ん︑SJ - 滋賀大学学術情報リポジトリ

a e

︿円九

gdt ︐ψ T

f t l o

W 時一一

and

W(F)

:=わいい)} . 1

^川 ^u^d^F⁽⁸⁾

to investigate the asymptotic behavior of the second component of the random vector An. For五xedF(t) and

ψ

(t)

，

we define

γ1(g)(8)

: =

F‑1 0

グ

(8) and

官2

メ ( ω

:戸=

1

^1ρ1)^曽^町

^υ ⁽

^ト山川^叩¹^ト^μ^川^叫^一^イ^叫^t^り朴)(

for 8 ε[0

，

1] and g εD[O

，

l ，] ^whereg*(8) = inf{t

，

1 : g(t)三8}. Since the transformationsγ1(・)^andi2(・)^a^r^e Hadamard differentiable at I(t) from Proposition 6.1.1 of Fen恥 lz(1983)

，

^t^h^e^f^uⁿ^c^tⁱ^oⁿ^a^lr(・)ⁱⁿ^d^u^c^e^d^on^D[O

，

1] by r(g) := W(g ⁰F) for g E D[O

，

1] is Hadamard differentiable at I(t) by the chain rule and the expression that r(g)

=

ⁱ²^{ⁱ¹⁽^g⁾^}^.^Note t^h^a^t^t^h^e^d^e^rⁱ^v^a^tⁱ^v^e

τ

^f⁽^g⁾^o^f

r(g) at I(t) is given by

的

)=‑1=

州仰

4..I!. THE IFRA ALTERNATIVE

We obtain from the conditions (4.2.7) and (4.2.9) η1/21{VV叫 ‑ W(P)} ‑{W(

行)‑

W{T(pT)}1

=η1/2

L

^{O O}^判^S⁽^s⁾^}

' l

^S⁽^u⁾^d^u^d^{叩 )}

ニ川{S(T)}

1

^T^S⁽^s⁾^d^s

⁺

^π^1/2

L

^{O O}^曹^{^S⁽^s^)}的⁾^d^s

三

d

内/片/2

弘い Mμ

J^仰川L^F^バ^珂^曹^叫^{S珂貯S(

σ ^η

^例

^T ^{小 )}

=0匂Ip(

い

η0) a邸sη→∞.

Therefore the Hadamard differentiability of

r (

・)implies that asη→ ∞

where

η1/2{VV叫 ‑ W(P)} = rf(仇)

+

op(n^O)

=ん(T間 )

‑ l

^T^h²⁽^t⁾^{丸仰}

仇(t)_:=η1/2{F~0 P‑1(t) ‑p^T0 p‑1(t)} for 0 ::; t ::; 1. Hence Remark 2.2 of Gill (1983) yields that

ρT

η1/2{VV叫 ‑ W(P)} = ‑

I

^h2(t)Hn(t)dM(t)

+

op(π。) asη→∞・

By applying Lemma 3.1.1 of Section 3.1 to the first component of the random vector An' it is seen that

山

{ l

^{え榊}

Therefore the random vector A耳 isasymptotically equivalent to Bn from Theo‑

民m 4.4 of Billingsley (1968).

Since each component of the random vector Bn is represented as the stochastic integral with respect to the square integrable martingale M(t)

，

and the functions hi(t)'s and the process H.叫(t)are predictable

，

Theorem 2.1 of An‑

dersen et al. (1982) together with the condition (4.2.8) implies that Bn converges in distribution as n→ ∞ to a normal distribution with zero mean vector and

30 CHAPTER ‑，. TESTS FOR IFR AND IFRA

dispersion matrix {σi，;h9，;9' where

的

j : = f

^ん⁽^t⁾^h^;(^t⁾^d

Hence we can conclude the proof from Corollary 3.3 of Ser:si時 (1980)and some calculations. •

COROLLARY 4.2.4. Suppose that the df 's F(t)

，

G(t) and the function曹2(t) satisfy the conditions of Theorem 4.2.3. Then we have asπ→ ∞

九叫ん(れ)‑と主主;→d

N ( O ，生 ) ，

l μ F J

where ^{内 : J}and σ~ :J denote the correspondings to those given in the equations (4.2.10) and (4.2.11) ofTheorem 4

ユ

3for the function曹2(t)

，

respectively. COROLLARY 4.2.5. Suppose that under the null hypothesis冗othe censoring df G(t) and the function雪2(t)satisfy the conditions (4.2.6)

イ

4.2.9)of Theo‑

rem 4.2.3. Then n¹/2 L2(ψ2) converges in distribution出冗 → ∞ toa normal distribution with mean zero and variance

σ

ι わ

^2{S

^附

²⁽^仰)，

where

引ト ' l

^w²⁽^s⁾^d^s

This corollary shows that under the null hypothesis the asymptotic vari‑ ance σ!^:^J of the statistic L2(れ)defined in the equation (4.2.2) depends on the unknowns μand G( t). Similarly as in the case of the statistic L₁(ゆ1

，

s)we can find a consistent estimator

手 : イ〆 ω ^{崎山伐}

^t⁾

This helps us to construct the asymptotically exact test based on L₂₍ψ2)'

Now we compare the efficacy of the test statistics L₁₍仇

，

αど1

，

>

，

and L₂₍ψ2) for the alternatives (i)‑(iv) listed in Section 4.1 under the propor‑ tional censoring model with the censoring parameterλ. In this situation from

4.~. THE IFRA ALTERNATIVE 31

Corollaries 4.2.2 and 4.2.5 we may take μ = 1 and 0 <λ< 1. And the asymp‑

totic variances of the s凶 ablynormalized versions ofthe statistics Ll (‑u a

，

β) and L₂₍ψ2) under 行oare given by

イ:= s ( : r ^[ ² ^a ^s ^: ¹ ^‑

^λ+2u‑

^ふ ⁺

¹⁾

^{り山}

and

σ ; イ〆

⁽^t⁾^{九，}

respectively. Then we have the following efficacies of the statistics Ll

( u ぺ

β) against the alter口rna

eff{L

t {

♂，β)} = (αβlnβ

) 2 / ( v 4

σi)， for (ii)

eff{Ll(包a

，

β)}

=

{αβ(β̲ 1)}2/(v⁴σ

i ) ，

for (i垣)

α+1 2 α . eff{L

t {

♂，β)} =

{一一一一一一一一}/イ

v v+1 v +β and for (iv)

( s ‑

l)lnν‑αslns eff{L

t {

包a

，

β)}= 一

2(V ‑s)2

σ ?

