On testing More IFRA Ordering-II

On testing More IFRA Ordering-I I Muh yiddin Izadi 1 Baha-El din Khaledi Department of Statisti cs, Razi U niv ersity , Kermansha h, Iran Chin-Diew Lai Institut e of F undam en tal Sciences - Statisti cs and Bioinform atics, Massey Univ ersity , P almersto n North, New Zeala nd 1 Corresp ond ing author Abstract Supp ose F and G are tw o life distribution functions. It is said t ha t F is more IFRA than G (written b y F ≤ ∗ G ) if G − 1 F ( x ) is starshaped on (0 , ∞ ). In this paper, the problem of testing H 0 : F = ∗ G against H 1 : F ≤ ∗ G and F 6 = ∗ G is considered in b oth cases when G is kno wn and when G is unknow n. W e prop ose a new test based on U-statistics a nd obtain the asymptotic distribution of the test statistics. The new test is compared with some w e ll kno wn tests in the literature. In addition, w e apply our test to a real data set in the con text of reliabilit y . Keyw ords : Asymptotic normality , star order, increasing f ailure rate a v erage, Pitman’s asymptotic efficiency , U-statistic. 1 In tro d uction Let X b e a lifetime of an appliance with densit y function f , distribution function F and surviv al function ¯ F . Let also denote F − 1 as the righ t con tin uous in v erse function of F . X is said to b e IFRA ( incr e asing failur e r ate aver age ) if ˜ r F ( x ) = R x 0 r F ( t ) dt x is nondecreasing in x ≥ 0 which is equiv alen t to that − log ¯ F ( x ) x is nondecreasing in x ≥ 0 where r F ( x ) = f ( x ) ¯ F ( x ) . It is of considerable interes t to pro ducers and use rs of the appliances to ev aluate the sev erit y of av erage failure risk at a particular p oint of time and to see if ˜ r F ( x ) is either increas ing or dec reasing in time. That is, it is of pra ctical impor t a nce to c haracterize the aging class of underlying random lifetimes. In particular, since the IFRA class of a ging is one o f the most impor t a n t aging classes, testing that the distribution F has a constan t hazard rate against the h yp othesis that F is IFRA has b een studied extensiv ely in the literature; see for example, Deshpande (1983), Ko c har (1985), Link (198 9), Ahmad (20 00) a nd El- Bassioun y (2003) a mong others. In f a ct, F is IFRA if and only if E − 1 λ F ( x ) x is nondecreasing in x ≥ 0 or equiv alen tly ˜ r F ( F − 1 ( u )) ˜ r E ( E − 1 λ ( u )) is nondec reasing in u ∈ (0 , 1) where E λ is an exponential distribution with mean λ . T his implies that F ages faster than E , i.e., F is more IFRA than E λ . In order to ev aluate the p erfor ma nce of an appliance, we need to compare its aging b eha vior with some distributions other than exp onential distribution such as the W eibull, gamma, linear failure rate or ev en an unkno wn distribution G . The notion of the star or der that establishes an equiv alen t class of distributions is one of the useful to ols fo r t his comparison. Let Y b e another non-negativ e random v aria ble with distribution function G . W e s ay that X is less than Y with resp ect t o the star or der (w ritten b y X ≤ ∗ Y or F ≤ ∗ G ) if G − 1 F ( x ) is starshap ed o n [0 , ∞ ); tha t is, G − 1 F ( x ) x is nondecre asing in x ≥ 0. It is kno wn that F ≤ ∗ G ⇔ ˜ r F ( F − 1 ( u )) ˜ r G ( G − 1 ( u )) is nondecreasing in u ∈ (0 , 1) , (1.1) where ˜ r F and ˜ r G are failure rate av erage functions of F and G , resp ectiv ely . Using (1 . 1), the relation X ≤ ∗ Y is interpreted as X ages faster than Y and it is said that X is more IFRA than Y (cf. Ko c har and Xu, 2011 ). It is ob vious that if F ≤ ∗ G and G ≤ ∗ F then F ( x ) = G ( ax ) for all x ≥ 0 and some a > 0. In this case, we say F = ∗ G . 