Лекции по теории статистических выводов. Володин И.Н. - 36 стр.

UptoLike

ВУЗ: 

КФУ | Казань

Составители: 

Рубрика: 

∞
X
n=0
P
θ
(ν
i
> n) = E
θ
ν
i
2
W (θ) = {w
i
(θ) = E
θ
ν
i
/E
θ
ν, i ∈ I,
X
i ∈I
w
i
(θ) = 1}
Υ. Φ
θ, λ ϕ δ
ϕ
= δ
ϕ
(X)
W
ϕ
(θ) = {w
ϕ,i
(θ), i ∈ I}
ν θ ∈ Θ
E
θ
ν >
ϕ∈Φ
ϑ∈Θ
I(θ, ϑ |δ
ϕ
)
P
i ∈I
w
ϕ,i
(θ) I(θ, ϑ |ξ
i
)
. (3.6)
Θ ⊆ R
m
E
θ
ν >
ϕ∈Φ
ϑ∈Θ
I(θ, ϑ |δ
ϕ
)
P
i ∈I
w
ϕ,i
(θ) I(θ, ϑ |ξ
i
)
>
ϕ∈Φ
λ
m
(θ; ϕ) >
ϕ∈Φ
lim
k∆θ k→0
J(θ, θ + ∆θ |δ
ϕ
)
P
i ∈I
w
ϕ,i
(θ) J(θ, θ + ∆θ |ξ
i
)
, (3.7)
λ
m
(θ; ϕ)
det
"
i (θ |δ
ϕ
) − λ
X
i ∈I
w
ϕ,i
(θ) i (θ |ξ
i
)
#
= 0. (3.8)
E
θ
ν
τ
ν
, δ
τ
ν
X
(τ
ν
)

P
ρ
.
    Íåòðóäíî ïîêàçàòü, ÷òî
                                 ∞
                                 X
                                         Pθ (νi > n) = Eθ νi
                                  n=0

è, òåì ñàìûì, óáåäèòüñÿ â ñïðàâåäëèâîñòè ôîðìóëû (3.5).                                     2

    3.3. Íèæíèå ãðàíèöû äëÿ ñðåäíåãî îáúåìà âûáîðêè.                                        Èç óñòàíîâ-
ëåííûõ â Ïðåäëîæåíèÿõ 3.1 - 3.3 ñâîéñòâ ìåð èíôîðìàöèè ñëåäóþò îñíîâ-
íûå ðåçóëüòàòû äàííîãî ïàðàãðàôà.
    Ââåäåì ìíîæåñòâî
                                                                       X
               W (θ) = {wi (θ) = Eθ νi /Eθ ν, i ∈ I,                          wi (θ) = 1}
                                                                       i ∈I

îòíîñèòåëüíûõ ñðåäíèõ îáúåìîâ íàáëþäåíèé â ñòàòèñòè÷åñêîì ýêñïåðèìåí-
òå êàæäîé èç ñëó÷àéíûõ âåëè÷èí êëàññà Υ. Ïóñòü Φ  êëàññ ãàðàíòèéíûõ
(ïî θ, d èëè λ ) ñòðàòåãèé ϕ ñ ðåøàþùèìè ôóíêöèÿìè δϕ = δϕ (X) è
ìíîæåñòâîì Wϕ (θ) = {wϕ,i (θ), i ∈ I} îòíîñèòåëüíûõ ñðåäíèõ îáúåìîâ íà-
áëþäåíèé.

    Òåîðåìà 3.1. Äëÿ ëþáîé ãàðàíòèéíîé ñòðàòåãèè ñ ìîìåíòîì îñòàíîâêè
ν   ïðè ëþáîì   θ∈Θ          ñïðàâåäëèâî íåðàâåíñòâî

                                                          I(θ, ϑ | δϕ )
                       Eθ ν > inf sup P                                         .                 (3.6)
                                  ϕ∈Φ ϑ∈Θ                wϕ,i (θ) I(θ, ϑ | ξi )
                                                  i ∈I

    Åñëè   Θ ⊆ Rm     è ñòàòèñòè÷åñêèé ýêñïåðèìåíò ðåãóëÿðåí, òî

                                          I(θ, ϑ | δϕ )
           Eθ ν > inf sup P                                     > inf λ m (θ; ϕ) >
                      ϕ∈Φ ϑ∈Θ            wϕ,i (θ) I(θ, ϑ | ξi )   ϕ∈Φ
                                  i ∈I

                                                 J(θ, θ + ∆θ | δϕ )
                       inf     lim       P                                  ,                     (3.7)
                      ϕ∈Φ k ∆θ k→0              wϕ,i (θ) J(θ, θ + ∆θ | ξi )
                                         i ∈I
ãäå   λ m (θ; ϕ)    íàèáîëüøèé êîðåíü óðàâíåíèÿ
                        "                                                       #
                                                 X
                     det i (θ | δϕ ) − λ                 wϕ,i (θ) i (θ | ξi )       = 0.          (3.8)
                                                 i ∈I
                                                                                                    
    Äëÿ äîñòèæåíèÿ íèæíèõ ãðàíèö                  Eθ ν     íåîáõîäèìî, ÷òîáû     τν , δτν X(τν )
áûëà äîñòàòî÷íîé ñòàòèñòèêîé äëÿ ñåìåéñòâà                          ðàñïðåäåëåíèé Pρ .


                                                    36

Страницы