For the test statistic L₂₍ψ2) we consider the weight function to be of form ψ2(包)=包P

，

ρ >‑1. Corollary 4.2.5 implies that

σ ( 1 B(1‑λ

，

2ρ+3l+

2，ρ ・ーー .(ρ+2)2¥(ρ

+

1)2(2p + 3)2(1 ‑λ) ^I +1ρ( )2 + 4B(l‑λ，4p + 7) 2B(1‑，A'p + 2)

(2p + 3)2 一 (p + 1)2(2ρ+ 3)

+ ω

倒(υ1一

λ λ ，

勾い+刊叫4的) 姐倒(υ1ト一

λ

川3ρ+什叫5吟)

一 .

(ωρ+1

り

)(σ2ρ+3幻)2 (ρ+ 1)(2ρ+ 3) }'

where B(

， . ・ )

denotes the Beta function. Here we consider the two test statistics

32 CHAPTER 4. TESTS FOR IFR AND IFRA

Table 4.2

Efficacie$ 01 the IFRA‑te$t $tαtutic$

叫 enthe cen$oring parameterλ =

ふ

αndi A1ternative

Statutic (i) (ii) (iii) (i

り

λ= 1/10

L₁₍包1，1.03) 1.2677 0.3168 0.0626 0.4775 L₁₍包 ¥1.95) 1.2841 0.2985 0.0606 0.4967 L₁₍u1.5， 3.42) 1.0506 0.1083 0.0329 0.4981 L2(^包0) 1.1945 0.2404 0.0554 0.4 729 L2(^包¹⁾ ¹^.⁰⁸¹⁰ ⁰^.³⁵⁹⁴ ⁰^.⁰⁶⁴³ 0.5756

入 =3/4

L₁₍包1，1.03) 0.7012 0.1752 0.0346 0.2641 L₁₍包1，1.95) 0.7421 0.1725 0.0348 0.2870 L₁₍包1.5，3.42) 0.8561 0.1600 0.0268 0.4058 L2(^包0) ₀_.₇₅₄₅ ₀_.₁₅₁₈ ₀_.₀₃₄₉ ₀_.₂₉₈₇ L2(^包¹⁾ ⁰^.⁵¹⁸⁴ ⁰^.¹⁷²⁸ ⁰^.⁰³⁰⁸ ⁰^.²⁷⁶⁰

L2(

旬。 )

and L2(U¹). Kumazawa (1988b) discussed the statistic A~ equivalent to L2(u^O). Then the asymptotic variances of the s凶 ablynormalized versions of L₂₍1

川

andL₂₍u¹) under冗。 aregiven by

2 1 2 4 1 1

=一一ー一一一一一一一一一一+一一一一一ー一一一一一+

2，0 ‑ 9(7ー λ) 3(6‑λ) ， 3(5一入) 4‑λr 4(3一入) and

σ~1=~( _‑ l+B(1-λ， 52++~B(l 一入， 11)_一 _一一 _一

2，1 ‑

9 ¥

100(1ー λ)¹ 4 ^{1 1} 25 B(1‑λ，3) ^I 2B(lー λ，6) 2B(1一入，8)¥

一一一・

respectively.

4.J!. THE IFRA ALTERNATIVE 33

And the efficacies of

L

₂₍11.0

) and

L

₂₍11.1

) are given: for the alternative (i)

eff{L2(J)}=(‑;

叫 2

川 / σ ; ，。

唱 77̲ ̲ 33 ̲ ̲ 2 ‑ eff{L2(包l)}

=

(一一90 ‑l‑‑n2

+ー

^，90 ‑‑ln3

+ー

^，15 ln5)2/σ;，u for (ii)

ef f{ L2( 11.O)} =

( 土

)2/σ

ふ

唱 19

‑

eff{L2(ピ)}= _{(一一 r~}

σん /

1350 for (iu)

e

ザ引州

η f 削 f パ

{υωL

‑

ef f{L2(ピ)}

=

(一一)2/σ

ん

2520 and for (iv)

角4穐ZHσ ︐

︐ ︐ ︐ ︐ ︐ ︐

句z a

︑ . . _︐

︐ v h

n v

' ・

・

一

J句 + / / 向

︒ 由j n q o z i n d i

一ほ

' E

‑ A 4・・﹄

‑ F d︐ .

+ +

9B 9 a n n E A EI

唱i to

h u ‑ ‑

凸RU

一向 ︐

唱i

‑a ・{ . 向 4‑ AH 4

一一

t︐ ︐

︑

︐ tt︑

一一一一

‑ J 1 J

︑••

︐︐︑

︐ ••

︐

n u

噌且

制叫包

︐ ︐

・

z︑

︐︐

・

1︑

今4笥ZM

L L

r4tr4t

t J I J Z J Z J

e e

respectively.

Table 4.2 shows the efficacies of the IFRA‑test statistics L₁₍11.¥ 1.03)

，

L1(11.¥ 1.95)

，

L(^包u

，

必)

and L2(¹¹^.^O⁾^f^o^r^t^h^e^a^l^t^e^rⁿ^a^tⁱ^v^e^s^lⁱ^s^t^e^dⁱⁿ^S^e^c^tⁱ^oⁿ⁴^.¹ and some values of the censorI時 parameterλ. For the statistic L₁(包a

，

s)we choose the values of αand

s

so as to maxImize its efficacy against a particular alternative. Since 6n :=

1 : 7 = 1 6 d

πis an estimate of P(X1三

Ud= 市，

we rec・

ommend the L1(11.1

，

1.95)‑test statistic for small values of6n and the L1(11.1

，

1.03)‑ test statistic for large values of 6n in the sense of the Pitman asymptotic relative efficiency.

v o r

e

P a h c

Tests f o r NBU and NBUE

5.1. The NBU Alternative

In this section we are interested in testing the null hypothesis

π 。 :

F(t) = 1 ‑exp( ‑tjμ) for t ~ 0 (μunspecl五ed) versus the alternative

冗3 : F(t) is NBU

，

but not exponential

，

under the random censorship model.

