Лекции по теории статистических выводов. Володин И.Н. - 87 стр.

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α
α(c) = P
0
( p
1
(X) > c p
0
(X) ).
P
0
P
0
,
X, p
0
(x) = 0.
1 −α(c), c ∈ R,
L(X) = p
1
(X)/p
0
(X),
α(c), c ∈ R,
P
0
( L(X) = c ) = α(c − 0) − α(c), α(−∞) = 1, α(+∞) = 0.
α, 0 < α < 1, c
0
α(c
0
) 6 α 6 α(c
0
− 0)
ϕ(x) =
1, p
1
(x) > c
0
p
0
(x) ,
γ, p
1
(x) = c
0
p
0
(x) ,
0, p
1
(x) < c
0
p
0
(x) ,
(7.3)
γ =
α − α(c
0
)
α(c
0
− 0) − α(c
0
)
.
γ α(c
0
−0)−
α(c
0
) = P
0
( p
1
(X) = c
0
p
0
(X) ) = 0, ϕ
E
0
ϕ(X) = P
0
( L(X) > c
0
) +
α − α(c
0
)
α(c
0
− 0) − α(c
0
)
P
0
( L(X) = c
0
) = α.
α
k = c
0
.
c
0
α(c) = α
c. (c
0
, c
00
) P
0
X ∈ C, C = {x : p
0
(x) > 0 c
0
< L(x) < c
00
},
P
0
(C) = α(c
0
) − α(c
00
− 0) = 0. p
0
(x) > 0, x ∈ C,
µ(C) = 0. P
1
Îí òàêæå óäîâëåòâîðÿåò (7.1), êðîìå ñëó÷àÿ, êîãäà ñóùåñòâóåò êðèòå-
ðèé ðàçìåðà ñòðîãî ìåíüøå α è ìîùíîñòè 1.
  Ä î ê à ç à ò å ë ü ñ ò â î. ( i ) Ðàññìîòðèì ôóíêöèþ

                         α(c) = P0 ( p1 (X) > c p 0 (X) ).
Ïðè âû÷èñëåíèè P0 âåðîÿòíîñòåé ìîæíî îãðàíè÷èòüñÿ òîëüêî ìíîæåñòâà-
ìè, ïðèíàäëåæàùèìè íîñèòåëþ ðàñïðåäåëåíèÿ P0 , óáèðàÿ èç íèõ òî÷êè
âûáîðî÷íîãî ïðîñòðàíñòâà X, ãäå ôóíêöèÿ ïëîòíîñòè p 0 (x) = 0.  ñèëó
ýòîãî çàìå÷àíèÿ 1 − α(c), c ∈ R, ÿâëÿåòñÿ ôóíêöèåé ðàñïðåäåëåíèÿ ñòàòè-
ñòèêè îòíîøåíèÿ ïðàâäîïîäîáèÿ L(X) = p1 (X)/p 0 (X), òàê ÷òî ôóíêöèÿ
α(c), c ∈ R, íå âîçðàñòàåò, íåïðåðûâíà ñïðàâà è
        P0 ( L(X) = c ) = α(c − 0) − α(c),        α(−∞) = 1, α(+∞) = 0.
  Äëÿ çàäàííîãî óðîâíÿ çíà÷èìîñòè α,      0 < α < 1, îïðåäåëèì c0 èç
ñîîòíîøåíèé α(c0 ) 6 α 6 α(c0 − 0) è ââåäåì êðèòåðèé
                        
                        
                          1, åñëè p 1 (x) > c0 p 0 (x) ,
                        
                 ϕ(x) =    γ, åñëè p 1 (x) = c0 p 0 (x) ,       (7.3)
                        
                         0, åñëè p (x) < c p (x) ,
                        
                                              1      0    0
ãäå
                                      α − α(c0 )
                            γ=                       .
                                  α(c0 − 0) − α(c0 )
  Ïîñòîÿííàÿ ðàíäîìèçàöèè γ îïðåäåëåíà âñþäó, êðîìå ñëó÷àÿ α(c0 −0)−
α(c0 ) = P0 ( p1 (X) = c0 p 0 (X) ) = 0, òàê ÷òî êðèòåðèé ϕ îïðåäåëåí ïî÷òè
âñþäó. Ðàçìåð ýòîãî êðèòåðèÿ
                                          α − α(c0 )
      E 0 ϕ(X) = P0 ( L(X) > c0 ) +                      P0 ( L(X) = c0 ) = α.
                                      α(c0 − 0) − α(c0 )
  Òàêèì îáðàçîì, êðèòåðèé âèäà (7.2) çàäàííîãî ðàçìåðà α (êðèòåðèé,
óäîâëåòâîðÿþùèé ðàâåíñòâó (7.1)) ñóùåñòâóåò, åñëè ïîëîæèòü k = c0 .
  Çàìåòèì, ÷òî çíà÷åíèå êðèòè÷åñêîé êîíñòàíòû c0 îïðåäåëÿåòñÿ, ïî ñó-
ùåñòâó, åäèíñòâåííûì îáðàçîì, êðîìå ñëó÷àÿ, êîãäà α(c) = α äëÿ öåëîãî
èíòåðâàëà çíà÷åíèé c. Åñëè (c 0 , c 00 ) òàêîé èíòåðâàë, òî P0 âåðîÿòíîñòü
ñîáûòèÿ X ∈ C, ãäå C = { x : p 0 (x) > 0 è c 0 < L(x) < c 00 }, ðàâíà íóëþ:
P0 (C) = α(c 0 ) − α(c 00 − 0) = 0. Òàê êàê p 0 (x) > 0, êîãäà x ∈ C, òî ïî-
ñëåäíåå ðàâåíñòâî âîçìîæíî ëèøü â ñëó÷àå µ(C) = 0. Íî âåðîÿòíîñòü P1

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