THE TWO−SAMPLE KUIPER TEST
BYKosABuRo HIRAKAWA’
1・ Introduction・ Let X=(Xl, X,,… ,.Ylm)and ]r=(}コ, 】夕,… , }「h) be two in− dependent samples of sizes m and n of independent observations from two popula. tions with continuous c.d.f.’s取x)and F,(Y)respectively. When one wish to test 寸he nu皿hypothesis H・:F・(x)≡F・(x) (1) against the alternative 1五(1):・F,(x)≠1『』(大), ・ (2) the Kolmogorov−S㎡ヒnov(K−S)test is often used. Many results of the study about it have been published. While Kuiper[9]proposed an adaptation of the K−S statis− tic and・derived its limiting distribution in the case〃1=n. Abrahamson[1]explored the Bahadur relative e伍ciency of the Kuiper test with respect to the K−S test based ◆nthe limiting. distributions and Raghavachari[11]recently derived the㎞血lg distributions of the K−S statistic and of the Kuiper statistic under the alternative. In this paper we give tables fbr the twぴsample Kuiper test fbr m=〃=2(1)100 and investigate the powers of the test based on samples with practical siZes. Also, the powers of the Wilcoxon[12]test equivalent to Mann−Whitney[10] test and of ’the正laga [6] test suitable to test、石陥(1) against H・②:F・(x)≡F・(x・rθ) (θ≠q) ’ (3) and the powers of the Ansari−Bradley(A−B)[2]test and of the Kamat[8]test suitable against . 」H,(3): F,(x一θ)≡F,(e(x一θ)) (c≠0, 1) (4) 二are investigated in order to compare with those of the Kuiper test. The powerS are cst㎞ated by the Monte Carlo method. 2.Tables fbr the Kuiper test. Letτl be the Kuipet statistic defined in(13)and 苦(α)be the smallest value of integer瓦’s satisfying ’Pr{T,≧kl五lb}≦α, (5) that is, T,(α)be the critical value of theα一level Kulper test. For the use of the Kuiper test we first compute the values of Pr{T,=k}f()rん=1,2,…, n and sam− .ple sizes〃1=n=2(1)100. The number of all ways from O(0,0)to M(h, n)without ・the restriction’by any sloped丘ne in Fig.1is(2n)!/(h!)2..Let、Fiゴbe the number ◎fways from O to M exceeding neither straight lines g:γ=x−i nor乃:y=x十プ, t [99]、 100
K.HIRAKAWA
G,∫be the number of ways from O to M which attain both g and h,克(α,β)be the number of ways from O to、alattice point P(α,β)under. the restriction by g and h. If we putん(α,β)=O fbr apoint P(a,β)outside the belt among g and h, then we have ・{鞭::蕊1−1)’(、)
provided Fo, o=O and民i=O江ガく0・‘@Summmg up Fig.1 G《」’sSatisfying the equatio皿ま十ノ== k and dividing by the totality bf ways, we obtain the wallted probability, i.e. . Pr{T,・=k}=(n!)2ΣG‘∫/(2n)!・ ・ (7) 「 お’=鳶 In TablC l values of Pr{Tl=k}equal to or more than O.0001’are given onIy fbr n』10,20,30,50,100,and in Table 2 values of the upPer tail probabilities Pr{Tl≧k} equal to or less than O.2 are given f()r same〆s・ Table l Ptobabilities for t血e KUiper statistic ゐ M(4n) y π .﹂ 9 P(α,β) 0(o,o) ‘’ ロ エm=n