Koul (1978a) considered the parameter

1

⁰⁰

1 ^{∞ 附}

⁾^}^'^I^t^{^S⁽^t⁾^}

一州 + り } 川 μ

= ψ ( ^い ^仲

⁸

^吋)州一 1

⁰⁰

1

⁰⁰^'^I^t^{^S⁽^s

⁺ ^の

^}^d^F^(s⁾^d^F⁽^t⁾

as a measure of the deviation of F( t) Itom exponentiality towards the NBU alternatives and developed the class of the test statistics in the uncensored case. Here the weight function世(・)is assumed to be nondecreasing. This parameter with

ψ

(t) = t was五rstinvestigated in Hollander and Proschan (1972).

For the testing problem based on the censored observations (Xj

， 8 d ，

1三 t三n

，

^Kumazawa (¹⁹⁸⁷^a⁾^p^r^o^p^o^s^e^d^t^h^e^c^l^a^s^s^o^f^t^h^e^s^t^a^tⁱ^s^tⁱ^c^s

山):= 1

1

^T^ψ{

^{丸山山}

⁽⁵^.¹^.¹⁾

5.1. THE NBU ALTERNATIVE 35

which corresponds to the the Koul's (1978a) NBU statistic in the uncensored case. The statistic M₁(ψ) withψ(t) = t was considered in Chen

，

Hollander and Langberg (1983a) using a modi五edKaplan‑Meier estimator of F(t).

THEOREM 5.1.1 (Kumazawa (1986c

，

1987a)). Suppose that the weight function ψ(t) is continuous and piecewise di長rentiablewith bounded derivatives. And suppose that the df F(t) is absolutely continuous and that the df's F(t) and G( t) satisfy the conditions

J ^九

²⁽^t⁾^d^C⁽ ⁽⁵^.¹^.²⁾

and

η1/2ψ{S(T)}→

o

in probability asπ→∞・ (5.1.3) Then the sequence of the問包πn1/

刊

2^{M1(世利)一W(F

円

)}c∞on^附 r堵ge白S 1血ndistribution as

η→ ∞ to a normal rv B with zero mean and variance E[B2 ，]where W 例 :戸=

l

^co^∞

l ψ

^co^∞^叫^仰

^{ ^σ

^似仰^刷

^珂

^的

^仰

^υ ⁽

^8+t

^り ⁾ ^阿

^り ^M ^柑刷川 ^μ ⁾ ^川阿附

^d^削仰^σ^F

^町 ^叩

⁽^t

B:=

‑ l

^co

l

^co^{仰 + 仰}^+t

^州

⁸⁺^什^t)}dF

^附 ⁰

+21

∞co

ρ

t^{却 ‑}^Z ⁸⁾^{卯一り}^dF

^{山川}

^t⁾^}^d^F⁽^t⁾

and Z(t) is the limiting process of Zn(t) given in Lemma 3.1.2 of Section 3.1. PROOF: To apply Theorem 3.2.2 of Section 3.2

，

we五rstnote that the induced functional r(g) := W(g 0 F) for 9 E D[O

，

1] can be expressed as a composition of Hadamard differentiable transformations. For五xedF(t) and ψ(t)

，

we define

and

γ1(9

t }

(8) ^:=^F‑1^o⁹^t⁽⁸⁾

，

γ2(92)(8

，

t):= 92(8)

+

92(t)

，

i'a(9a

，

91)(8

，

t) :=ゆ[1‑91⁰F{9a(8

，

t)}]

判(93):=

1 かい，仰

，

where 91ε D[O

，

1 ，] 92 E L¹[0

，

1 ，] 93ε L¹[0

，

1] x [0

，

1 ，] 0三8

，

t < 1 and

36 CHAPTER 5. TESTS FOR NBU AND NBUE

gt(s) = inf{t

，

1 : g(t)三s}. Then from Propositions 6.1.1

，

6.1.2 and 6.1.6 of Fernholz (1983) the above transformationsγ1(・)‑γ4 ( .) are all Hadamard differentiable at I(t). Therefore r(g) 判 Oγ'3{γ2 0 γ1(g)

，

g} is Hadamard differentiable at I(t) by the chain rule of Proposition 3.1.2 of Fernholz (1983).

Next we have

η1/21{M1(ゆ)‑W(F)} ‑{W(F~) ^‑W(FT)}I 三山

J 川

211T

ι よ ^ル ¹ ^t ^〆 ^戸 ^ψ

^叫

^川

^仰

^{ ^σ ^例

^削^S

^珂(川

^仰⁸

+η1/2IL~ 1~ 附 +

^t⁾^}^d^F⁽^s^)dF(^中

+ポ/²11

L~t 仰仰F(s)dF例|

三3η1/2ゆ{S(T)}

Hence the desired result follows from Theorem 3.2.2 of Section 3.2 and some calculations. •

We consider the weight function ψ(包)=♂ asa special ca日

COROLLARY 5.1.2 (Kumazawa (1987a)). Let世(包)= 包ぺ α ど1. Suppose that under the null hypothesis 1io the censori時 dfG(t) satisnes the conditions (5.1.2) and (5.1.3) of Theorem 5.1.1. Then n1/2{Md♂) ‑(α+ 1)‑2} converges in distribution as n

→∞

to a normal distribution with mean zero and variance

1~

fa{S(t)}dC(

where

ん(t):=α2(α+ 1)‑4{(α+ 1) ln t + 1 }t2a+².

From Lemma 2.4 ofKumazawa (1987a)

，

the asymptotic variance of M d u^a)

，

α三1

，

^under t^h^eⁿ^u^l^l^h^y^p^o^t^h^e^sⁱ^s^may be c^oⁿ^sⁱ^s^t^eⁿ^t^l^y^e^s^tⁱ^m^a^t^e^d^by

9 3 イム向一 ) 凶

Using this estimator

，

we can construct the asymptotically exact test based on

5.1. THE NBU ALTERNATIVE 37

Table 5.1

Efficacie! 01 the NB U・te!t!tatutic! when the^的 l!oringparameterλ =

占

^and2

Alternative

Statutic

i り。

ⁱ^} ^{ⁱⁱⁱ^} ^{ⁱ^v^}

λ = 1/10

M₁₍包1) 1.2676 0.3169 0.0625 0.4774 M₁₍包1.5) 1.1560 0.1849 0.0481 0.4862 M₁₍包2) 1.0366 0.1151 0.0364 0.4737

M₁₍^包1) ₀_.₇₀₀₉ ₀_.₁₇₅₂ ₀_.₀₃₄₆ ₀_.₂₆₄₀ M₁₍包1.S) ₀_.₈₀₆₂ ₀_.₁₂₈₉ ₀_.₀₃₃₅ ₀_.₃₃₉₁

M₁₍包2) 0.8075 0.0897 0.0283 0.3690

the statistic M₁(

♂ ) ，

α三1.

