Лекции по теории статистических выводов. Володин И.Н. - 11 стр.

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ϕ
c
i |x
(t
n
)
i
ξ
i
, X
i
(n + 1)
ϕ
s
ϕ
c
,
ρ
I = {I, a
s
}.
ρ
i |X
(t
n
)
= ϕ
c
i |X
(t
n
)
· ϕ
s
a
c
|X
(t
n
)
, i ∈ I;
ρ
a
s
|X
(t
n
)
= ϕ
s
a
s
|X
(t
n
)
.
ρ = (ϕ
s
, ϕ
c
) ρ
ϕ
s
ϕ
c
ν τ
ν
= (ι
1
, . . . , ι
ν
).
n ν
t
n
= (i
1
, . . . , i
n
) τ
ν
ν
i
, i ∈ I,
X
i∈I
ν
i
= ν,
F
i
, i ∈ I.
ν τ
ν
,
P
n
ν = n |X
(t
n
)
o
= ϕ
s
a
s
|X
(t
n
)
n−1
Y
k=1
ϕ
s
a
c
|X
(t
k
)
,
P
n
τ
ν
= t
n
|X
(t
n
)
o
= ϕ
s
a
s
|X
(t
n
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n−1
Y
k=1
ϕ
s
a
c
|X
(t
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ϕ
c
i
k+1
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,
t
n
∈ I
n
, n = 0, 1, . . . ,
øàãå ýêñïåðèìåíòà c âåðîÿòíîñòüþ ϕc i | x(tn )                              âûáèðàåòñÿ èíäåêñ i ñëó-
                                                                       

÷àéíîãî ýëåìåíòà ξi , íåçàâèñèìàÿ êîïèÿ Xi êîòîðîãî áóäåò íàáëþäàòüñÿ
íà (n + 1) -ì øàãå.
   ñëó÷àå ðàíäîìèçèðîâàííûõ ïðàâèë ϕs è ϕc , êîãäà ïåðåõîäíûå âåðî-
ÿòíîñòè îòëè÷íû îò íóëÿ è åäèíèöû, âûáîð ñîîòâåòñòâóþùèõ äåéñòâèé â
ýêñïåðèìåíòå îñóùåñòâëÿåòñÿ ïðèìåíåíèåì ïðîöåäóðû ðàíäîìèçàöèè.
  Åñòåñòâåííî, ýòè ïðàâèëà ìîæíî îáúåäèíèòü â îäíî ïðàâèëî óïðàâëåíèÿ
ρ ñòàòèñòè÷åñêèì ýêñïåðèìåíòîì, îïðåäåëèâ ïîñëåäîâàòåëüíîñòü ïåðåõîä-
íûõ âåðîÿòíîñòåé ñ óïîðÿäî÷åííûõ âûáîðî÷íûõ ïðîñòðàíñòâ íà ôàçîâóþ
àëãåáðó ñîáûòèé ïðîñòðàíñòâà I = {I, as }. Êàæäàÿ ïåðåõîäíàÿ âåðîÿò-
íîñòü ýòîãî ïðàâèëà îïðåäåëÿåòñÿ êàê
                                                                                
                      (tn )                     (tn )                        (tn )
              ρ i|X         = ϕc i | X         · ϕ s ac | X                              , i ∈ I;
                                                           
                                  (tn )                 (tn )
                          ρ as | X        = ϕ s as | X          .
  Ïàðà ïðàâèë ρ = (ϕs , ϕc ) èëè îáùåå ïðàâèëî ρ íàçûâàåòñÿ óïðàâëåíèåì
ñòàòèñòè÷åñêèì ýêñïåðèìåíòîì.
  Ñ ïðàâèëàìè ϕs è ϕc ñâÿçûâàþòñÿ äâå õàðàêòåðèñòèêè óïðàâëåíèÿ: ìî-
ìåíò îñòàíîâêè ν è óïðàâëÿþùàÿ ïåðåìåííàÿ τν = (ι1 , . . . , ιν ). Ðåàëè-
çàöèÿ n ìîìåíòà îñòàíîâêè ν îïðåäåëÿåò îáúåì âûáîðêè, à ðåàëèçàöèÿ
tn = (i1 , . . . , in ) óïðàâëÿþùåé ïåðåìåííîé τν  èíäåêñû ñëó÷àéíûõ ýëå-
ìåíòîâ, íàáëþäàåìûõ â ñòàòèñòè÷åñêîì ýêñïåðèìåíòå. Óïðàâëÿþùàÿ ïåðå-
ìåííàÿ îïðåäåëÿåò îáúåìû âûáîðîê
                                                   X
                                  νi , i ∈ I,                     νi = ν,
                                                            i∈I

èç ñîîòâåòñòâóþùèõ ðàñïðåäåëåíèé Fi , i ∈ I.
  Ðàñïðåäåëåíèÿ ν è τν , ñîãëàñîâàííûõ ñ ñîîòâåñòâóùèìè ïîñëåäîâàòåëü-
íîñòÿìè ñèãìà-àëãåáð, îïðåäåëÿþòñÿ óñëîâíûìè âåðîÿòíîñòÿìè
              n             o                   n−1
                                                  Y                
                      (tn )               (tn )
             P ν = n|X        = ϕ s as | X            ϕs ac | X(tk ) ,
                                                                      k=1

    n                 o                   n−1
                                            Y                                  
                (tn )               (tn )                 (tk )             (tk )
   P τν = tn | X        = ϕ s as | X            ϕ s ac | X       ϕc ik+1 | X        ,
                                                        k=1
                                           n
                                   tn ∈ I , n = 0, 1, . . . ,

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