 59

0.25 + 0.5 > 0.3

c = 59

0.25 + γ · 0.5 = 0.3 γ = 0.1 II H 0 F F H 0 IF

F I

H 0 0.9

0.1

β = P 1

L1

L0> c

+ γ · P 1

L1

L0= c

=

25

36 + 0.1 · 10

36 =

26

36 = 0.72.



7 k 7 − k H 0 : k = 3 H 1 : k = 4

X

X = 0, 1, 2, 3

α = 0.2

L0(x) =

3x

4

3−x

73

, L1(x) =

4x

3

3−x

73

.

0 1 2 3

L1 135 1235 1835 435L0

435

1835

1235

135

L1

L0

14

23

32

4

1/35 < 0.2

1/35+12/35 > 0.2

c = 3

2 0.2 = 1

35 + γ · 12

35 γ = 0.5

1/2

1/2

1 − β = P 1(H 0 ) =

1

35 +

12

35 +

1

2 · 18

35 =

22

35.

X ∼ Exp(λ)

H 0 : λ = 1

2 H 1 : λ =

13 α = 1/8

X = (0, ∞)

L0(x) = 1

2e−

12x, L1(x) =

1

3e−

13x,

L1(x)

L0(x) =

2

3ex2− x

3 = 2

3ex6 .

α = 1

8 = P 0

2

3eX6 > c

= P 0

X > 6 ln

3c

2

= 1 − F 0

6 ln

3c

2

= e−3 ln 3c

2 =

2

3c

3

.

c = 4

3 c

α = 1

8 = P 0

2

3eX6 > c

= P 0 (X > d) = 1 − F 0 (d) = e−

d2 ,

d = 6ln2 = 4.16

4.16

H 0 : λ = 1

2 H 1 : λ < 1

2

H 1 : λ = λ1(< 1

2)

λ

H 0 : λ = 2

H 1 : λ = 1

α = 0.05

n N (m, 1) H 0 : m = 0 H 1 : m = 1

α = 0.05

→ u



→ t

→ F

u

X 1,...,X n ∼ N(m, σ) σ m

a) H 0 : m = m0

H 1 : m = m0b)

H 0 : m ≤ m0

H 1 : m > m0c)

H 0 : m ≥ m0

H 1 : m < m0

u = X − m0

σ · √

n H 0∼ N (0, 1).

α

a) X k = |u| > u α2 b) X k = u > uα c) X k = u < −uα

uδ

Φ(u

δ) = 1

−δ

Φ(x)

∆n(m) := m−m0

σ

√ n

β n(m) = P m(|u| > uα2

) = P m

X − m0

σ

√ n

> u α2

= 1 − P m

−uα

2<

X − m0

σ

√ n < uα

2

=

1−P m

−uα

2<

X − m

σ

√ n +

m − m0

σ

√ n < uα

2

= 1−P m

−uα

2− ∆n(m) <

X − m

σ

√ n < uα

2− ∆n(m)

=

1 − Φ(uα2

− ∆n(m)) + Φ(−uα2

− ∆n(m)) = Φ(−uα2

+ ∆n(m)) + Φ(−uα2

− ∆n(m)),

X

−m

σ √ n ∼ N (0, 1)

β n(m)

β n−1(m) < β n(m) (m = m0) β n(m) n→∞−→ 1

β n(m) > α (m = m0)

limm→±∞β n(m) = 1

X 1,...,X 16 ∼ N (m, 1) X = 0, 1 H 0 : m = 0 H 1 : m =0

α = 0, 1

u

0.95 = 1 − α

2 = Φ(uα

2)

⇒ uα2

= 1.65.

u = 0,1−01 √ 16 = 0.4

|0.4| ≤ 1.65 H 0

H 0

uα2

= 0.4 ⇒ Φ(uα2

) = 0.66 = 1 − α

2 ⇒ α

2 = 0.34 ⇒ α = 0.68

.