Now we compute the efficacies of the test statistics M₁(u^a)

，

α三1

，

against the alternatives (i)‑(iv) listed in Section 4.1 under the proportional censorIn model. From Corollary 5.1.2 we may take μ 1 and 0 < 入 <1

，

and the asymptotic varIance under 冗ois found to be equal to

σ2. (2α‑λ+ 1)3(2α2+2α+ 1)

a ・一 (2α + 1)3{(α + 1

) 2

+ (α ‑λ)}' Some calculations yield that for the alternative (i)

eff{M₁(包a)}= α 2 (α+ 1)6σ2' for (ii)

for (iu)

and for (iv)

CHAPTER 5. TESTS FOR NBU AND NBUE

eff{M₁(包Q)}= α 2 (α+ 1)8

ぺ '

eff{M₁₍包Q)}= α

{ln(α+ 1)‑αp eff{M₁₍包Q)}=

α2(α+ 1)4σ2

Table 5.1 shows the efficacies of the test statistics M

t {

u1)

，

t { 包 1 .

5)and M

t { ポ )

for the alternatives (i)・(iv)and some values of the censoring parameterλWe recommend the M₁(u¹)‑test statistic for the testing problem in the sense of Pitman asymptotic relative efficiency.

Remark. Joe and Proschan (1983

，

1984) obtained some results on the de白cr何附e印ωaSln 100α‑pe何 entile(0 < α < 1

り )

residual and the new better than used with respect to the 100α‑pe^陀 entileaging properties and developed the statistic for testing exponentiality against these life distributions in the uncensored case. And Hol‑ lander

，

Park and Proschan (1985

，

1986) introduced the new better than used αt time to aging property and considered the problem of testing exponentiality ver‑ sus this aging property in the uncensored and the censored case. Under the random censorship model Kumazawa (1988a) proposed the classes of the test statistics generalizing their statistics to accommodate the censored data

，

and derived the asymptotic distributions of the statistics under some milder condi‑ tions.

5..2. THE NBUE ALTERNATIVE 39

5.2. The NBUE Alternative

We develop a test of the null hypothesis

π 。 :

F(t) = 1 ‑exp( ‑tjμ) for tと

o

(μunspecl五ed) against the alternative

冗4 : F(t) is NBUE

，

but not exponential

，

on the basis of the possibly right censored data (Xi，ん)， 1 ::; i三n，de:fined in Section 2.2.

Kumazawa (1986a) introduced the class of the test statistics

based on the measure

f T

^曹¹⁽^t⁾

え

(t)dt

N1(

ψd:=

^{】市}

戸

f o

.L ψ1( ^S)Sn( s )ds

f

^{仇 ( 伽}

(5.2.1)

of exponentiality against the NBUE life d

f '

s using weight function 1h(t ，)where Wl

(ト ' l

^'^l^/^J^l⁽^S⁾^d^s

The measure with ψ1 (t)三 constαntwas considered in De Souza Borges

，

Prosd即 1and Rodrigues (1984) for the above testing problem in the uncen‑ sored case. Note that we reject the null hypothesis冗。 infavor of 冗4for small values of the statistic N₁₍

ψ d .

The parameter

1

⁰⁰^'^l^/^J²^{^S⁽

^州

^F^F⁽^t⁾^一

1

^t^S⁽^s⁾^d^s^}dF

as a measure ofthe deviation of F(t) towards the NBUE alternatives with weight functionれ(t)was considered in Kumazawa (1986e)

，

^and t^h^e^c^l^a^s^s^o^f^t^h^e^t^e^s^t

statistics

~fh{丸 (t)} 必丸 (s)dsdFn(t)

N2(れ): ー〈 μ叫

(5.2.2)

40 CHAPTER 5. TESTS FOR NBU AND NBUE

was discussed. Hollander and Proschan (1975) used this measure withれ(t)三

constant and proposed the resulting statistic in the uncensored case. In the censored case Koul and Susarla (1980) generalized the statistic given in Hollander and Proschan (1975) based on a modi五edKaplan‑Meier estimator

，

and gave the asymptotics of their statistic under some strong regularity conditions.

Noting that a life df F(t) is NBUE if and only ifthe scaled TTT‑transforms Hi¹(t) defined by the equation (4.1.1) of Section 4.1 satisfy Hi¹(t)三tfor all t E [0

，

1] from Theorem 2.4 of Kle色jる(1982a)

，

we may consider

JU ︑_E

吋r I

rJ t

唱目ゐ

rjh

ψ ︐ 一一

as a meaSl悶 ofNBUE‑ness with weight function ψ( t). Then change of variable formula in multiple integral shows that

f ∞ ^ψ

^{^S⁽^t⁾^}J~ ^S⁽^s⁾^d^s^d^F⁽^t) (1

a2(ψ) ^{J U} T~-'-I"'JU-'-I---'I

I

tψ(l‑t)dt

，

μF Jo

which also yields the N2(れ)‑statistic.The measureム2(ゆ)with世(t)三 constant was used by Kumazawa (1988b) and investigated the resulting statistic _A~un‑ der the random censorship model. Based on the same property

，

Klefsjる(1983) proposed the statistic Aa

，

^which i^s^known t^o^be t^h^e^c^u^m^u^l^a^tⁱ^v^e^T^T^T^‑^s^t^a^tⁱ^s^tⁱ^c

discussed in Ba山 wet al. (1972)

，

^Cl^時 ^t^e^r6

，

^on t^e^s^tⁱⁿ^g^a^g^aⁱⁿ^s^t^t^h^e^IFR a^l^t^e^rⁿ^a^‑

tives and which is equivalent to the Hollander and ProschaI内 (1975)statistic. Note that we reject 冗。 infavor of 冗4^f^o^r^l^a^r^g^e^v^a^l^u^e^s^o^f^N2(ψ2)'

On the basis of the fact that the NBUE property is expressed by means of the mean residualli長 eF(t)defined in the equation (2.1.2) of Section 2.1

，

the test statistic

ぬ戸品丸刈

⁽⁵^.²^.³⁾

was introdl悶 din Kumazawa (1989b)

，

^where

T < 一

< 一nu

︒

伊1

鋸

u w

‑

一S

一ハり

︿円

九一

︿以

T一

﹁﹄刷一

︿向一一

Kumazawa (1987b) discussed the asymptotic behavior ofthe suitably normalized version of e_ton the五xedinterval [0

，叫，

0<包 <TH

，

under the ra吋 omcensor‑

5.1l. THE NBUE ALTERNATIVE 41 ship. The statistic Na may be considered as a natural extension of the statistic given by Barlow and Doksum (1972) and Koul (1978b) in the uncensored case

，

and we reject冗oin favor of冗4for large values of N_a_・

In order to derive the asymptotic distribution of the statistic N₁₍仇)， we first assume that

ψ

l(t) is not constant on the unit interval

[ 0 ，

1].