10 20 30 50 100 kand P忙{Tl=k} 2 .0111 8 .0127 3 .1416 9 ..OO17 4 .3228 10 .0001 5 .2962 6 .1584 7 .0554 3 .0033 9 ;0995 15 .0001 4 .0440 10 .0491 5 .1483 11 .0204・ 6 .2304 12 .0071 7 .2274 13 .0021 8 .1677 14 .0005 3 .0001 9 .1780 15 .0056 4 .0034 10 .1359 16 .0021 5 .0305 11 .0901 17 .000762nδ・11
.0974 .0529 ..0002 7 .1669 13 .0278 19 .0001 8 .1953 14 .0131 5 .0007 11 .1501 17 .0222 23 .0003 6 .0077 12 .「1349 18 .0127 24 .0001 7 .0324 13 .1095 19 .0069 25 .0001840
12
.0755 .0817 .0035 9 .1196 15 .0566 ’21 .0017 10 .1465 16 .0366 22 .000873951
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.0001 .・O828 .0740 .0127 .000884062
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.0013 .0987 ..0600 .0085 .000495173
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.0061 .1063 .0470 .0056 .00020628411223
.0180 .1058 .0355 .0035 .’nOOI 11 1723
29
・35 .0374 .0988 .0260 .0022 .0001 12 .0608 18 .0875 24 .0185 ・30 .0013 Kuiper[9]provided an asymptotic fbrmula fbr the、distribution of∨/7 Vs,語withレ∼,。・・ Trln as 。c .。
P・・’・{’・f’iiT・V・・.・n≦・}−1一Σ・(・j…−1)〆・’+、:(1+Σノ・e・(2j…一・)・一・“・’)+・(n−・)・. ∫=1 tt’. ・ゴ=1 (8)
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THE TWO−SAMPLE KUIPER TEST
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106 K.HIRAKAWA
Can we obtain good approXimations to exact probab盗ties by the use of this fbr− mula?For compad㎎with exact values of Pr{Tl≧k}given in Table 2, we com− pute the丘rst term and the total of の co Σ・②・b・−1)・−js・2+、:(1+Σノ…②…一・)・一・2・2) (・) ∫=1 ∫=1 substituting k/∼/7 into c and show them in Table 3. 丁油1e 3 Comparison of apProximations鋤d exact冊1ues of Pr{T・≧k} 40 60 80 100克①②③
〃①②③
ん①②③
た①②③
11 .4183 .4907 .4871 12 .2779 .3389 .3346 13 .1716 .2179 .2134 14 .0988 .1311 .1265 15 .0531 .0739 .0694 16 .0267 .0392 .0348 17 .0125 .0196 .0153 18 .0055 .0092 .0050 15 .2579 .3058 .3029 16 17 .1737 .1117 .2114 .1398 ’.2083 .1367 18 .0687 .0885 .0855 19 .0405 .0538 .0508 20 .0228 .0314 .0285 21 .0123 .0176 .0147 22 .0064 .0095 ’.0067 18 .2107 .2474 .2452 19 .1471 .1761 .1738 20 .0993 .1213 .1190 21 .0648 .0809 .0787 22 .0410 .D523 .0501 23 .0251 0.32’9 .030724
.0149 .0200 .0178 25 0.086 .0118 .0097 21 .1625 .1901 .1883 22 23 .1155 .0800 .1373 .0966 .1354 .0948 24 .0540 .0663 0.645 25 .0355 .0444 0.426 26 .0228 .0290 .0273 27 .0143 .0185 .0168 28 .0087. .Ol16 .oo99 Notes:①Exact value.②Approximation by the f辻st terln of(9).③Approx血atlon −by(9). If we compute using c=(k−0.5)/ン可with continuity correction O.5, those approxi− mations will more increase.. 3.C…i蜘町・f th・K・ip・・t・・t・L・t恥)・nd・F・(x)・be c・ntinu・u・c可’・・f populations π1 and π2 respectively,81,楊(x)and 32,外α)be emp血…ical c4r’s obtained 血om samples described in the introduction. De丘ne the two−sample Kuiper statisticby
