st.dvi

9

3 5................................... 5............................. 5....................................... 5.................................. 7......................................................................... 3............................. 4................................... 7 F t.................................. 0 3 9....................................... 9................................ 9 -.............................. 30.................................... 3....................................... 33....................................... 34...................................... 35............... 35............................ 36....................... 37 3.................................... 38................................. 38................................. 40 4 43................................... 43.................................. 43.................................... 44

4.................................... 45................................. 45.................................... 45.................................... 46................................... 46................................... 47 3 χ...................................... 48.................................... 48................................... 49 5 5...................................... 5................................... 5................................... 53.................................... 54

5. x,..., x 00 0 x<0 5 6 0.03 6 0.03 0 x<0 5 8 0.04 4 0.07 0 x<30 5 0.06 6 0.3 30 x<40 35 0.05 47 0.35 40 x<50 45 36 0.8 83 0.45 50 x<60 55 49 0.45 3 0.66 60 x<70 65 30 0.5 6 0.8 70 x<80 75 0.05 83 0.95 80 x<90 85 0.055 94 0.97 90 x 00 95 6 0.03 00 00.000 : x, x,..., x

6 50 40 30 0 0 0 0 30 40 50 60 70 80 90 00 00 80 60 40 0 00 80 60 40 0 0 0 30 40 50 60 70 80 90 00.: ( ) ( ) x x + x + + x x G x x x ( + + + ) x H x x x x x x x l+, l + m x l + x l+, l (max{x i} + mi{x i }) 40km 50km v d v d d 40 + d 50 ( + 40 50 )

. 7.: v ( 40 + ) 50 v 44.44 : x, x,..., x ( x x + x x + + x x S {(x x) +(x x) + +(x x) } S S () z i xi x S, z, z,..., z 0 T i 0z i +50 0 S (x i x)+50 50, 00 0..

8 0 30 40 68.8% 60 70 80 95.45% 99.73%.3: () () 0 50 6 0 99 00 () () 00 6 50 + 00 50 8 (0 50) + (00 50) 8 65 5 65 0 0 (0 50) + 50 30 5 50 50 000 0 (00 50) + 50 70 5

. 9 00 00 99 (0 ) + (00 ) 00 99 9.949 99 + 99 00 99( + 99) 00 0 0 (0 ) + 50 48.994 9.949 000 0 (00 ) + 50 49.5 9.949 99

(statistical iferece). ( f) μ, σ X, X,...,X.: (populatio) (sample) (parameter) μ σ N(μ, σ )

() f(x) μ, σ f(x) X, X,..., X (sample mea) X X + X + + X (sample variace) S (X X) +(X X) + +(X X) (ubiased sample variace) U (X X) +(X X) + +(X X) (statistic).. f μ σ X μ σ / E[X] E[X ]+E[X ]+ + E[X ] μ μ ( X + X + + X ) V [X] V V (X + X + + X ) {V (X )+V (X )+ + V (X )} σ σ... f μ σ U σ E[U ] E[(X X) ]+E[(X X) ]+ + E[(X X) ] E[(X X) ]E[(X μ X + μ) ] E[(X μ) ] E[(X μ)(x + μ)] + E[(X μ) ] σ E[(X μ) (X μ)+(x μ)+ +(X μ) ]+ σ σ E[(X μ) ]+ σ σ σ + σ

. 3 σ σ σ E[U ] E[(X X) ] σ σ X, U (estimator) S U U U. (X,X,...,X ) - { } (.) p(x) (π) / det(v ) exp (x m, V (x m)) m (m,m,...,m ) V (V ij ) m i E[X i ], V ij Cov(X i,x j )E[(X i m i )(X j m j )]. (X,X,...,X ) ϕ(ξ,ξ,ξ ) ϕ(ξ,ξ,...,ξ )E[exp{ iξ j X j }], (ξ,ξ,...,ξ ) R. (.) (.) ϕ(ξ,ξ,...,ξ ) exp{i ξ j m j + V jk ξ j ξ k }, j,k (.) V V X, X,..., X X j (X,X,...,X ) - (X,X,...,X ) -