u

X 1, . . . , X n1 ∼ N (m1, σ1) Y 1, . . . , Y n2 ∼ N (m2, σ2)

σ1, σ2

m1, m2

a) H 0 : m1 = m2

H 1 : m1 = m2b)

H 0 : m1 ≤ m2

H 1 : m1 > m2c)

H 0 : m1 ≥ m2

H 1 : m1 < m2

u = X − Y σ21n1

+ σ22n2

H 0∼ N (0, 1).

t

X 1, . . . , X n ∼ N (m, σ) m, σ

u

t = X

−m0

S ∗n · √ n H 0

∼ tn−1,

S ∗n =

1n−1

(X i − X )2

f f = n − 1

a) X k = |t| > tf (α2 ) b) X k = t > tf (α) c) X k = t < −tf (α)

tf (δ )

F (tf (δ )) = 1 − δ

F

f

t

tf (α2 ) tf (α) t

t

X 1, . . . , X n1 ∼ N (m1, σ)

Y 1, . . . , Y n2 ∼ N (m2, σ)

m1, m2

σ

u

t = X − Y

(n1 − 1)S ∗n12 + (n2 − 1)S ∗n2

2·

n1 · n2 · (n1 + n2 − 2)

n1 + n2

H 0∼ tn1+n2−2.

f f = n1 + n2 − 2

t m1 = m2

X − Y ∼ N (0, σ2/n1 + σ2/n2),

1

σ(X − Y )

n1 · n2

n1 + n2∼ N (0, 1).

1

σ2

(n1 − 1)S ∗n1

2 + (n2 − 1)S ∗n22

∼ χ2n1−1+n2−1,



χ2n1−1 χ2

n2−1

t

t

t = X − Y

S ∗n1

2

n1+

S ∗n22

n2

H 0≈ tf ,

f

f = (g1 + g2)2

g21n1−1 +

g22n2−1

,

gi = S ∗ni2/ni f

F

X f 1

Y

f 2 χ2

Z = X/f 1Y/f 2

(f 1, f 2) F

F f 1,f 2 f 1 f 2 E (Z ) = f 2f 2−2

F

X 1, . . . , X n1 ∼ N (m1, σ1) Y 1, . . . , Y n2 ∼ N (m2, σ2) m1, m2 σ1, σ2

a) H 0 : σ1 = σ2

H 1 : σ1 = σ2b)

H 0 : σ1 ≤ σ2

H 1 : σ1 > σ2c)

H 0 : σ1 ≥ σ2

H 1 : σ1 < σ2

F = S ∗n1

2

S ∗n22

H 0∼ F n1−1,n2−1.

f 1 = n1

−1 f 2 = n2

−1

a)X k = F < F f 1,f 2(1 − α2

)

F > F f 1,f 2(α2

) b)X k = F > F f 1,f 2(α) c)X k = F < F f 1,f 2(1 − α),

F f 1,f 2(δ )

G(F f 1,f 2(δ )) = 1 − δ

G

(f 1, f 2)

F

F

H 0

F = S ∗n1

2

S ∗n22 =

1n1−1 ·

(n1−1)S ∗n1

2

σ21

1

n2−1 · (n2−1)S ∗n2

2

σ22

∼F n1

−1,n2

−1.

F < F f 1,f 2(1 − α/2) ⇔ 1

F f 1,f 2(1 − α/2) <

1

F ∼ F f 2,f 1 ,

F f 2,f 1(α/2) =

1

F f 1,f 2(1 − α/2).

X k =

1

F > F f 2,f 1(α/2) F > F f 1,f 2(α/2)

.