Vlm,”=sup{s1,m(x)−S2,”α)}−inf{S1,勿(x)−5「2,蕗ぴ)} x ズ ・nd・d・pt th・・司・・h・・w・・ejec・H・・兄ぴ)≡F・(・)・if・Vm,・−Vm,・(・);・・h・rwi・r・㏄・p・ 王lo, where、 Vm,ヵ(α)is theα一1evel critical value, i.e. Vm,外(α)=inf[{んlPr(V.,蕗≧k)≦α}∩{k.1丘is possible value ofレ三,ち}]. Then, this test is consistent. The proof is as fbllows:It is known[13],[5]that sup l 51,m(x)−F1α)1→0 ・with probabiHty 1. (1旬THE TWO−SAMP工E KUIPER TEST
107 (10)implies that fbr arbitraryε>0, 血nPr{sup lS1,励(x)−F1(x)1<ε}=L (11) 堺→m ∫ Samely, ㎞Pr{sup IS2,”(x)−F2(x)1<ε}=1. (12) 錫→oo 夕 Using notationsδ+,δ一andδde丘ned by δ+−sup{F、ω一・F,(x)},δ一一inf{F・(x)−F・(x)},δ一δ+一δ一we have
Pr{lV.,”一δ1<ε}=Pr{lsup{(S1,頒(x)−F1(x))一(S2,”(x)−F』(x))十F,(X)−F2(x)} s −inf{(Si,91(x)−F1(x))一(S2,籏(x)一・F2(x))十Fl(x)一」F2(x)} 一(δ+一δ一)1<ε} ≧・・{・e・1・ ω一F・(・)1<号・nd・e・1・2…(・)−F2(・)1<号}・ Since two samples are mutually independent, it fbnows that ・・{IVm,一・1<・}≧m・・国・e・1・い(・)−F・(・)1<号}…{・e・1・…ω一F・(・)1<号}]・ Sinceδ=O under H』andδ>O under」酊1ω, using(11)and(12)we have 血nPr{V.,刊くεlHo}=1, 鴬二and
㎞Pr{昨.。一δ1<εIH・(・)}−1 withδ>0・ τニニ Usingδ/2 as a value ofε, it follows that 恕・・ト<』凪}−1・ ”−づco ㌍・・{Vm,・〉』昂…}−1・ 外→oo Vm,外(a)<δ/2 fbr su丘iciently large m, n and fixedα. Therefbre, we obtain 血mPr{v,。,.>Vm,。(α)}=1・ =: This implies that the test is consistent. 4. Alternati▼e tests for comparing with the Kuiper test. The powers of the test statisticsち∼T6 written in(14)∼(18)below, are going to compared with that of the Kuiper test. Let}伊‘=1, xF 1,γ輌=O if the輌一th smanest observation in the combilled sample of size IV=〃十πbelongs to X, and let w‘=−1, xi=0,γ‘=1if it belongs to ヱ We denote the smaUest and the Iargest of integers j’s satisfying xi=1 by a and b respectively, those of integers satisfying y‘=1 by c and d respectively. We deal with fbnowing statistics. t ハ T,=maxΣ}v‘−minΣ}v‘, (Kuiper) (13) 1≦1≦N《=1 1≦‘≦N‘=1 \108
K.HIRAKAWA
1 T2=max 1ΣWi 1, 1≦1≦Ni=1 N N T』一・1ΣiXi−E(ΣiXi)1, i=1 i=1 T)= T, 一 1ΣiXi+ T6= Ic−1−1V+dlc−1十1V−b
a−1十N−d
la−−1−N+bl Nt2 N Σ ‘=1 《=N’2十1c−1十N−d
lc−1−1V+blla−1−N+dl
a−1十1V−b
江XlXN =・1 if xlYN=1 if XNツ1=・1 if y、γN=1, N’2 (K−S) (Wilcoxon) (Haga) (N−i+1)x‘−E{Σix,+ Σ(N−i+1)Xi}1 ’=1 ‘=N’2+1 Ts and Ts are Bradley, s statistics respectively. if xlXN =・1 if XIYN=1 if XNY1==1 ifγ、γN=1. (Kamat) (A−B) (14) (15) (16) (17) (18) the deviations from the means of the original Wilcoxon’s and Ansari一O…○×**…*×○…0