1 Izadi and Khaledi (20 1 2) ha v e considered the problem of tes ting the n ull hypothesis H 0 : F = ∗ G aga inst H 1 : F ≤ ∗ G and F 6 = ∗ G . They prop osed a test based on k ernel densit y estimation. In this pap er, w e further study this problem of testing in the one- sample as w ell as the t wo-sample problem a nd prop ose a new simple test based on a U-statistic. In b o t h cases, w e compare the new prop osed test with some w ell kno wn tests in the literature. It is found that our test is comparable to the others. T o e stablish our new test w e need the following lemm a. Lemma 1.1 L et X 1 , X 2 ( Y 1 , Y 2 ) b e two indep endent c opies of the r ando m varia b le X ( Y ) with distribution function F ( G ) an d let µ (2) F = E [max { X 1 , X 2 } ] ( µ (2) G = E [max { Y 1 , Y 2 } ] ) wher e E [ . ] is the exp e ctation op er ator. If F is mor e IFRA than G , then µ (2) F µ F ≤ µ (2) G µ G wher e µ F ( µ G ) is the e xp e ctation of F ( G ). Pro of: W e kno w that more IFRA order is scale in v ariant. Th us, X ≤ ∗ Y implies µ Y µ X X ≤ ∗ Y . No w, the require d result follo ws from T heorem 7.6 of Ba rlo w and Prosch an (1981 , page 122). Remark 1.1 The ab ove lemma has b e en pr ove d by Xie and L ai (1996) under the c ondition that F is m o r e IFR than G (for definition, se e Shake d and Shantikumar, 2007, p. 2 14) which is str onger than mor e IFRA or der. No w, let δ F = µ (2) F µ F , δ G = µ (2) G µ G and δ ( F , G ) = δ F − δ G . (1.2) It is obv ious that if F = ∗ G , then δ ( F , G ) = 0 and if F ≤ ∗ G and F 6 = ∗ G , then it follows from Lemma 1.1 that δ ( F , G ) < 0. That is, δ ( F , G ) can b e considered as a measure of departure from H 0 : F = ∗ G in fav or of H 1 : F ≤ ∗ G and F 6 = ∗ G . So, our test statistic is based on the estimation of δ ( F , G ). The organization of this pap er is as follows . In Section 2, w e pro p ose the new test for the case when G is kno wn. The case when G is unknown is studied in Section 3. In Section 4, the p erformance of our test is ev alua ted and compared. 2 2 The One-Sample Proble m Let G 0 b e a know n distribution function and X 1 , . . . , X n b e a random sample from an unkno wn distribution F . Now b y us ing the measure (1.2), the test statistic ˆ δ ( F , G 0 ) = ˆ δ F − δ G 0 is used for testing H 0 : F = ∗ G 0 against H 1 : F ≤ ∗ G 0 and F 6 = ∗ G 0 where ˆ δ F = X X i 6 = j max { X i , X j } n ( n − 1) ¯ X (2.3) and ¯ X is the mean of the random sample . In the next theorem, w e obtain the asym ptotic distribution of ˆ δ ( F , G 0 ) b y using the standard theory of U-statistics. Theorem 2.1 Supp ose E [max { X 1 , X 2 } − δ F 2 ( X 1 + X 2 )] 2 < ∞ . As n → ∞ , n 1 / 2 [ ˆ δ ( F , G 0 ) − δ ( F , G 0 )] is asymptotic al ly normal with m e an 0 and varianc e σ 2 F = 4 µ 2 F × V ar  X F ( X ) + Z ∞ X tdF ( t ) − δ F 2 X  . (2.4) Under H 0 , n 1 / 2 ˆ δ ( F , G 0 ) is asymptotic al ly normal with m e an 0 and varianc e σ 2 0 = σ 2 G 0 . Pro of: First note that ˆ δ F − δ F = X X i 6 = j  max { X i , X j } − δ F 2 ( X i + X j )  n ( n − 1) ¯ X = X X i 6 = j φ ( X i , X j ) n ( n − 1) ¯ X , 3 where φ ( X i , X j ) = max { X i , X j } − δ F 2 ( X i + X j ) . Let define T ∗ = X X i 6 = j φ ( X i , X j ) n ( n − 1) . By the standard theory of U-statistics, if E [ φ 2 ( X 1 , X 2 )] < ∞ , as n − → ∞ √ nT ∗ n σ ∗ d → N (0 , 1) where σ 2 ∗ = 4 × V ar ( φ 1 ( X )) and φ 1 ( x ) = E [ φ ( x, X ) ] . No w b y the strong la w of large num b ers w e ha ve ¯ X a.s. → µ F and hence, by Slutsky theorem √ n [ ˆ δ ( F , G 0 ) − δ ( F , G 0 )] is asymptotically no r ma l with mean 0 and v ariance σ 2 F = σ 2 ∗ µ 2 F . Under H 0 , δ ( F , G 0 ) = 0 and σ 2 0 = σ 2 G 0 .  