4 X, X,..., X (X,X ) 0 ϕ(ξ,ξ ) exp{i ξ j m i + V jj ξj } exp{iξ m + V ξ } exp{iξ m + V ξ } ϕ X (ξ )ϕ X (ξ ).. X N(μ, σ ) Z X μ σ N(0, ) X, X N(μ,σ ), N(μ,σ ) X + X, N(μ + μ,σ + σ ) X, X,..., X N(μ, σ ), X X + + X E[exp{iξ(X μ)/σ}] E[exp{ iξμ/σ} exp{i(ξ/σ)x}] N(μ, σ /). exp{ iξμ/σ} exp{i(ξ/σ)μ + (ξ/σ) σ } exp{ ξ }. Z N(0, ) X + X E[exp{iξ(X + X )}] E[exp{iξX }]E[exp{iξX }] exp{iξμ + σ ξ } exp{iξμ + σ ξ } exp{iξ(μ + μ )+ σ + σ ξ }. X + X N(μ + μ,σ + σ ) X. N(μ, σ /).. α>0 (.3) Γ(α) 0 x α e x dx

. 5 α, β>0 (.4) B(α, β) 0 x α ( x) β dx Γ(α+) αγ(α) Γ(+)! (.4) x si θ dx si θ cos θ π/ π/ B(α, β) si (α ) θ ( si θ) β si θ cos θdθ si α θ cos β θdθ 0 0 B(, )π.3. (.5) f α,β (x) Γ(α)β α xα e x/β, x 0 0, x < 0 Ga(α, β) [0, ] (.6) B(α, β) xα ( x) β Be(α, β) (.4) 0 f α,β (x) dx Γ(α)β α Γ(α) 0 Γ(α) 0 Γ(α) 0 0 x α e x/β dx (x/β) α e x/β dx x y α y dy e y y α e y dy (y x/β, dy dx/β).4. (.7) f α,β f α,β f α +α,β.

6 x 0 f α,β f α,β(x) x y,y 0 x 0 f g(x) f(x y)g(y) dy Γ(α )β (x α y)α e (x y)/β Γ(α )Γ(α )β α +α (x y) α y α e x/β dy Γ(α )β α yα e y/β dy Γ(α )Γ(α )β α +α e x/β (x tx) α (tx) α xdt 0 Γ(α )Γ(α )β α +α e x/β x α +α ( t) α t α dt 0 B(α,α ) Γ(α )Γ(α ) β α +α x α +α e x/β B(α,α )Γ(α + α ) f α +α Γ(α )Γ(α ),β(x). x<0 f α,β f α,β(x) 0 f α,β f α,β(x) dx B(α,α )Γ(α + α ) Γ(α )Γ(α ) B(α,α )Γ(α + α ). Γ(α )Γ(α ) (y tx, dy xdt) f α +α,β(x) dx f α,β f α,β(x)dx dx f α,β(y) dy f α,β(x y)f α,β(y) dy f α,β(x y) dx. B(α,α )Γ(α + α ) Γ(α )Γ(α ) (.8) B(α, β) Γ(α)Γ(β) Γ(α + β)

. 7 α β ( B ), Γ( ) Γ(). B(, )π Γ( ) π X, Y f, g f g X + Y E[F (X + Y )] F (x + y)f(x)g(y) dx dy u x + y, v y x x (x, y) (u, v) u v y y u v 0 dx dy (x, y) (u, v) du dv du dv F (u)f(u v)g(v) du dv F (u) f g(u) du.5. Ga(/, ) χ ( χ ().6. X N(0, ) X χ () E[F (X )] 0 0 0 0 F (x ) e x / dx π F (x ) e x / dx π F (y) e y/ dy π y F (y) y (/) e y/ dy π F (y) e y/ dy Γ(/) /y(/) (y x,dyxdx dx ydy)

8 0 F (y)f /, (y) dy..7. X, X,..., X N(0, ), X + X + + X χ (). χ () N(0, ) + N(0, ) + + N(0, ) }{{} X j f /, X + X + + X f /, f /, f /, f /,..8. X, X,..., X N(μ, σ ), Y (X σ j μ) χ (). X j f /, σ (X j μ).7 ( Xj μ ), X j μ σ σ N(0, ).9. X, X,..., X N(μ, σ ), Y (X μ) σ / χ ().. X N(μ, σ /).6.0. X, X,..., X N(μ, σ ), Y (X σ j X) U χ ( ). σ Y X μ 0,σ Y j l jk X k k Y, Y,..., Y (l jk ) Y, Y,..., Y. X + X + + X Y + Y + + Y.