α 1 F 1/F

F = max

S ∗n1

2

S ∗n22 ,

S ∗n22

S ∗n12

F n1−1,n2−1(α/2)

F n2−1,n1−1(α/2)

F

χ2

X 1, . . . , X n

∼N (m, σ)

m, σ

a) H 0 : σ = σ0

H 1 : σ = σ0b)

H 0 : σ ≤ σ0

H 1 : σ > σ0c)

H 0 : σ ≥ σ0

H 1 : σ < σ0

χ2 = (n − 1)S ∗2n

σ20

H 0∼ χ2n−1.

f = n − 1

a)X k = χ2 < χ2f (1 − α

2 ) χ2 > χ2f (

α2 ) b)X k = χ2 > χ2

f (α) c)X k = χ2 < χ2f (1 − α),

χ2

f (δ ) G(χ2

f (δ )) = 1

−δ G f χ2

χ2

F = S ∗2

n

σ20

H 0∼ F n−1,∞

F

F n−1,∞

χ2n−1

1 1.9 0.7 1.22 0.8 −1.6 2.43 1.1 −0.2 1.34 0.1 −1.2 1.35 −0.1 −0.1 06 4.4 3.4 17 5.5 3.7 1.88 1.6 0.8 0.89 4.6 0 4.6

10 3.4 2 1.2

α = 0.01

H 0 : m1 = m2 H 1 : m1 = m2

A−B

t H 0 : m = 0 H 1 : m = 0

S ∗n = 1.23 X = 1.58 t = X−m0

S ∗n

√ n = 4.06 t9(0.005) = 3.35 |4.06| > 3.35

t X = 2.33 Y = 0.75

S ∗A2 = 4

S ∗B

2 = 3.8



F F = 1.1 F 9,9(0.025) = 4.03 t

t = X − Y

(10 − 1)S ∗A2 + (10 − 1)S ∗B

2·

10 · 10 · (10 + 10 − 2)

10 + 10 = 1.78.

t18(0.01/2) = 2.89

|1.78| < 2.89

H 0

χ2

A1, A2,...,Ar P (Ai) = pi n

ν i

Ai

χ2 =r

i=1

(ν i − n · pi)2

n · pi

n −→ ∞ r − 1 χ2

pi

s(< r − 1)

ˆ pi

χ2

=

ri=1

(ν i−

n

· ˆ pi)

2

n · ˆ pi n −→ ∞ r − s − 1 χ2

α

n

χ2

A1, A2,...,Ar

H 0 : P (Ai) = pi i = 1, . .., r , H 1 : ∃i : P (Ai) = pi.

n

ν i

Ai

χ2 =r

i=1

(ν i − n · pi)2

n · piH 0≈ χ2

r−1.

f = r − 1

X k = χ2 > χ2f (α),

χ2f (δ )

G(χ2

f (δ )) = 1 − δ

G

f

χ2

χ2



χ2

A1, A2,...,Ar

H 0 : ∃ϑ : P (Ai) = pi(ϑ)∀i, H 1 : ∃ϑ : P (Ai) = pi(ϑ)∀i,

ϑ

s

n

ν i

Ai

ϑ ϑ ˆ pi = pi(ϑ)

χ2 =r

i=1

(ν i − n · ˆ pi)2

n · ˆ piH 0≈ χ2

r−s−1.

f = r − s − 1

npi

nˆ pi

H 0 : P (Ai) = 16 ; i = 1, . . . , 6

n = 60

1 2 3 4 5 6

ν i 8 7 14 12 10 9

n · pi 10 10 10 10 10 10

χ2 = (8 − 10)2

10 +

(7 − 10)2

10 +

(14 − 10)2

10 +

(12 − 10)2

10 +

(10 − 10)2

10 +

(9 − 10)2

10 = 3.4.

f = r − 1 = 5

χ2

5(0.1) = 9.24

3.4 < 9.24 H 0

H 0 : X i ∼ ( p)

p

1 2 3 4

ν i 324 57 14 5

X =

1 · 324 + 2 · 57 + 3 · 14 + 4 · 5

400 =

500

400 =

5

4

ˆ p = 1X

=

0.8

A1 = 1, A2 = 2, A3 = 3, A4 = 4, . . .

ˆ p1 = 0.8, ˆ p2 = 0.2 · 0.8 = 0.16, ˆ p3 = 0.22 · 0.8 = 0.032, ˆ p4 = 1 − ˆ p1 − ˆ p2 − ˆ p3 = 0.008.