THEOREM 5.2.1 (Kumazawa (1986a)). Suppose that the weight function

ψ

t{t) is nonnegative and right continuous. And suppose that the df's F(t) and G(t) satis今 theconditions

l

^{T H}^h^;⁽^t⁾^d^C⁽^t⁾^<

^∞

⁽⁵^.²^.⁴⁾

and

ポ/2hi(T)→

o

in probabi五tyasη→ ∞ (5.2.5) for i = 1

，

2 and 3

，

where T is the largest observation of the X_i

，包

a o

Ju e e

e o r

︑

JM︐a

︑

_{. E}

︑ ・ l f s

s i

‑‑

︐ ︐

E︑ 唱

C υ A V

∞

f I J t f I J t 一一一一

a ' b a v b

唱A崎ZM

L H L M

and

ha(t):=

1

⁰⁰^'^l^'^t^{^s⁾^S⁽^s⁾^d^s

Then we have as n→ ∞

d

刊ぺ/川

q ^巾

²^{オ(いい川州(仲叶ゆ}

〆ん 2 . ̲

^{戸 =}

( p 去志 f

^含^伝^;^主

J γ{ h~誓砦;ヂi ⁺ ^E 守 f 子 2 ^一 ^l ^し ^E 守 f ^評 ^剖リ ^l ^レ r ^μ ^戸 ^二 ^〕 ²

^釘 ^弘 ^削 ^C ^印例 ^仰 ⁽ ^州

^例⁽^t^t

^の

⁾

PROOF: By applying the same method as given in the proof of Theorem 4.2.3

42 CHAPTER 5. TESTS FOR NBU AND NBUE

of Section 4.2

，

it is seen that the random vector A ^叫 :=戸

μ/

n¹

片刈

J

^夙附

^州

⁽^t^t

is asymptotically equivalent to Bn

ベ f

^{九州}

Since the random vector Bn converges in distribution as π→ ∞ to a normal distribution with zero mean vector and dispersion matrix {σi，j _{h\~i ，j~3}with

σi，j

イ

^E^hⁱ⁽^t⁾^h^j⁽^t^{附，}

the desired result follows from Corollary 3.3 of Serfl.i時 (1980).•

Next we consider the test statistic given by

j f t g n (

似

μ 3

in the case ofψ1 (t)三 constαnt.This statistic is also considered as a test statistic for testing against the HNBUE alternatives in Section 6.2 and treated in a more general framework: we have the following result from Theorem 6.2.5 of Section 6.2.

COROLLARY 5.2.2 (Kumazawa (1986a)). Suppose that the df 's F(t) and G(t) satisfy the conditions

l

^TH

^{好例}

and

η1/2h_i(T)→

o

ⁱⁿ^p^r^o^b^a^bⁱ^lⁱ^t^yasπ→∞，

for i = 1 and 2

，

where

5.~. THE NBUE ALTERNATIVE

ん(t):=

1

⁰⁰^S⁽^s⁾^d^s

and

ん(t):=

1

⁰⁰^s^S⁽^s⁾^d^s

Then we have as n

→∞

ぺ

^rsE(s)ds

^一与)→

^‑^‑^‑⁺^d^N

川

⁽⁰

，め

L μ λ μ云J where μ2 = h2(0) and

σ2 .̲

. 1

^J^E^{²^μ²^h¹⁽^t⁾

^一

^μ^Fh²⁽^t⁾^P^d^C⁽^t⁾

一

μ手

The asymptotic behavior of the statistic N₁(ψ1) under the null hypothesis can be summarized as follows.

COROLLARY 5.2.3 (Kumazawa (1986a)). Suppose that under the null hypothe‑ sis 1πio the censoring df G(t) and the weight function仇(t)satisfy the conditions (5.2.4) and (5.2.5) of Theorem 5

ユ

¹^.Then n¹/2{N1(ψ1) ‑1} COI附 rgesin dis‑ tribution as n

→∞

to a norma1 distribution with mean zero and variance

u²:=

1

⁰⁰

1 { ←乎)タ附

The similar methods as given in earliers show that

(j~

^:=

1 ¹ ^{ ^一割、

⁽^t

^一

⁾^d^e⁽^t⁾

is a consistent estimator of ^σ2given in Corollary 5.2.3

，

where

ん:= 1

^T^ψ¹

^州

In order to derive the asymptotic distribution of N2(ψ2) from Theorem

44 CHAPTER 5. TESTS FOR NBU AND NBUE

4.2.3 of Section 4.2

，

we set

and

叫(t

← J ^仇州川 ^ω ^υ ⁽

^州榊^s

^り ^M

^柿

^μ ⁾ ^凶

^d^白^s

川州(例ttの):=

i =

^的 ⁾^d^s

ん(t)

1 = : =

^ψ^[^2{S

^{州山バ}

^S⁽^s⁾^}^]^S⁽^s⁾^d^s

COROLLARY 5.2.4 (Kumazawa (1986e)). Suppose that the df's F(t)

，

G(t) and the weight function

ψ

2(t) satis今theconditions of Theorem 4.2.3 of Section 4.2. Then we have as n

→∞

where

and

η1/2 ~

N:川)一色↓→

^dN(O，

u !