. 9. Y X (X + X + + X ) 3. Y, Y,..., Y N(0, ),. Y (X j X) Xj X Y j Y Y + Y + + Y χ ( ) Y Y X Y X (X,X,...,X ) (Y,Y,...,Y ) X + X + + X Y (X + X + + X ) (l jk )... (l jk ) (X,X,...,X ) p(x,x,...,x ) (Y,Y,...,Y ) q(y,y,...,y ) p(x,x,...,x )q(y,y,...,y ) (y,y,...,y ) (x,x,...,x ). }{{} p(x,x,...,x ) π e (x +x + +x )/ q(y,y,...,y ) π e (x +x + +x )/ π e (y +y + +y )/ Y, Y,..., Y N(0, ) π e y j /.

0 N(μ, σ ) Z j X j μ σ N(0, ) Z (Z + Z + + Z ) X μ σ X j X X j μ + μ X Z j Z σ σ σ Y (X σ j X) (Z j Z). F t.. m, ( m ) m ( G m, (z) B( m, ) z m m ) m+ z +, z > 0 (.9) 0, z 0 (m, ) F F (m, ) 0 F.. X, Y m, χ (.0) Z X/m Y/ (m, ) F

..3. X Y, X p(x), Y q(y), Y>0 Z ax by (.) r(z) 0 ( byz ) p q(y) by a a dy. [ ( ax )] E[f(Z)] E f by ( ax ) f p(x)q(y)dxdy 0 by z ax by,u y y u, x b a yz b a uz x x b (x, y) (z, u) z u u b z a a bu y y a z u 0 dx dy (x, y) (z, u) dz du bu dz du a ( buz ) f(z)p q(u) bu du dz 0 a a ( buz ) f(z)dz p q(u) bu 0 a a du. }{{} r(z) Z G m, (z) Z X/m Y/ X my X p(x) m Γ( m e x/ )xm Y q(y) Γ( )y e ( m ) p 0 yz q(y) my dy ( m ) m m Γ( m )Γ( ) yz e m yz y e y m ydy y

G m, (z) m + m Γ( m )Γ( ) ( m u ( + m z)y ) m du ( + m z)dy u y dy 0 m+ m+ m+ Γ( ) ( m Γ( m )Γ( ) B( m, ) ( m z m y m + e (+ m z)y dy y 0 }{{} y ( + m z) u dy y du u (+ m ) m+ z u m+ e u du u ( m + ) (+ m z ) m+ Γ ) m ( z m + m ) m+ z ) m z m ( + m ) m+ z. F F (m, ) χ (m)/m χ ()/.4. X, X,..., X m, Y, Y,..., Y : Z {(X m X) + +(X m X) } {(Y Y ) + +(Y Y ) } U X UY ( ) (m, ) F σ (Xi X) (m )U X χ (m ) (.0) σ σ (Yj Y ) ( )U Y χ ( ) (.0) σ σ UY UX χ (m )/(m ) χ ( )/( ) F (m, ).5. X, X,..., X : i.i.d., N(μ, σ ) Y (X μ) U F (, ).

. 3 ( )U (X μ) χ () (.9) σ σ σ (Xi X) χ ( ) (.0) (X μ) U χ () F (, ). χ ( )/( ).6. t (.) f(x) ) B( (+ x +, ) t().7. X N(0, ), Y χ (), X Y X Y t t() N(0, ) χ ()/ [ ( X )] ( x ) E F F y e x Y 0 π Γ( )y e y dy dx t x y,u y x t u, y u x x (x, y) (t, u) t u u t u u y y t u 0 F (t) e t u 0 π Γ( )(u) e u udu dt { } + πγ( ) F (t) e t + u u du dt 0 }{{}

4 v ( + t )u dv ( + t )du ( e v v ) 0 ( + t ) ( + t )dv ( ) + + + t + e v v + dv ( + + Γ( + ) πγ( B(, ) ) + t ( F (t) + t ( F (t) 0 ) + ( + Γ + t ) + dt ) + dt ).8. X, X,..., X : i.i.d., N(μ, σ ) U (X μ) U t( ). U (X μ) N(0, ), σ (.) ( )U χ ( ), σ (.0) (X μ) U (X μ) σ ( )U σ (X μ) t( ). U N(0, ) t( ). χ ( )/( ).9. Z t() Z F (,). X N(0, ), Y χ (), X Y Z X, Z X Y Y/ χ () χ ()/ F (,). t F