1 c ○…○×**…*○×…× 1 コ ゆロ ○×* ×…×○**…*×○…0 4 N l a ば ハ「 ×…×○**…*○×…× め N l a め ハr Obseτvation belonging to X Observation belonging to γ Observation belong血g to either X or r Fig.2 It is known that the test T, is co皿sistent against all altematives H1ω, T3 is con− 、iSt。nt・if・P,α’〈(〉)Y’1H。)≠1/2 P・・vid・d th・t X’andγ’・・e i・d・卿d・nt・and°m ■ variables from the populationsπ1 andπ2 respectively, and Ts is conslstent agalnst H,{3}. 5.S叩P1.m。。鍋・al・d・ti…f・・iti・al仙臼・L・t T・(a・) and R・(⑳)be the critical value and the critical region of theα一1evel test based on Ti・i・e・ Ti(α、)−nf[{kl・Pr(Ti≧k)≦α・}∩{姫i・th・p・ssib1・val…f瑚・ R‘(αi)一{’1’≧Ti(α1)}・ 1。。,d。, t。。。mp。,e th・p・w・…fth・t・・t・w・wi・h t・…Ti(α・)f・・同・2・’”・ 6;αF1.0,炉0.2,α、−0.1,α、−0.05,α・−0・02・α・−0・01;・−10・20・30・50・The n㏄d。d va1。。、。f T,(α1)訂。 f。。nd i・T・bleS 2−1∼2−3・T2(⑳)’・ar・aU av姐・b1・丘。m血。 p。輿[4コ孤d」SA T・bles[7コ. We ca輌d曲…fT・(α・)and TKα・)
in[7コf。,1≧3. V・1・…fT,(α・)are・9i・・n・nly・f・・〃≦20・∫≧3 in[12コ・[7コ紐d T、(a、).ar。9i。。n f…−10,’≧3・in[2コ・We ca1・㎡・t・th・lack・f曲・・and 9ive 血。m i。 T。bles 4.a,牛b孤d牛c ad・㎞9伽・・f・・〃−40・Since・it・h…bee・・h㏄k・d by硬記1。。、 th・t th・di・垣b頭・…f・T・ and T・・an be・PPt・Xim・t・d by n°mal di,垣b。d。n,, we ca1。u1。t・d T,(α)・nd Ts(α)f…−20,30・40・50・・i・gth・n・ma1THE TWOSAMPIE KUIPER TEST
109 apProximation with cont血uity correction and counting fractions as one. Table 4−a Ti(.20) and Pr{Ti≧Ti(・20)} (i=4,6) 4(Haga)6(KamaO
10 5 .15049 5 .10346 20 5 .16355 5 .14133 30 5 .16802 5 .15352 40 5 .17028 5 .15952 50 63 O9 1つ﹂ 7.ノ011
. ● ξ﹂ま﹂ Table 4−b Approximate percentage poin鶴of the Wilcoxon statisticm=n
1020
30ヨ50
α .20 18 *18 t.19032 48 88 134 187 .10 23 °*23 t.08921 62 °*62 112 172 240 .05 27 °*27 ★.04326 73 °*73 134 205 285 .02 32 °*31 t.01854 87 °*86 158 243 338 .01 35 °*34 t.00893 96 °*95 175 269 375一Mean
n(2n十1)/2 105 410 915 1620 2525 Varjance n2(th+1)112 175 1366.67 4575 10800 21041.67 Notes:*Exact critical value. ★Exact probability. °Value fbund血[12]and l7]. Table 4∼c Approximate percen血ge poiロt80f the A−B s包tisticm=n
10 20 R0ヨ50
Mean
n(n+1)/29
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4802270
1 49 88 135 188 210 465 820 1275 Variance n2(n2−1)1 12(2n−1) 43.42 341.03 1142.80 2698.73 5258.84 Notes:*Exact critical value. ★Exact probabhity. °Vahle fbund口1[2]. 6.Compari80ns of powers. Using the Monte Carlo method two samples with equal s泣es are generated from two populations under altemative hypotheses. The ◎ombinations of the]吐st populationπ1 from which sarnple(Xl, x2,… , Xn) iS drawn and the second populationπ2丘om which sample Oコ, Y2,…,}㌦)is drawn are shown in Table 5. U(a,ク)denotes the unifbm distributio皿on an interval[a,坦 and C(ッ)denotes the chi−square distribution with d.f」〃. C’(6)is the dist亘bution of†Z62 whereズ62 is the chi−square random variable with d.£6.110 Table s Combinations of pop皿lations π1 π2 N(0,1) N(1,1) m(0,0.52) n1(0.5,0.52) {1(0,1) σ(0.4,1.4) ミ(0.16,0.84) ミ(0.3,L1) C(2) C(4) C(3) C’(6)
