Лекции по теории статистических выводов. Володин И.Н. - 35 стр.

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ν E
θ
ν < ∞.
I
θ, ϑ |X
(τ
ν
)
=
X
i ∈I
E
θ
ν
i
· I(θ, ϑ |ξ
i
). (3.5)
J (θ, ϑ |X)
ϑ, U (θ) ϑ = θ.
ρ
p ( X |θ ) = p
ρ,τ
ν
X
(τ
ν
)
|θ
=
ϕ
s
a
s
|X
(τ
ν
)
ν−1
Y
k=0
ϕ
s
a
c
|X
(τ
k
)
ϕ
c
τ
k+1
|X
(τ
k
)
·
ν
Y
k=0
f
ι
k
(X
ι
k
|θ).
I
θ, ϑ |X
(τ
ν
)
= E
θ
ln
p
ρ,τ
ν
X
(τ
ν
)
|θ
p
ρ,τ
ν
X
(τ
ν
)
|ϑ
= E
θ
ν
X
k=0
ln
f
ι
k
( X
ι
k
|θ , )
f
ι
k
( X
ι
k
|ϑ )
.
{χ
i |x
(τ
n−1
)
, i ∈ I}, n = 1, 2, . . .
χ
i |x
(t
n−1
)
= 1, (n−1)
x
(t
n−1
)
n
X
i
ξ
i
, χ
i |x
(t
n−1
)
=
0 χ
I
θ, ϑ |X
(τ
ν
)
=
∞
X
n=1
X
i ∈I
E
θ
χ
i |X
(t
n−1
)
ln
f
i
(X
i
|θ)
f
i
(X
i
|ϑ)
=
X
i ∈I
∞
X
n=1
E
θ
χ
i |X
(t
n−1
)
E
θ
ln
f
i
(X
i
|θ)
f
i
(X
i
|ϑ)
=
X
i∈I
∞
X
n=0
P
θ
(ν
i
> n) · I(θ, ϑ |ξ
i
)
ìîìåíò îñòàíîâêè                ν êîòîðîãî êîíå÷åí: Eθ ν < ∞. Òîãäà
                                                X
                                          (τν )
                                I θ, ϑ | X       =  Eθ νi · I(θ, ϑ | ξi ).                                               (3.5)
                                                                     i ∈I

 ñëó÷àå ðåãóëÿðíûõ ñòàòèñòè÷åñêèõ ýêñïåðèìåíòîâ àíàëîãè÷íîå óòâåð-

æäåíèå ñïðàâåäëèâî äëÿ ôèøåðîâñêîé èíôîðìàöèè                                                 J (θ, ϑ | X)         ïðè âñåõ

ϑ,   ïðèíàäëåæàùèõ íåêîòîðîé îêðåñòíîñòè                                            U (θ) òî÷êè ϑ = θ.
     Ä î ê à ç à ò å ë ü ñ ò â î. Êàê è â ïðåäëîæåíèè 3.2, äîñòàòî÷íî äîêàçàòü ñïðà-
âåäëèâîñòü ôîðìóëû (3.5) òîëüêî äëÿ ðàçëè÷àþùåé èíôîðìàöèè.
     Äëÿ ýêñïåðèìåíòà ñ óïðàâëåíèåì ρ
                                                                                          
                                                                                (τν )
                                      p ( X | θ ) = p ρ,τν                      X         |θ =

                                ν−1
                                Y                                                               Yν
                        (τν )                                (τk )                         (τk )
         ϕ s as | X                       ϕ s ac | X                     ϕc τk+1 | X                ·   fιk (Xιk | θ).
                                    k=0                                                               k=0
Ñëåäîâàòåëüíî, ðàçëè÷àþùàÿ èíôîðìàöèÿ
                                                                     ν
                                          p ρ,τν X(τν ) | θ        X      fιk ( Xιk | θ , )
         I θ, ϑ | X(τν )            = Eθ ln                    = Eθ     ln                   .
                                            p ρ,τν X(τν ) | ϑ               fιk ( Xιk | ϑ )
                                                                     k=0

     Îïðåäåëèì ïîñëåäîâàòåëüíîñòü ñåìåéñòâ èíäèêàòîðíûõ ôóíêöèé
                                                            
                                                   (τn−1 )
                       {χ i | x     , i ∈ I}, n = 1, 2, . . .

ïîëàãàÿ χ i | x(tn−1 ) = 1, åñëè ýêñïåðèìåíò íå áûë îñòàíîâëåí äî (n−1) -ãî
                      

øàãà âêëþ÷èòåëüíî ñ ðåçóëüòàòîì x(tn−1 ) è áûëî ïðèíÿòî ðåøåíèå íà n -ì
øàãå íàáëþäàòü êîïèþ Xi ñëó÷àéíîãî ýëåìåíòà ξi , è ïîëàãàÿ χ i | x(tn−1 ) =
                                                                          

0 â ïðîòèâíîì ñëó÷àå.  òåðìèíàõ èíäèêàòîðíûõ ôóíêöèé χ ðàçëè÷àþùàÿ
èíôîðìàöèÿ ïðèíèìàåò âèä
                             X ∞ X                   f (X | θ)
                       (τν )                   (tn−1 )       i   i
          I θ, ϑ | X          =      Eθ χ i | X         ln              =
                                                           fi (Xi  | ϑ)
                                n=1                     i ∈I
      ∞                                                             ∞
     XX             
                                 (tn−1 )
                                                    fi (Xi | θ) X X
                Eθ χ i | X                     Eθ ln             =     Pθ (νi > n) · I(θ, ϑ | ξi )
                                                     fi (Xi | ϑ)
     i ∈I n=1                                                      n=0              i∈I
(â ñèëó êîíå÷íîñòè ìîìåíòà îñòàíîâêè ñóììû â ïðàâîé ÷àñòè ýòîãî ðàâåí-
ñòâà ìîæíî ïåðåñòàâëÿòü).


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