. 5 F F.0. X B(N,p) 0 r N (r + ), (N r), x 0 p (q p) q (.3) P (X 0)+P (X )+ + P (X r) x 0 F (, )(x) dx (N r), (r + ), x q p (.4) P (X r +)+P (X r +)+ + P (X N) x F (, )(x) dx (.5) P (X 0)+P (X )+ + P (X r) N! r!(n r )! p y r ( y) N r dy. p [ y r ( y) N r dy N r yr ( y) N r ] N r pr ( p) N r + r N r p + r N r p p y r ( y) N r dy y r ( y) N r dy. N! r!(n r )! N! r!(n r )! p y r ( y) N r dy N! r!(n r)! N r pr q N r N! + (r )!(N r)! N! y r ( y) N r dy r!(n r )! p k ( ) N p r i q N r+i N! + r i (r k )!(N r + k)! i p p y r ( y) N r dy. y r k ( y) N r+k dy

6 k r, r i j (.6) N! r!(n r )! p y r ( y) N r dy r ( N j r ( N j r ( N j j0 ) p j q N j + N! (N )! ) p j q N j +( p) N ) p j q N j p ( y) N r+k dy (r + ), (N r) r, N r N! r!(n r )! Γ(N +) Γ(r + )Γ(N r) + Γ( ) Γ( )Γ( ) B( F y y x x+ x +, dy ( x + ) dx, ). y p p x, x + p x + p x p x( p) p x p p ( ) y r ( y) N r x ( dy x 0 x + x + / / x ( x + ) dx ( + )/ (.6) / / B(, ) x 0 x 0 x ( x + ) dx ( + )/ F (, ) (.3) ) ( x + ) dx

. 7 (.4) p, q Y B(N,q) P (X r +)+P (X r +)+ + P (X N) P (Y 0)+P (Y )+ + P (Y N r ) (.3)

9 3. f(x; θ) f(x, θ) θ θ X, X,..., X X, X,..., X ˆθ g (X,X,...,X ) θ ˆθ ˆθ ˆ E[ˆθ ]θ ˆθ V (ˆθ )E[(ˆθ θ) ] (mea square error) F {f(x; θ)} θ θ (likelihood fuctio) l(θ) log f(x; θ). l(θ) x l(θ; x) l l(θ) f(x, θ) f(x; θ), l(θ) f(x, θ) f(x; θ) ( ) f(x, θ). f(x; θ) θ l E f(x, θ) ( x f(x, θ) X

30 3.. {f(x; θ)} (.) I(θ) E[ l(θ) ] f(x, θ) f(x; θ) dx θ (Fisher s iformatio) (.) (.3) E[ l(θ)] d dθ d dθ E[ l(θ)] 0, E[ l(θ)] I(θ) f(x, θ) f(x; θ) dx f(x; θ) f(x, θ) dx f(x, θ) dx 0, [ ( f(x, θ) E[ l(θ)] E f(x; θ) d dθ f(x, θ) f(x; θ) )] f(x, θ) f(x; θ) f(x; θ) dx E[ l(θ) ] f(x, θ) dx E[ l(θ) ] I(θ). f(x, θ) dx I(θ) - X, X,...,X f (x,x,...,x ; θ) f(x j ; θ) l l (θ; x,x,...,x ) log f (x,x,...,x ; θ) l(θ; x j ).

. 3 θ l (θ; x,x,...,x ) f (x,x,...,x ; θ) f (x,x,...,x ; θ) l(θ; x j ). S S l (θ; X,X,...,X ) E[S ]E[ l (θ; X,X,...,X )] f (x,x,...,x ; θ)dx dx R d dθ 0. R f (x,x,...,x ; θ)dx dx θ E[ˆθ ] g(x,x,...,x )f (x,x,...,x ; θ) dx dx. R θ g(x,x,...,x ) f (x,x,...,x ; θ) dx dx R E[g(X,X,...,X ) l (θ X,X,...,X )] E[ˆθ S ]. f(x x,...,x ; θ) I (θ) E[ l (θ) ]V(S )V( l(θ; X j )) V ( l(θ; X j )) I(θ).. (Cramer-Rao) ˆθ Cramer-Rao (.4) θ V (ˆθ ) I (θ) I(θ). (.5) l (θ) I (θ)(g(x,x,...,x ) θ) (efficiet equatio)