1 2 3 ≥ 4n · ˆ pi 320 64 12.8 3.2



χ2 = (324 − 320)2

320 +

(57 − 64)2

64 +

(14 − 12.8)2

12.8 +

(5 − 3.2)2

3.2 = 1.95.

f = r − s − 1 = 4 − 1 − 1 = 2

α = 0.05

χ22(0.05) = 5.99 1.95 < 5.99 H 0

χ2

pi

r

A1,...,Ar

s B1,...,Bs

H 0 :

P (Ai ∩ Bj) = P (Ai) · P (Bj)

i, j

n ν ij Ai ∩ Bj ν i• =

sj=1

ν ij

Ai

ν •j =

ri=1

ν ij

Bj

P (Ai) = pi

P (Bj) = q j Ai ∩ Bj r · s

χ2 =

ri=1

sj=1

(ν ij − n · piq j)2

n · piq j

H 0≈ χ2r·s−1.

pi, q j

ˆ pi = ν i•

n , q j =

ν •jn

.

r

−1

pi

pr = 1−r−1

i=1

pi s

−1

q j

χ2 =

ri=1

sj=1

(ν ij − n · ˆ piq j)2

n · ˆ piq j=

ri=1

sj=1

(ν ij − ν i•ν •jn )2

ν i•ν •jn

H 0≈ χ2f ,

f = r · s − (r − 1 + s − 1) − 1 = (r − 1)(s − 1)

r = s = 2

χ2 = n · (ν 11ν 22 − ν 12ν 21)2

ν 1•ν 2•ν •1ν •2.



ν ij = ν i•ν •j

n

ν 11 + ν 12 = ν 1• ν 11 + ν 12 =

ν 1•ν •1

n +

ν 1•ν •2

n =

ν 1•n

(ν •1 + ν •2) = ν 1•.

ν 12 − ν 12 = −(ν 11 − ν 11) ν 21 − ν 21 = −(ν 11 − ν 11) ν 22 − ν 22 = ν 11 − ν 11

χ2

ν 11 − ν 11 = ν 11 − 1

n(ν 11 + ν 12)(ν 11 + ν 21) =

ν 11ν 22 − ν 12ν 21

n .

n

1

ν 1•ν •1+

1

ν 1•ν •2+

1

ν 2•ν •1+

1

ν 2•ν •2=

n2

ν 1•ν •1ν 2•ν •2.

√ n·(ν 11ν 22−ν 12ν 21)√

ν 1•ν 2•ν •1ν •2

H 0≈ N (0, 1)

u

u

H 0

H 1

30 20 50

70 80 150

100 100 200

α = 0.05

25 25

75 75

χ2 = 200·(30·80−70·20)2

100·100·50·150 = 2.67.

(2 − 1) · (2 − 1) = 1

χ2

1(0.05) = 3.84

α = 0.05

√ 200·(30·80−70·20)√

100·100·50·150 = 1.63.

u0.05 = 1.65

X Y A1, . . . , Ar

H 0 : X Y

P (X ∈ Ai) = P (Y ∈ Ai)

i

X

n

ν i

Ai Y

m

µi Ai

i

ˆ pi =

ν i+µi

n+m

χ2 =

ri=1

(ν i − nˆ pi)2

nˆ pi+

ri=1

(µi − mˆ pi)2

mˆ pi

H 0≈ χ2f ,

f = (r − 1) + (r − 1) − (r − 1) = r − 1

r − 1 r − 1

χ2 =r

i=1

ν in − µi

m

2

ν i + µi· n · m

H 0≈ χ2r−1.



α = 0, 05

1 2 3 4 5 6 r = 6

ν i 7 11 8 10 8 6 n = 50µi

16 11 20 19 18 16 m = 100

χ2 =6

i=1

ν i50 − µi

100

2

ν i + µi· 50 · 100 = 3.58.