K.㎜KAWA
and those probabMties for i=1,2,… ,6are not equal for丘xed l. E[f‘(α釧1五]are shown in Table 6. we give in Tables 7−1∼7−80bserved values of fi(α1)for 1≧3 under」ff the graphs of fi(α1)for n=20,50 ill Figs.3−1∼3−8. shOws that there one has not Ti(αJ)satisfシingα1≦Pr{Ti≧Ti( We observed fbnowing aspects. The procedure for generating samples of sizes n=10,20,30,50are repeated 500 t㎞es fbr each case. 1潟t f‘(α’)be the frequency that Ti fe皿 on Ri(α1)then the power Pr{Ti∈・R‘(α釧H1}can be es・ timated by fi(αt)/m, where m is a reptition number of the pro㏄dure. Since in our research m=500, Var(f‘(α‘))=50qρ(1−p)when the tme power is p. We wish to observe f‘(αt)(輌=1,2,…,《9 under the common condition Pr{Ti∈Ri(α1)岡=α for a皿i,s. However, we can not find in genera1, the value of Ti(α1) satisfying the equation Pr{Ti≧ Ti(αi)IHo}=αfbr givenα, because Ti is discrete variable. Thus Pr{Ti∈R《(αt)1」Ho}<αl in general The values of For the purpose of comparing of the powers land show The bar−in the、αt−column al)}<α↓一、. 1° The case when昂(2}is tme. (1)π、:N(0,1)andπ2:N(1,1) (a) It is observed that ノ)(α1)>f2(α‘)〉ノ1(α‘)>f4(α‘)〉ノ)(α1)〉」f5(α‘) for 1=3,4,5,6;n==50. (b) The Kuiper test T, is somewhat inferior to T2 but it hold the power close to O.7 forα=0.05, n=30 and O.9 forα=0.05, n=50. (2) π1:U(0,1)andπ2:σ(0.4,1.4) The relation betw㏄n these populations is similar to that between N(0,1)and ]V(1,1)ill the viewpoint也at the value of sup{F1(x)−F,(x)}−inf{F1(x)−F2(x)} x ご are nearly equal for both combinations. But the asp㏄t of tails is different from the case(1)above. (a)We observe that ノ≧(⑳)≧ノ)(α1)≧f2(α1)≧ノ1(α1)>f6(αi)≧f5(⑳) for 1=3,4,5,6; n=10,20,30,50. (b) The Kuiper test is somewhat inferior to T. 2° The case when H,(3)is血e. (1) π1:N(0,1)andπ2:1V(0,0.52) (a) In this case f5(αJ)〉ノ三(α1)〉ノ1(α」)〉ノ》(∠吻)〉ノ)(α1)〉ノ)(ai) for 1=3,4,5,6;n=50. (b) The power of T, seems to be much smaller than that of Tl. /THE TWOぶAMPLE KUIPER TEST
111 Table 6 Va1ロes of①Ti(a)and②E[ノi(α)] 10 20 30 50乃答乃
乃乃乃㍗.恥恥
五克冗九
乃孔㌫冗九
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一一
27 21.6 7 21.4 14 19.2 6 21.6 11 15.1 9 16.8 73 25.0 8 13.6 37 25.0 7 17.8 13 24.9 11 17.3 134 25.0 8 14.6 67 25.0 7 21.7 17 24.1 14 19.6 285 25.0 8 15.5 143 25.0 7 24.9 0.02 ① ②007101107