3 3 E[S ]0 Schwarz Cov(S, ˆθ )E[S ˆθ ] E[S ]θ E[S ˆθ ]. Cov(S, ˆθ ) V (S )V (ˆθ )I (θ)v (ˆθ ). (.4) Schwartz S, ˆθ S aˆθ + b Cramer-Rao b I (θ) 0a + bθ a bθ. S b(ˆθ θ). Cov(S, ˆθ )bv (ˆθ ) b I (θ)..3. (.4) /I (θ) Cramer-Rao I (θ)v (ˆθ ) ˆθ (efficiecy) 00 %.4. ˆθ 00% ˆθ (.5) l (θ; x,...,x ) I (θ) dθ g(x,...,x ) θi (θ) dθ + c(x). (.6) f (x,...,x ; θ) exp{a(θ)g(x,...,x )+b(θ)+c(x,...,x )}. a(θ) I (θ) dθ, b(θ) θi (θ) dθ c(x,...,x ) θ

. 33.5. f (.7) f (x,...,x ; θ) h(g(x,...,x ),θ) k(x,...,x ) ˆθ g(x,...,x ) (sufficiet statistics) k(x,...,x ) θ θ.. N(μ, σ ) σ f(x; μ) πσ e (x μ) /σ l(μ) log(πσ ) (x μ) σ l(μ) (x μ)/σ. [ ] (X μ) I (μ) E[ l(μ) ]E σ. (X,X,...,X ) { } (x j μ) f (x,...,x ; μ) πσ exp σ μ σ 4 S l (μ; X,...,X ) X j μ σ I(μ)(X μ). X μ X, U. :. :

34 3 3. : 4. : ε>0 P ( ˆθ >ε) 0 as. 5. :.. p p r X, X,..., X p i.i.d. P (X j )p, P (X j 0) p X + X + + X r L(p) ( ) L(p) P (X + X + + X r) p r ( p) r. r L(p) p p X, X,..., X L(p) log L(p) ( ) log L(p) log + r log p +( r) log( p) r p d dp log L(p) r p r p r( p) ( r)p p( p) r p p( p). 0 p r log L(p) p r ˆp r (X + X + + X ).3. N(μ, σ ) f (x,...,x ; μ) { πσ exp } (x j μ) σ

. 35 μ, σ μ l (μ, σ ) log π log σ (x j μ) σ. μ, σ 0 ν l (θ, σ ) x j μ σ 0, σ l (θ, σ ) σ + (x i μ) (σ ) 0 μ x + + x x, σ (x j μ) (x j x). (μ, σ ) X X + + X, S (X X) + +(X X). S. θ [L, U] α L, U P (L θ U) a. L U α 0.99 0.95 [L, U] σ

36 3 α Z α π e x / dx α Z α Z α 00α% α L X σz α/ U X + σz α/ 3.: Z α [L, U] α X N(μ, σ /) (X μ) N(0, ) ( σ ) (X μ) P Z α/ Z α/ α σ ( P X σz α/ μ X + σz ) α/ α. Z 0.05.96 Z 0.005.56 (.576).. 0.5 5 X.03 95% [.007,.399]. X ± σz 0.05 0.5.96.03 ± 5.03 ±.96 0.03 ± 0.96 X,X,...,X N(μ, σ ).

. 37 σ U {(X X) + (X X) + +(X X) } σ (X j X) χ ( ) χ 00α% χ α ( ) ( P χ α/ ( ) σ ( ( )U P χ α/ ( ) σ α/ 0 χ α/ χ α/ 3.: ) (X j X) χ α/ ( ) α ( )U χ α/ ( ) ) α. α/ ( [ α) ] ( )U χ α/ ( ), ( )U χ α/ ( ) X,X,...,X N(μ, σ ) X (X + + X ) U {(X X) + +(X X) } (X μ) N(0, ) σ ( )U χ ( ) σ (X μ) U (X μ) σ ( )U σ N(0, ) t( ) χ ( )/( ) (X μ) t( ) U