.，)，

t μF J

μψ2

イ ∞ 曽

²^{^S⁽^S⁾^}

^的

⁾^d^s

σ2 .̲

J ; H {

μψlh

1

⁽^t⁾^‑^μFh2(t)PdC(t)

わ・一 μF

COROLLARY 5.2.5 (Kumazawa (1986e)). Suppose that under the null hypothe‑

515冗othe censori時 dfG(t) and the weight function

ψ

2(t) satis今theconditions (4.2.6)

イ

4.2.9)ofTheorem 4.2.3 ofSection 4.2. Then n¹/2{N2(ψ2)^ーν}converges in distribution as n

→∞

to a norma1 distribution with mean zero and variance

σ ι : = 1 =

^[^v^‑^W2{S

^附

where

A consistent estimator

ν:=

1

^J^.^w²⁽^s⁾^d^s

u! :=

I

^[^v^{一世}

2 { 丸

⁽^t^ー

) } ] 2

S!(t‑)de(t)

5.~. THE NBUE ALTERNATIVE 45

can be c∞onstruded f企romthe pr目e吋viousdiscussions and we can obtain an asymp‑

totical均l

(σ5.2.2幻)by using this estimator.

We need the following lemma to give the asymptotic distribution of the test statistic N₃defined in the equation (5.2.3).

LEMMA 5.2.6 (Kumazawa (1989b)). Suppose tnat tne df 's F(t) and G(t) satisfy tne conditions

l

^{T H}

^可ダ ^内

²^刊^勺⁽^附^t^t

l

^{T H}^句¹^E^V^h^μ

^内内仰

²^刊^勺⁽^例^切^t^t^吟^t^附⁾

^μ

^dC(^d^印州^t^吋⁾

^< ^∞

⁽⁽⁵⁵^.^.²²^.^.⁶⁷⁾⁾

and

が/2h(T)

→ o

in probab出tya.sn

→∞，

(5.2.8) wnere

仰):=

1 =

^S⁽⁸⁾^d⁸

Tnen tne stocnastic process

丸 (t):=が/2

{ 丸刈

ゐr0三t三T converges weakly in D[O

， T H ]

a.sη

→∞

^t^o^a^G^a^u^s^sⁱ^a^.ⁿ^p^r^o^c^e^s^s

B(t) with zero mean and covariance function

引軸C

包 JU

n J

包n J

R T

f I

︐ ︒

‑ J

一一

B a O

wnere

f

^h⁽⁸⁾ ^c^r^t

J

¹ ^h⁽包)^，h(8)h(包) g，(包):= 1{‑u.くけ{一一一 S(8)( ‑l{u>け一一 + 一一γ一.

ー .lμF . J 一 ' μ F μ F

PROOF: We have for 0

<

f

.T Z.司(包)dh(包) Bπ(t)

=

‑S(t)

z .

^叫⁽^t^)̲^{Jt ‑}^，^.^，

I"'n

(5.2.9)

46 CHAPTER 5. TESTS FOR NBU AND NBUE

+h(t)

f :

^Z^叫(包⁾^d^h⁽^包^{) π1}^/²^h⁽^T⁾^h⁽^t)^，ⁿ¹^/²^h⁽^T⁾

一一

μ F μ n μ F μ n μ叫

with Zn(t) =η1/2{Fn(t) ‑F(t)}jS(t). Hence Lemma 3.1.2 of Section 3.1 to‑ gether with the Cramer‑Wold Device and the Slutsky's Theorem implies that the limiting process of Bn(t) can be expressed as

L~H Z(包)dh(包)， h(t)

f ; H

Z(包)dh(包)

‑S(t)Z(t) ‑

μ F μ

予

1

^{T H}

^{白州包}

^{) = 帆}

where Z(t) is the limiting process of Zn(t). Therefore some calculations yield the desired result. I

THEOREM 5.2.7 (Kumazawa (1989b)). Suppose tlzat undeI tlze nulllzypotlzesis

冗otlze ceI^問 Il昭 dfG(t) satisfies tlze conditions (5.2.6)

イ

5.2.8)of Lemma 5.2.6. Tlzen we have as n

→∞

1/2 N3

rσ

⁽^t⁾¹ T.'I/‑'¥TJ"

T I 1 J

一一~ ‑ d sup IW{

一一 }‑

F(t)W(l) 1

，

九 O<t<∞l σ (

∞

‑ ' ‑ / " " ‑ ' J

wlzeIe W(t) denotes a standaId Gaussian pIocess witlz zeIo mean and covaIiance function E{W(s)W(t)}

=

s八t

，

(1"2 (t)

イ

^S²⁽⁸⁾^d^C⁽

and

ま : イ勾

⁽⁸^‑⁾^d^e⁽⁸⁾

PROOF: Note that we are in the situation TH

=

TF ∞. Lemma 5.2.6 shows that the limiting process of Bn(t) given in the equation (5.2.9) under the null hypothesis冗ois given by

的 ) = 1

⁰⁰^{¹^{^{叩 }‑}^F⁽^t

^州。(包)

Then it is seen that the stochastic process {B(t) : 0三t<

∞}

has the same distribution as the process {W {σ(t)} ‑F(t)W {σ(

∞)} :

0三t<

∞}.

Hence we can conclude the proof from the Continuous Mapping Theorem and the fact that

5.~. THE NBUE ALTERNATIVE

u ;

ⁱ^s^a^c^oⁿ^sⁱ^s^t^eⁿ^t^e^s^tⁱ^m^a^t^o^r^o^f^u²⁽^∞). .

In the uncensored case the variance σ2(t) becomes to F(t)

，

so we have

2 1 2 0 p ( 苧斗

^=P(ofp

^川

= P ( ょ

??1{W(t)‑tW(1)}三z)

= 1 ‑exp( ‑2z2) for all z三0

，

which can be also derived by the result of Barlow and Doksum (1972) since in this situation

手 !

⁼

2 :

⁷⁼¹ⁿ^l⁽^:⁼^:⁺^l⁾^h^a^s^t^h^e^lⁱ^mⁱ^tⁱⁿ^g^v^a^l^u^e¹.

Here we assume that under the null hypothesis F(t) the censori時 dfG(t) satisfies Foψ‑1(8 )三 8for all 8ど

o

^wⁱ^t^h~(8) :=σ(8)/σ(∞): this condition holds for the proportional censori時 modelgiven by G(t) = Sλ(t) with 0 < λ <1. Then we have for all z > 0

民 P ( 竺 ! ト

^)=P(ofF^∞^[W{^少⁽^t⁾^}‑F⁽^t⁾^W⁽¹⁾^]

斗

=P(oi??1{W(t)‑Fop‑1(t)W(1)}三z) 三P(WW(t)‑tW似

The asymptotic distribution of the suitably normalized version of N₃under the null hypothesis for arbitrary G(t) can not be evaluated and the above expression would be useful to determine the critical point of the N₃‑test.

