Лекции по теории статистических выводов. Володин И.Н. - 12 стр.

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θ ∈ Θ
P
θ
(ν = n) =
X
t
n
∈I
n
Z
X
t
n
ϕ
s
a
s
|x
(n)
"
n−1
Y
k=1
ϕ
s
a
c
|x
(n)
#
p
t
n
x
(n)
|θ
dµ
t
n
x
(n)
,
P
θ
(τ
ν
= t
n
) =
Z
X
t
n
ϕ
s
a
s
|x
(n)
"
n−1
Y
k=1
ϕ
s
a
c
|x
(k)
ϕ
c
i
k+1
|x
(k)
#
p
t
n
x
(n)
|θ
dµ
t
n
x
(n)
,
t
n
∈ I
n
, n = 0, 1, . . .
X = X
(τ
ν
)
= (X
ι
1
, . . . , X
ι
ν
)
(X, A).
X
p
ρ,t
n
x
(n)
|θ
= ϕ
s
a
s
|x
(n)
"
n−1
Y
k=0
ϕ
s
a
c
|x
(k)
ϕ
c
i
k+1
|x
(k)
#
p
t
n
x
(n)
|θ
,
x
(n)
∈ X
t
n
, t
n
∈ I
n
, n = 0, 1, . . . ; θ ∈ Θ,
µ
t
n
.
P
ρ
= {P
ρ,t
, t ∈ T} = {{P
ρ,t
( ·|θ), θ ∈ Θ}, t ∈ T}
θ ∈ Θ (X, A)
P
ρ
(P
ρ
, G)
G
λ
∈ G
X,
p
λ,t
n
x
(n)
=
Z
Θ
p
ρ,t
n
x
(n)
|θ
g
λ
( θ )dχ(θ), x
(n)
∈ X
t
n
, t
n
∈ I
n
,
êîòîðûå îïðåäåëÿþò èõ ðàñïðåäåëåíèÿ ïðè êàæäîì çíà÷åíèè ïàðàìåòðà
θ ∈ Θ ïîñðåäñòâîì èõ óñðåäíåíèÿ ïî ðàñïðåäåëåíèÿì íàáëþäàåìûõ ñëó-
÷àéíûõ ýëåìåíòîâ:
                                                       Pθ (ν = n) =
                                              "                         #
        X Z                                  n−1
                                                Y                                          
                    ϕs as | x(n)                       ϕ s ac | x (n)           (n)
                                                                          p tn x | θ dµtn x (n)
                                                                                                  ,
       tn ∈I n                                  k=1
             Xtn

                                                      Pθ (τν = tn ) =
                          "                                                 #
Z                        n−1
                            Y                                                                
      ϕs as | x(n)                        ϕ s ac | x (k)
                                                          ϕc ik+1 | x (k)           (n)
                                                                              p tn x | θ dµtn x (n)
                                                                                                      ,
                                 k=1
Xtn
                                                tn ∈ I n , n = 0, 1, . . .
    Òàêèì îáðàçîì, ñòàòèñòè÷åñêèé ýêñïåðèìåíò ñîñòîèò â íàáëþäåíèè ñëó-
÷àéíîãî âåêòîðà (ñëó÷àéíîé âûáîðêè) X = X(τν ) = (Xι1 , . . . , Xιν ) ñ èç-
ìåðèìûì ïðîñòðàíñòâîì çíà÷åíèé (âûáîðî÷íûì ïðîñòðàíñòâîì) (X, A).
Ðàñïðåäåëåíèå X îïðåäåëÿåòñÿ ôóíêöèÿìè ïëîòíîñòè
                                                     "                                                   #
                                                  n−1
                                                       Y                                                              
             (n)                              (n)                       (k)                    (k)                  (n)
 p ρ,tn x |θ = ϕs as |x                                      ϕs ac |x             ϕc ik+1 |x                 p tn x |θ ,
                                                       k=0

                                 x(n) ∈ Xtn , tn ∈ I n , n = 0, 1, . . . ; θ ∈ Θ,
ïî ìåðå µtn . Ñîîòâåòñòâóþùèé ýòèì ïëîòíîñòÿì êëàññ
                     Pρ      = {Pρ,t , t ∈ T} = {{Pρ,t ( · | θ), θ ∈ Θ}, t ∈ T}
ñåìåéñòâ (ïî ïàðàìåòðó θ ∈ Θ ) ðàñïðåäåëåíèé íà (X, A) îáû÷íî íàçûâàåò-
ñÿ ñòàòèñòè÷åñêèì ýêñïåðèìåíòîì, îäíàêî â äàííîì êîíòåêñòå ýòî ìî-
æåò ïðèâåñòè ê íåäîðàçóìåíèÿì ñ ïîíÿòèåì ñòàòèñòè÷åñêîãî ýêñïåðèìåí-
òà êàê ïîñëåäîâàòåëüíîñòè îïðåäåëåííûõ äåéñòâèé ïî íàáëþäåíèÿì ñëó-
÷àéíûõ ýëåìåíòîâ. Êëàññ Pρ áîëåå åñòåñòâåííî íàçâàòü ñòàòèñòè÷åñêîé
ìîäåëüþ. Òîãäà ïàðà ñåìåéñòâ (Pρ , G) áóäåò íàçûâàòüñÿ ñòàòèñòè÷åñêîé
ñòðóêòóðîé.
    Åñëè àïðèîðíûå ðàïðåäåëåíèÿ Gλ ∈ G íå âûðîæäåíû, òî â ðàìêàõ ñòà-
òèñòè÷åñêîé ñòðóêòóðû îñîáî âàæíóþ ðîëü èãðàåò ìàðãèíàëüíîå ðàñïðå-
äåëåíèå ñëó÷àéíîé âûáîðêè X, îïðåäåëÿåìîå ôóíêöèÿìè ïëîòíîñòè
                                Z                         
                   (n)                               (n)
      p λ,tn x               =            pρ,tn x          | θ gλ ( θ )dχ(θ),         x(n) ∈ Xtn ,            tn ∈ I n ,
                                   Θ
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