3
’−0100/4818
4 12 158 9 80 9 19 16 338 9 170 9 7.3 6.2 9.3 5.5 8.0 7.6 6.1 10.0 7.1 10.0 8.2 7.8 10.0 7.8 10.0 ・5.2 6.7 5.8 10.0 8.4 10.0 6.8 0.01 ① ②0/84.078
今﹂11
21一う09Q’
110/414
15 13 175 10 88 10 20 17 375 10 188 10 .9 1.0 4.4 2.8 4.9 2.2 4.9 2.0 5.0 3.7 5.0 3.5 4.4 3.3 5.0 4.1 5.0 2.4 3.2 2.9 5.0 4.5 5.0 3.4 Note:−shows the lack of value by the reason of discreteness. (2) π1: U(0,1)andπ2:σ(0.16,0.84). (a) Tbe orders of fi(α;)are as fbllows: f6(αハ)>fi(α1)>f5(α1)>f2(α1)>f4(α‘)〉ノ)(α,) f()r 1=3,4;n=50 alld ノ)(α1)>f5(α1)〉ノ1(α1)>f2(α1)〉ノ)(dVl)>f3(at) fbr 1=5,6;n=50. (b) The esth皿ate of the power of T2 is moderately small even fbr n=50. (c)The estimates of the powers of T,, T, and T』are much sma皿er than those of T,, T』and Tl fbr 1=3,4,5,6; n=20,30,50. 3° The case when H,(1}is true, but th②and H,(3}are not true. The orders of義(α1)fbr fbur cases are as fbllows:. (1) π1:ハ∼(0,1) andπ2:N(0.5,0.52). .112
K.HIRAKAWA
Table
7−1 Values of諺(al)in the case of IV〔0, 1) and IV(0, 1) fi・(α) 500 0.10 0.05 0.02 0.01 10 20 30 50売莞莞恥三乃芸玉莞莞莞一莞緩莞
133 一 47 9 219 一 116 34 302 232 180 118 277 228 126 93 12 7 4 3 一 25 11 5 297 207 一 127 417 364 275 193 456 . 434 379 327 354 281’ 247 218 22 9 2 1 115 72 46 23 一 343 一 190 457 428 381 326 487 470 435 406 386 315 279 242 22 9 3 2 139 109 49 35 474 453 403 368 494 489 471 447 500499
496 490 419 366 328 295 15 8 1 1 166 132 74 60 400 300 200 100 ・一,一一氏=≠Q0一π=50
0_010.02 0●05 0.10 Grap血s of f‘(a)in the case of IV(0, 1) and N(1, 1) Fig.3−1 αTable
7−2 Values of fi(ai)in the case ofσ(0,1).and こノ(0・4,1●4) fi(α) 500 0.10 0.05 0.02 0.01 10 20 30 50 克 克 聡 乃 既 九 欠1 乃 冗 九 先 既 τ1 処 乃 九 先 恥 τ1 乃 答 処 跣 九 199 316 405 463 9 395 480 497 500 13 39 498 499 500 8 75 500 500 500 500 8 117魏55袈鑑2⑲援鑑罎錺魏4翫
19483 290 320 2 2 394 474 500 1 10 75K00116
4.415一認魏o舶
19 W1iω12
22
14 po〃 T3 po〃P5
2244
獅携珊09霧魏0茄
400 300 200 100 0 ・一一一一氏≠Q0一力=50
0.01 0◆02 0.05 0.10 Graphs ①f j6(a)in t血e case Of σ(0, 1) and σ(0・4, 1.4) Fig.3−2 αTHE TWO−SAMPLE KUIPER TEST
i113 Table 7−3 Values of fi(ai)in the caSC of ハ1{0, 1) and ハ「i(0, 0.52) fi・(α) 500 io20
30 50 乃 先 聡 乃 既 鑑 万 乃 冗 乃 鑑 石一 τ1 乃 冗 ㍗ 鑑 恥 欠1 乃 .7る 九 既 恥 0.01 0.05 0.02 0.01 136 42 co 31 185 279 117 53 61 347 360 162 61 79 433 438 473 272 61 121 490 485⑲16器讐召裂㌃3630叢撰”59揚
7391043
41
67
25 12 3 206 224 50 16 16 ’316 306 394 83 16 42 450 423 11Q6351⑳2146346106⑳512砂57%554291994
111
2 2︵∠ 3413
400 300 200 100 0 :一一一一一n= 20 −/n=50 0.010.02 0.05 0.10 Graphs ofノ}(α)in the caSe of IV(①, 1) and IV(0, 0.52) Fig.3−3a
Table
7−4 Values ofノ…(al)in the case ①f Iノ(0,1) and こ1(0.16,0.84) ノ}(α) 500 1020