N ow we shall compare the efficacies of the test statistics N₁(仇)and N₂₍れ) for the alternatives (i)‑(iv) given in Section 4.1 under the proportional censoring model. For the selection of the weight function we take仇(t)=

ψ

2(t) = t^a^• Then Corollaries 5.2.3 and 5.2.5 imply that μ = 1 and the censoring parameter λsatisfies 0 < 入 <1: here we assume α >ー1/2for the statistic N₁(u^a⁾and α > ‑1 for N₂(u^a). And the asymptotic variar悶 sunderピチoare gi ven by

σ2 r(2α+ 3) ‑2(1 ‑λ)叶 1+(1一入)2a+2

and

48 CHAPTER 5. TESTS FOR NBU AND NBUE

Table 5.2

Efficacies 01 the NB UE‑test statistics

叫印 thecensori句 pαrameterλ =

ム

αηd2 A1ternative

Statistic (i) (ii) (iii) (iv) λ= 1/10

N₁₍包0) 0.7218 0.7217 0.0451 0.1804 N₁₍包0.5) 0.3785 0.5195 0.0241 0.0831 N₁₍包1) 0.2307 0.4101 0.0144 0.0455 N₂₍包0) 1.2474 0.6490 0.0721 0.3874 N₂₍UO.5) 1.3319 0.5711 0.0728 0.4408 N₂₍包1) 1.3732 0.5056 0.0711 0.4776

λ= 3/4

N₁₍包。 ) 0.0100 0.01 0.0006 0.0025

N₁(u^O.5) 0.0019 0.0026 0.0001 0.0004 N_1(U1) 0.0003 0.0006 0.0000 0.0000 N₂₍包0) 0.1863 0.0969 0.0107 0.0578 N₂₍包0・5) 0.2738 0.1174 0.0149 0.0906 N₂₍包1) 0.3497 0.1287 0.0181 0.1216

2 1

(1‑λ)(α+ 2)2 ^I ^n¥1 ̲^，ⁿ ^，¥+ In • 二、、 }/(α+ ¹⁾²

，

respectively. Then some calculations yield that: for the a1teロ凶ive(i) r'(α+2) ^"¹2 ^I2

eff{N1(♂)}

=

{ γ + } z / σ ?

，

r(α+ 2) ln(α+2) 12 eff{N₂₍^包Q)}= {

(α+ 1)(α+ 2)σ2 for (ii)

5.~. THE NBUE ALTERNATIVE

for (iu)

and for (iv)

(α+ 1)2 eff{N1(♂ )}=‑37‑

，

eff{N₂₍包a ) } = 1 ^ー (α+ 2)4σ2

(2^a+1 ̲ 1)2 eff{N1(包a)}=22a+4

イ '

eff{N₂₍♂)} =

~

l2(α+ 2)(α+ 3^nI ^，

<\\~

^， "，‑)σ2

J ~

(α+ ¹⁾²

eff{N₁₍^包a ) } = 2 (α+ 2)2σi

fln(α+ 2 ) 1

) 2

eff{N2(ua)}={‑ 〉/I σ2

l (α+1)2 (α+1)(α+ 2)

J

^I ^孟

respectively

，

^whereγis t^h^e^E^u^l^e^r^'^s^c^oⁿ^s^t^aⁿ^t^.

Table 5.2 gives the efficacies of the test statistics N₁(u^O)

，

^N1 (^包0.5)

，

^N1(^包1

， )

N₁₍^包²⁾

，

^N2(^包0)

，

^N2(^包0.5)

，

^N2( U1

) and N₂₍^包2)_f_o_r_t_h_e_a_l_t_e_r_n_a_t_i_v_e_s and some val‑ ues of the censoring parameterλHere we recommend the use ofthe N₂₍^包¹)‑test for the testing problem in the sense of the Pitman asymptotic relative efficiency.

Chapter 6 Tests f o r DMRL and HNBUE

6.1. The DMRL Alternative

Kumazawa (1988b) considered the measure oi dispersion from exponential‑ ity to DMRL 日長 df'sF(t) given by

J 1 1 1 ‑

]~(1- り(1-t){1-s-Jt}ah，巧

1(8) 1‑H

F

¹⁽^t⁾

and proposed the test statistic

A 4

ior testing the null hypothesis 冗。:F(t) = 1 ‑exp( ‑tjμ) ior t三

o

(μunspecl五ed) against the alternative

冗5 : F( t) is DMRL

，

but not exponential

based on the Kaplan‑Meier estimator Fn(t). Here the property that the scaled TTT‑transiorms H;1(t) de五nedin the equation (4.1.1) oi Section 4.1 satisiy that {1 ‑H;1(t)}j(1 ‑t) is nonincreasing in t E [0

，

1] ior DMRL li長 df'sirom Theorem 2.5 oi Kle色UO(1982a) is used in defining the above measure. By using weight iunction仇(t)this measure can be generalized as

f 1 f 1 1‑H

F

¹⁽⁸⁾ ^1‑^H

F

¹⁽^t⁾

ム: = J o J .

⁽¹^{ー仰一川}⁽¹^{一肌}^(1‑t){ ^1‑s ^‑ ^1‑t

d

^'^I^t¹⁽^t⁾^d^t^‑

J o

^OO^曹^d^S⁽^t⁾^}

J ;

^S⁽⁸⁾^d⁸^d^F⁽^t⁾

μF

6.1. THE DMRL ALTERNATIVE 51

where

wdt)

叫

φ

^{仇榊}

The method of replaci時 F(t)by the Kaplan‑Meier estimator Fn(t) suggests to construct the test statistic

rwd 丸

(t)}

J ; J E ^山

⁾^d^s^d^Fⁿ⁽^t⁾

Pl(

ψd:=

μn (6.1.1 ) for the above testing problem. Kle色jる(1983)developed the test statistic A4 in the uncensored case by using the same property of the scaled TTT‑transforms H

p

¹(t) and the A₄‑statistic may be considered as the corresponding to P₁₍

ψd

with仇(t)三 constαπtin the uncensored case. Note that we reject旬。 infavor of 冗5for small values of

P

₁(

ψ d .