38 3 α/ α/ t 00α% t α ( ) t α/ ( ) t α/ ( ) [ ] (X μ) P t α/ ( ) t α/ ( ) α U 3.3: μ P [ X U t α/ ( ) μ X + U ] t α/ ( ) α [ α X U t α/ ( ), X + U ] t α/ ( ) 3. X,...,X m N(μ,σ ) Y,...,Y N(μ,σ ) X Y () X m (X + + X m ) Y (Y + + Y ) X N(μ, σ m ) Y N(μ, σ )

3. 39 [ P μ μ [ P X Y X Y N(μ μ, σ m + σ ) Z X Y (μ μ ) N(0, ) σ + σ m Z α/ X Y (μ μ ) σ m + σ Z α/ σ m + σ Z α/ μ μ X Y + ] α ] σ m + σ Z α/ α α [ ] σ X Y m + σ σ Z α/, X Y + m + σ Z α/ () ( X Y N(μ μ, m ) + σ ) σ ( σ σ) U (Xi X) + (Y j Y ) m + (m )U X +( )U Y m + (m + )U σ (m )U X σ U X Y + ( )U Y σ χ (m ) + χ ( ) χ (m + ) Z X Y (μ μ ) ( m + )σ N(0, ) T Z (m+ )U σ m+ X Y (μ μ ) t(m + ) U ( m + )

40 3 [ α X Y U m + t α/(m + ), X Y + U m + ] t α/(m + ) (3) Welch T X Y (μ μ ) U X m + U Y T φ t φ ( m U X + U Y ) φ ( U m X ) m + ( U Y ) α [ U X Y X m + U Y U t α/(φ), X Y + X m + U ] Y t α/(φ) X,...,X m N(μ,σ ) Y,...,Y N(μ,σ ) X m (X + + X m ) Y (Y + + Y ) U X m {(X X) + +(X m X) } U Y {(Y Y ) + +(Y m Y ) } (m )U X σ χ (m ) ( )U Y σ U X /σ U Y /σ χ ( ) χ (m )/(m ) χ ( )/( ) F (m, )

3. 4 α P [F α/ (m, ) U X U Y σ σ F α/ (m, )] α P [ U Y F UX α/ (m, ) σ σ U Y F UX α/ (m, )]. σ /σ [ ( α) U Y F UX α/ (m, ), U ] Y F UX α/ (m, ) F F (m, ) F (, m) F α (m, ) F α (, m) F α (m, )

43 4... 80 39 6 p p 6 H 0 : p 6 H : p 6 p 6 X 80 X Bi(80, 6 ) E[X] 30, V(X) 80 p ( p) 80 6 5 6 55. 30 X 30 c H 0 X 30 >c H 0 c α P ( X 30 >c)α α 0.005, 0.0 W {x; x 30 >c} c X N(30, 5 ) Z N(0, ) ( X 30 P ( X 30 >c)p 5 > c ) ( P Z > c ) α. 5 5 α 0.05 P ( Z >, 96) 0.05 c.96 c.96 59.8 P ( X 30 > 9.8) 0.05. 5 X<30 9.8 0. or X>30 + 9.8 39.8

44 4 0. 30 39. 4.: 39.. : H 0 : H 0 H. ( ). 3. (α 0.05, 0.0) 4. 5. H 0 : θ θ 0 H : θ θ θ θ 0 P ( ˆθ θ 0 >c)α c 0 H 0 : θ θ 0 H : θ>θ 0 H 0 : θ θ 0 H : θ<θ 0 θ θ 0 P (ˆθ θ 0 >c)α c θ θ 0 P (ˆθ θ 0 <c)α c {x; x θ 0 >c}, {x; x θ 0 >c}, {x; x θ 0 <c}

. 45 H 0 H 0 ( α) ( ). X, X,..., X : [A] σ H 0 : μ μ 0 H : μ μ 0 (X μ0 ) Z σ : W {z; z >Z α/ }. N(0, ). Z W Z W H 0 : μ μ 0 H : μ>μ 0 H 0 : μ μ 0 H : μ<μ 0 W {z; z>z α} W {z; z<z α}. [B] σ U (X j X) T (X μ0 ) t( ), U