χ26−1(0.05) = 11.1 3.58 < 11.1 H 0

Θ = Θ0 ∪ Θ1

n

X

supLn(X ; ϑ) : ϑ ∈ ΘsupLn(X ; ϑ) : ϑ ∈ Θ0 .

H 0

H 0

f 0 f 1

maxf 0(X ), f 1(X )f 0(X )

= max

1, f 1(X )

f 0(X )

.

H 0

T = 2

supϑ∈Θ

log Ln(X ; ϑ) − supϑ∈Θ0

log Ln(X ; ϑ)

.

σ ∈ Rq

( p − q )

τ

H 0 : σ = σ0, τ

, H 1 : σ = σ0, τ

n → ∞

T

χ2q σ

χ2

χ2

χ2



X 1, . . . , X n F

H 0 : F = F 0 H 1 : F = F 0

F n

Dn = sup

x∈R F n(x)

−F 0(x) .

Dn H 0 F 0 X (n)i X

(n)i+1

in

F 0

Dn = max0≤i≤n

max

|F 0(X

(n)i ) − i/n|, |F 0(X

(n)i+1) − i/n|

.

H 0

F 0 X i

F 0

F 0(X i) ∼ E (0, 1)

√ nDn

P (√

nDn < y) n→∞−→ K (y) =

+∞

i=−∞(−1)i · e−2i2y2 (y > 0).

K

n

α = 0.05

H 0

√ nDn > 1.36

n

H 1 : F (x) > F 0(x) ∀x H 1 : F (x) < F 0(x) ∀x

D+n = sup

x∈R

F n(x) − F 0(x)

,

D−

n = supx∈R

F 0(x) − F n(x)

.

H 0

F 0

√ nD±

n

P (√

nD±n < y)

n→∞−→ K 1(y) = 1 − e−2y2 (y > 0).

K 1 n

α = 0.05

H 0

√ nD±

n > 1.22

n

(X i, Y i)

i = 1, . . . , n

H (x, y)

H F (x) = limw→∞ H (x, w) G(y) = limz→∞ H (z, y)

H 0 :

H (x, y) = F (x)G(y) ∀x, y

K n =n−1i=1

nj=i+1

[I ((X i − X j)(Y i − Y j) ≥ 0) − I ((X i − X j)(Y i − Y j) < 0)] .

K n

K n H 0

F, G (X i, Y i) (F (X i), G(Y i))

K n F, G

F (X i) G(Y i) E (0, 1)

K n

n

u

n u



K n

K n

H 0

E 0(I ((X i − X j)(Y i − Y j) ≥ 0)) = P 0((X i − X j)(Y i − Y j) ≥ 0) = P 0(X i − X j ≥ 0 Y i − Y j ≥ 0)+

P 0(X i − X j ≤ 0 Y i − Y j ≤ 0) = 0.5 · 0.5 + 0.5 · 0.5 = 0.5.

E 0(I ((X i − X j)(Y i − Y j) < 0)) = 0.5 E 0(K n) = 0

K n

H 0

δ ij = [I ((X i − X j)(Y i − Y j) ≥ 0) − I ((X i − X j)(Y i − Y j) < 0)]

D20(K n) =

0

i<j

δ ij ,k<l

δ kl

=

i<j

k<l

0(δ ij , δ kl).

0(δ ij , δ kl) =

0 i, j ∩ k, l = ∅1

i = k, j = l

1/9

D2

0(K n) = 1 · n(n − 1)

2 +

1

9 · n(n − 1)(n − 2)

6 · 6 =

n(n − 1)(2n + 5)

18 .