30 50 乃克聡㍗乃九 万乃恥乃先 乃乃孔㍗先鑑 乃聡九既既 0.10 0.05 0.02 0.01 102 28 44 31 140 22 18 95 137860/517
3 ξ﹂く∨ 100一︶47二︶32
400 300 197 66 44 31 246 412 154 25 18 10 176 328 7 7 2 107 259532160
fJ 7’7’1
200 90 57 54 343 491 251 43 33 16 292 472 22 8 9 212− 411 108 10 6 3 160 353 100 451 15.2 63 101・ 449 .500 4.13 91 35 38 403 500 259 38 14 23 347 499 254 18 8 14 292 497 0.01 0.02 O.05 0.10 Graphs of/i(α)in the case of U(0,1) and U(0.16,0.84) ・皿9・_3=4.、 α114
K.HIRAKAWA
Table
7−5 Values of兎(al)in the case ①fN(O,1)anqN(O.5,0.52) .fi・(α) 500 10 20 50 乃乃乃九跣恥 五乃乃乃乃恥 万乃乃㍗既π 乃乃先乃先恥 0.10 0.05 0書02 0.0110577’9
65634
11甘皿−﹂11 106 96 ’95 156 74ヨ663753泣
2242︵﹂3
212688
400 300 355 347 305259
302 399 269 277 240 160 228 335 191 169 125 160 278 181 112 119 86 100 221 200/
,チ〆 /乙’ ,〃⑭
353 294 250 337 432 283 204 162 151 196 326プ
,1 /!2 406 350 316 399 458 281 217 195 253 378 100 489 489 441 416 472492
486 478 413 362 439 489 461 434 366 329 404 473瑞幾纏
0◆010.02 0.05 O.10 Graphs of fi(のin the case ofN(0,1)and N(O.5,0.52) Fig.3−5 αTable
7−6 Values of兎(al)桓t血e case ①fσ(0,1)and σ(0.3,1.1) ノ}(α) 500 1 0.10 0.05 0.02 0.01 10 30 30 50 万 乃 冗 九 鴛 恥 τ1 乃 冗 乃 先 恥 乃 乃 恥 乃 既 乃 τ1 乃 九ゴ 乃 先 恥 81 119 210 27449
151 290 356 473 70 100 365 428 49895
223 、 364 468 476 500 145 395 貌912430 1﹂’■且74
199 297 418 37 56 185 316 387 493 52 .164 314 442⑭
499
83 353 00’︶77︵∠ ⇔∠50/7 1 122 223 371 15 35 238 313 488 22 77 189 351 426 499 43 24839ω456
36 .58 168 320 7 17 59 165 256 466 12 45 132 306 388 499 3、1 203 400 300 200 100 0.01 0◆02 0.05 ,,,’’’”T 0.10 Graphs of fi(α)in the case ①fU(0,1)and こ1(0.3,1.1) Fig.3−6 αTHE TWOSAMPIE KUIPER TEST
115 Table 7−7 Valロes of fi(al)in the case of (て2) and C(4) fi(α) 500 ’ 10 20 30 \ 30 T6 乃先 乃乃 0.10 0.05 0.02 0.01 136 212 303 258 24 232 191 10 41 45 103 162 106 4 23 う一.82’︶37 ーエ︵∠061
400 300・279
403 447 371 44 133 189 333 412 281 27 97 253 345 238 8 61 104 161 289 186 2 33 200 457 490 3’99 33 164 353 426 478 335 15 124 383輔
301 374
219 340 410 265 2 50 100 477 497 500 429 64 252㈲
493 498 375 31 205 396 470 496 337 17 131 349 453 488 314 8 103 0.01 0.02 0.05 0.10 Graphs of兎(α)血the caSe of C(3) and C(4) Fig.3−7 α Table 7−8 values of fi(ai)in the case of C(3) and C’(6) fi・(α) 500ざ、
10 20 30 50 乃乃恥処鑑既 乃乃 的乃 乃乃万乃 0.10 0.05 0.02 0.01 ω菊7255680920/
3234
349259
111111
1
110784
400 300 123 76 71 64 159 140 67 46 44 28 109 82加215584
︵U14丁︵U15 3’11’14.︵∠ 200 87 90 70 201 205 120 47 48 28 150 152 7’︵U7・100122186
︹510118872
P︶41 100 230 129 104 94 287 285 186 90 7242
213 231 89 37 39 25 137 145 69 Q8 P9 P6 O0 P5111
O.010.02 0.05 0.10 Graphs of ri(a)in the case of C(3) and C’(6) Fig.3−8116