46 4 W {t; t >t α/ ( )}.. A 700 0 A 5% 68, 580, 66, 743, 75, 576, 653, 805, 98, 783 H 0 : μ 700 H : μ>700 X 77.9, U 0.659, t 0.05 (9).833 0(77.9 700) T 0.470 0 W {t; t>t 0.05 (9)} {t >.833}. 0.470.833 X, X,..., X : H 0 : σ σ 0 H : σ σ 0 ( )U Y χ ( ) σ 0 W {y; y<χ α/ ( ), y > χ α/ ( )} Y W Y W X, X,..., X m N(μ,σ ), Y, Y,..., Y N(μ,σ ). H 0 : μ μ H : μ μ [A] σ, σ Z X Y σ m + σ N(0, )

. 47 [B] σ σ W {z; z >Z α/ } m U m + { (X j X) + T X Y U [C] σ, σ [A] σ, σ m + t(m + ) W {t; t >t α/ (m + )} U X m U Y (Y j Y ) } m (X j X) (Y j Y ) T X Y U X m + U Y T φ t φ ( m U X + U Y ) φ ( m U X ) m + ( U Y ). Welch s test X, X,..., X m N(μ,σ ), Y, Y,..., Y N(μ,σ ). H 0 : σ σ H : σ σ H 0 F U X U Y W {f; f< F (m, ) F α/ (,m ), f > F α/(m, )}

48 4 3. χ χ?? [A] A A A k k k k H 0 : H : P (A i ) i i P(A i ) i X k ( i i ) i i W {x; x>χ α (k )} χ k χ [B] F θ : θ (θ,θ,...,θ t ) θ ˆθ A i P (A i ) i i (θ) θ ˆp i i (ˆθ) H 0 : H : F θ F θ X k ( i ˆp i ) i ˆp i W {x; x>χ α(k t )} k t χ μ, σ A i A i {a i <Z a i+ }

3. χ 49 A B A r B c A B B A B B B c A p p p c p A p p p c p..... A r p r p r p rc p r p p p c c : p i p ij r : p j p ij i r c : i p ij p ij P (A i B j ), p i P (A i ), p j P (B j ) A, B H 0 : p ij p i p j i,...,r; j,...,c H : B A B B B c A c A c..... A r r r rc r c : i : j : c r i r ij ij c i ij ˆp ij ij, ˆp i i, ˆp j j

50 4 X r i c ( ij i j ) i j χ ((r c)(c )) rc c, r

5 5. (x,y ), (x,y ),...,(x,y ) x Y : Y j a + bx i + ε i. 5.: y ax + b x i Y ε i error ε i N(0,σ ) y ax + b a, b a, b Y j N(ax j + b, σ ). i.e., (Y,Y,...,Y L(a, b) πσ exp { (y } j a bx j ) σ l(a, b) log L(a, b) log πσ (y j a bx j ) σ.

5 5 j (y j a bx j ) S x S y S xy (x j x), (y j y), (x j x)(y j y). (y j a bx j ) (y j y bx j + bx + y bx a) (y j y bx j + bx) + (y j y bx j + bx)(y bx a) + (y bx a) {(y j y) b(y j y)(bx j bx)+(bx j bx) } + (y bx a) Sy bs xy + b Sx + (y bx a) ( Sx b b S ) xy + S Sx y + (y bx a) ( Sx b S ) xy S xy + S Sx Sx y + (y bx a). b S xy, a y bx Sx a, b (.) ˆb S xy S x, â Y ˆbx S xy (x j x)(y j Y ).

. 53 â, ˆb.. (â, ˆb) E[â] a, E[ˆb] b, (+ x V (â) σ V (ˆb) σ Sx Cov(â, ˆb) σ x Sx S x ), Y j a + bx j + ε j Y a + bx + ε j ˆb (x j x)(y j Y ) S x j (x j x)(a + bx j + ε j a bx ε) S x j b(x j x)(x j x) b + S x j (x j x)ε j S x + j (x j x)ε j S x â Y ˆbx ( j a + bx + ε b + (x ) j x)ε j x Sx a + { x(x } j x) ε S j j x a + { x(x } j x) ε Sx j. j j (x j x)ε S x â, ˆb ε j E[ˆb] b, V (ˆb) (xj x) σ S 4 x S xσ S 4 x σ S x E[â] a,