τ = 0

τ = 2P ((X 1 − X 2)(Y 1 − Y 2) > 0) − 1

|τ | ≤ 1

X Y

τ = 0

N 1(i) = | j = i | X j < X i Y j < Y i |

N 2(i) = | j = i | X j > X i Y j < Y i |N 3(i) = | j = i | X j < X i

Y j > Y i |N 4(i) = | j = i | X j > X i Y j > Y i |

N 1(i) N 2(i)

N 3(i) N 4(i)

•(Xi,Y i)

N (i)

(X i, Y i)

Bn = 1

n

ni=1

N 1(i)

n · N 4(i)

n − N 2(i)

n · N 3(i)

n

2=

1

n

ni=1

( H n(X i, Y i) − F n(X i) Gn(Y i))2,

F n, Gn, H n

H n(X i, Y i) = N 1(i)/n, F n(X i) = (N 1(i) + N 3(i))/n, Gn(Y i) = (N 1(i) + N 2(i))/n.

Bn H 0 F, G nBn

n

α = 0.05 H 0 nBn > 0.058

n

X 1, . . . , X n Y 1, . . . , Y m

F

G

H 0 : F = G H 1 : F = G

W n,m =n

i=1

mj=1

I (X i ≥ Y j).



Z

N := n + m

Z

(N )1 < · · · < Z

(N )N

ri X (n)i

W n,m = r1 + · · · + rn − n(n+1)2

X (n)1 r1 −1 Y j X

(n)2 r2 −2 Y j X

(n)i ri− i Y j

W n,m H 0

H 0

X

Y

1/n+m

n

X W n,m

I (X i ≥ Y j)

n, m u n, m

u

W n,m

W m,n H 0

E 0(I (X i ≥ Y j)) = P 0(X i ≥ Y j) = 12 ⇒ E 0(W n,m) = n·m

2 W n,m H 0

δ ij = I (X i ≥ Y j)

D20(W n,m) = 0

n

i=1

m

j=1

δ ij ,

n

l=1

m

k=1

δ lk =

n

i=1

m

j=1

n

l=1

m

k=1

0(δ ij , δ lk).

0(δ ij , δ lk) =

0 i = l, j = k

1/4 i = l, j = k1/12

i = l, j = k

1/12

i = l, j = k

D2

0(W n,m) = nm1

4 + nm(m − 1)

1

12 + mn(n − 1)

1

12 =

n · m(n + m + 1)

12 =

n · m

4 · n + m + 1

3

n · m

4

δ ij P (X > Y ) = 1/2

|hatF n, Gm

Dn,m = supx∈R

F n(x) − Gm(x) .

Dn,m H 0

m·nm+n ·

Dn,m n, m

n, m

H 1 : F (x) > G(x) ∀x

D+n,m = sup

x∈R

F n(x) − Gm(x)

.

H 0

m·nm+n ·D+

n,m n, m

n, m

(X, Y )

X

Y



X

h2 = E [(Y − (aX + b))2]

h2 = E (X 2)a2 + b2 + 2E (X )ab − 2E (XY )a − 2E (Y )b + E (Y 2)

a 0

a =

E (XY )

−bE (X )

E (X 2) .

b 0

b = E (Y ) − aE (X ).

a∗ = cov(X, Y )

D2(X ) , b∗ = E (Y ) − a∗E (X ).

h2 Y

(X 1, . . . , X q, Y )

X = (X 1, . . . , X q) h2 = E [(Y − (aT X + b))2] aT = (a1, . . . , aq)

a∗ = cov(Y, X )Σ−1(X ), b∗ = E (Y ) − a∗E (X ).

cov(Y, X ) = (cov(Y, X 1), . . . , c o v(Y, X q))

Σ(X )

q ×q

(i, j)

cov(X i, X j)

Y

X

Y

Y 0.82X +4.06 Y 1.02X 2−1.25X +4.38

(X i1, . . . , X iq, Y i)

i = 1, . . . , n

h2 = 1n

ni=1(Y i − (aT X i + b))2

Y i = aT X i + b + i X i

Y i

X i i

N (0, σ2)

Y i N (aT X i + b, σ2)

a b a∗ b∗

σ2

σ2 = 1

n

ni=1

(Y i − (aT X i + b))2,

1

n − q − 1

ni=1

(Y i − (aT X i + b))2

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