And we may use

品川

as a meaSl町 ofDMRL‑ness oflife df F(t). The measure d2(α

，

s) with α=9=

o

was first considered by Hollander and Proschan (1975) and Chen

，

Hollander and La時 berg(1983b). Bergman and Klefsjる(1989)discussed d2(α

，

β) with α and βnonnegative integers. It is seen that the measure d2(α

，

s) with α= s is equal to dl (仇)with 'ltl (包)= u ，X<so in this case the both measures lead us to construct the equivalent test. Then some simple calculations yield that

where

and

h 州的 ( い Mα

= :

J ^わ f ^∞ ^コ

^州^g^酬^{^州^{^S^珂^伊^印^削^似^制^仰⁽^t^の^附^収^川⁾^リ^例^州

^μ

^川

^}

^d^t

^，

α1 : = ‑(s + 1)(s + 2)(α+β+ 3)'

句広士

2)(β+1)

(6.1.2)

(6.1.3) (6.1.4)

52 ^CHAPTER⁶^. TESTS FOR DMRL AND HNBUE

α+β+4

α3: =一(α

+

2)(s + 2)(α+β+3)" (6.1.5) The test statistic

f T

^g^{

^丸

⁽^t⁾^}^d^t

P₂⁽α

，

β) .IU .，~- ( 6.1.6)

p・n

may be COI凶 rudedby the use ofthe Kaplan‑Meier estimator Fn(t) and we rejed

π 。

ⁱ^nf^a^v^o^r^o^f^冗⁵^f^o^r^l^a^r^g^e^v^a^l^u^e^s^o^f^P2 (α

，

β). Cl悶 1

，

Hollander and La時 berg (1983b) and Bergman and Kle色UO(1989) considered the test statistic based on a modified Kaplan‑Meier estimator

，

and proved the asymptotic normality of the normalized version under some stronger conditions than those given in the below.

In order to derive the asymptotic distribution of the P₁₍ψd^司statistic

，

we apply Theorem 4.2.3 of Sedion 4.2. To this end we set

判 ← J ^則

⁸⁾^d⁸ ⁽⁶^.¹^.⁷⁾

COROLLARY 6.1.1. Suppose that the df 's F(t)

，

G(t) and the function雪1(t) satis今 theconditions of Theorem 4.2.3 of Section 4.2. Then we have as n→ ∞

where

and

η1/2~Pl(ψd 一色}→d ^N(O

，ぺ ) ，

t μF J 巴

内 :

= 1

^{0 0}^r^.^p^伊⁽⁸

^{州ム}

σ2:=fJH{μψ1 h1(t) ‑JLFh2(t)PdC(t)

ん(t): =

1

^{0 0}

^[ ^' ^l ⁱ ^d

^S(8)}S

^{仲バ}

^S⁽^s⁾^}^]S⁽⁸⁾^d^s

COROLLARY 6.1.2. Suppose that under the null hypothesis 冗othe df G(t) and the血nction雪1(t) satis今 theconditions (4.2.6)

イ

4.2.9)of Theorem 4.2.3 of Section 4.2. Then we have as n→ ∞

n¹^/²{Pl(仇)‑v}→d N(O， σ~J，

6.1. THE DMRL ALTERNATIVE 53

where

… ^J

^内 ⁾^d^s ⁽⁶^.¹^.⁸⁾

and

J u a zu

s

可EEEEEBEEBd

a︐b S ︐

v

FEE‑‑EEEEa﹄∞

ρ t ' ' ' o

2 一一 ψ

From this corollary the asymptotic variance

σ 3 "

under the null hypothesis

予'1

depends on the unknown parametersμand G(t). By the similar method as given in the previous sections we can construct a consistent estimator

3 3 1 イ

^[^v

^一品

⁽^t

^一 ⁾ ^} ^]

²^S!(t‑)^d^e⁽^t⁾

by the theory of counting processes.

Next we consider the asymptotic behavior of the test statistic P2(α

， s )

defiI凶 inthe equation (6.1.6). This result can be proved by Theorem 4.1.1 of Section 4.1 and stated as follows.

COROLLARY 6.1.3. Suppose that the df 's F(t) and G(t) satis今 theconditions

l

^{T H}

h~(t)dC(t)

^∞

⁽⁶^.¹^.⁹⁾

and

η1/2 h₁(T)→

o

ⁱⁿ^probab^出^tyasπ→∞， (6.1.10) where

ん(t):=

1

⁰⁰^S⁽^u⁾^d^匂

Then the sequence of the rv's

ポ/2

九 {

⁽^o^:^，^s^)‑

告 )

converges in distribution as n→∞ to a normal rv with zero mean and variance

1I F‑

‑ ' t w ‑

︐ ︐

••. ︑ ‑

一

︐α一

角4

‑

︑

︐ E

︑︑.. 一

︐ ︐ ‑

a︐

b E

︐ ︐

•. ︑ ‑

今必

‑

一4F

ドキュメント内滋賀大学学術情報リポジトリ (ページ 33-58)

ん︑SJ

:=わいい)} . 1

ψ

，

: =

グ

メ ( ω

1

υ (

，

，

，

，

，

，

=

τ

)=‑1=

行)‑

L

' l

1

+

L

d

弘い Mμ

σ η

T 小 )

い

r (

+

‑ l

I

+

{ l

え 榊

，

，

j : = f

，

N ( O ， 生 ) ，

ユ

，

イ

σ

ι わ

附

引 ト ' l

，

手 : イ 〆 ω 崎 山 伐

，

，

，

>

，

，

イ:= s ( : r [ 2 a s : 1 ‑

ふ +

り 山

σ ; イ 〆

( u ぺ

t {

) 2 / ( v 4

，

=

i ) ，

t {

{一一一一一一一一}/イ

( s ‑

t {

，

σ ?

，

，

+

+ ω

λ λ ，

λ

り

， . ・ )

^υ ⁽

⁺

σ ^η

^T ^{小 )}

^{え榊}

N ( O ，生 ) ，

^附

引ト ' l

手 : イ〆 ω ^{崎山伐}

イ:= s ( : r ^[ ² ^a ^s ^: ¹ ^‑

^ふ ⁺

^{り山}

σ ; イ〆

旬。 )

川 / σ ; ，。

一一

Ud= 市，

1 ^{∞ 附}

一州 + り } 川 μ

= ψ ( ^い ^仲

^吋)州一 1

⁺ ^の

^{丸山山}

J ^九

^{ ^σ

^珂

^的

^υ ⁽

^り ⁾ ^阿

^り ^M ^柑刷川 ^μ ⁾ ^川阿附

^町 ^叩