54 5 V (â) { x(x } j x) σ j S x { x(x j x) j S x { } + x Sx σ S 4 x ) (+ σ x, Sx Cov(â, ˆb) { (x Sx j x) x(x j x) S j x x(x j x) Sx Sx xsx Sx Sx σ x Sx j } + x (x j x) σ Sx 4 } (.) (.3) â a V (â) ˆb b V (ˆb) (â a) σ +x /Sx (ˆb b) σ/s x N(0, ) N(0, ) () σ σ σ (.4) e j Y j ˆbx j â a + bx j + ε j ˆbx j â a â +(b ˆb)x j + ε j σ ˆσ (.5).. (e,e,...,e ) (â, ˆb) (.5) ˆσ (â, ˆb) ˆσ /σ χ j e j

. 55 ε j 0 Cov(e j, â) Cov(ε j (â a) (ˆb b)x j ), â) Cov(ε j, â) Cov(â a, â) x j Cov(ˆb b, â) Cov(ε j, â) Cov(â, â) x j Cov(ˆb, â) ( σ x(x ) ) j x) (+ σ x σ x + x j σ 0. â, ˆb (.) a + bx j S x ( x(x j x) x Sx Sx S x + x jx S x ) S x Cov(e j, ˆb) Cov(ε j (â a) (ˆb b)x j ), ˆb) Cov(ε j, ˆb) Cov(â a, ˆb) x j Cov(ˆb b,ˆb) Cov(ε j, ˆb) Cov(â, ˆb) x j Cov(ˆb,ˆb) (x j x)σ S x + xσ S x σ (x Sx j x + x x j ) 0. x j σ S x â + x jˆb Y ˆbx + xjˆb Y + S xy S x (x j x). (.6) â a +(ˆb b)x j Y + S xy S x Y a bx + S xy S x (x j x) a bx j (x j x) b(x j x) Y a bx + S xy bsx (x Sx j x) Y S xy Y j a + bx j + ε j Y, Y,..., Y Y j Y j E[Y ] E[Y j ] (a + bx j )a + bx, j j

56 5 V (Y ) j V ( Y j ) V (Y j ) σ. j S xy S xy (x j x)(y j Y ) (x j x)y j E[S xy ] (x j x)e[y j ] (x j x)(a + bx j ) (x j x)bx j b (x j x)(x j x) bsx, V (S xy ) V ((x j x)y j ) (x j x) V (Y j ) (x j x) σ j σ Sx, Cov(Y,S xy ) (x j x)v (Y j ) (x j x)σ 0. j 0 j j j (Y a bx) σ N(0, ), (SxY bs x ) σs x N(0, ) (.7) (Y a bx) σ + (S xy bs x ) σ S x χ () σ ε j Y j (a + bx j ) (.4) e j a â +(b ˆb)x j + ε j ε j Y j (a + bx j ) }{{} (.6) â a +(ˆb b)x }{{} j + e j }{{} â a +(ˆb b)x j Y a bx + S xy bsx (x Sx j x)

. 57 (â a +(ˆb b)x j ) j j j { Y a bx + S } xy bsx (x j x) { S x (Y a bx) +(Y a bx) S xy bs x S x (Y a bx) + (S xy bs x) S 4 x S x (Y a bx) + (S xy bsx ). S x (x j x)+ (S } xy bsx) (x Sx 4 j x) σ (.7) e j (â a +(ˆb b)x j )(â a)ê +(ˆb b) e j x j (ˆb b) e j x j. e j x j (Y j ˆbx j â)x j (Y j Y )x j ˆb (Y j Y ˆbx j + ˆbX + Y ˆbx â)x j (x j x)x j (Y j Y )(x j x) ˆb 0. e j (â a +(ˆb ˆb)x j )0. ε j e j +(â a +(ˆb b )x j ) (x j X)(x j x) S xy S xy S x S x ε j j j e j + j (â a +(ˆb b)x j )

58 5 + ( )/σ χ () ( )/σ χ () (â, ˆb) e j ( )/σ χ ( ) ˆσ σ (.), (.3) σ ˆσ (.8) (.9) T a T b (â a) ˆσ t( ), +(x/sx ) (ˆb b) t( ) ˆσ/Sx a, b ˆσ /σ χ ( ) T a, T b t

59 [],,,,,, 99. [] R. V. Hogg ad A. T. Craig, Itroductio to Mathematical Statistics, Macmilla Compay, Lodo, 970. [3],,,,, 003.