A small v alue of ˆ δ ( F , G 0 ) indicates that testing H 0 against H 1 is significant. Th us, w e reject H 0 at lev el α if n 1 / 2 ˆ δ ( F , G 0 ) /σ G 0 < z α , where z α is α th quan tile of the standard normal distribution. In the case G 0 ( x ) = E λ ( x ) = 1 − exp( − λx ), x ≥ 0 and λ > 0, the problem is tes ting the null h yp othesis H 0 : F is an exp onen tial distribution against the alternativ e h yp othesis H 1 : F is IFRA and not exp o nential. It can b e sho wn that δ E λ = 3 2 and σ 2 E λ = 1 12 . By the a b o v e theorem, under H 0 , √ n ( ˆ δ F − 3 / 2 ) is asymptotically normal with mean 0 and v ariance 1 12 . Th us w e reject H 0 in fa v or of H 1 if √ 12 n ( ˆ δ F − 3 / 2) < z α . In the f ollo wing w e find the exact distribution o f ˆ δ F under t he hypothesis F is an exp o nen tia l distribution. First, no t e that w e can r ewrite ˆ δ F as ˆ δ F = 2 n X i =1 ( i − 1) X ( i ) n ( n − 1) ¯ X = n X i =1 c i : n D i n X i =1 D i (2.5) 4 where X ( i ) is the i th order statistic of X i ’s, D i = ( n − i + 1)( X ( i ) − X ( i − 1) ) , c i : n = 2 n X j = i ( j − 1) ( n − 1)( n − i + 1) and ass uming X (0) = 0. No w, by the same argumen t s as in Langen b erg and Sriniv asan (1979), w e will g et the follo wing result. Theorem 2.2 L et F b e an ex p onential distribution, then P { ˆ δ F ≤ x } = 1 − n X i =1 n Y j =1 j 6 = i c i : n − x c i : n − c j : n I ( x < c i : n ) (2.6) wher e I ( . ) is the usual indic ator function. By using Theorem 2.2, we tabulate the critical point of √ 12 n ( ˆ δ F − 3 / 2) under exp o- nen tia lit y f or small sample size s ( ≤ 40 ) in T able 1. So, for small sample size s, w e reject exp o nen tia lit y in fa v or of IFRA-ness if √ 12 n ( ˆ δ F − 3 / 2) is smaller than the critical p oint in T a ble 1 corresp onding with the lev el of significance chose n. El-Bassioun y (200 3) has considered the problem of testing expo nen tia lit y against IFRA- ness in the alternativ e and prop osed a class o f test. His test is based on the test statistics ˆ ∆ r +1 = 2 X X i 0, w e compare our prop osed test with the following w ell kno wn tes ts whic h are in the literature. Not e that in this case the problem is testing exponentialit y against IFRA-ness. Deshpande (1983): The test statistics is J b = 1 n ( n − 1) X X i 6 = j h b ( X i , X j ) , b ∈ (0 , 1) where h b ( x, y ) = 1 , if x > by ; 0, otherwise. Large v alues of J b are used to reject exp o nen tia lit y in fa v or o f IFRA- ness. It has been sho wn that under H 0 , n 1 / 2 ( J b − ( b + 1) − 1 ) is asymptotically normal with mean zero and v ariance 4 ξ where ξ = 1 4 { 1 + b b + 2 + 1 2 b + 1 + 2(1 − b ) b + 1 − 2 b b 2 + b + 1 − 4 ( b + 1) 2 } . Deshpande (1983) has recommended b = 0 . 9. 9 Ko c har (1985): H 0 is rejected for large v alues of T n = P n i =1 J ( i n +1 ) X ( i ) n ¯ X , J ( u ) = 2(1 − u )[1 − log (1 − u )] − 1 . (4.11) The asymptotic distribution of (108 n/ 17 ) 1 / 2 T n is the standard normal distribution. Link (1989): Large v alues of the test statistic Γ = 2 n ( n − 1) X X i z 1 − α . Ahmad has recommende d standard normal densit y as k ernel function and a n = n − 1 2 . First, w e in v estigate the accuracy of normal distribution as the limit distribution of the test statistics under H 0 . In o rder to do this, w e sim ulate the size of the t ests for nominal sizes α = 0.01, 0.05, 0.1 a nd large sample sizes n = 40(5)60(10) 7 0. In the sim ulation, 10000 samples are generated fr o m exponential distribution with mean 1. The calculated size is the prop or t ion of 10000 generated sample s that resulted in rejection of H 0 where the rejection regions hav e b een o btained b y the asymptotical distribution of test statistics. The sim ulated v alues w ere tabulated in T able 2. All sim ula tions w ere done b y R pac k age. F rom T able 2 , w e find that the tests b y Deshpande (1 983) and Ko c har (198 5) a re o v er sho ot the nominal sizes for all sample sizes. The sim ulated sizes of the tests due to Link (1 989) and Ahmad (20 0 0) are g reater than the nominal sizes but Link’s test alw ays dominates Ahmad’s test. It is clear from the con ten ts of T able 2 that the sim ulated size s of our new test are m uch closer to the nominal size s for all sample s izes. 10 T able 2: Sim ulated sizes of our test for differen t nominal sizes and larg e sample sizes n nominal s ize ( α ) n nominal s ize ( α ) 0.01 0.05 0.1 0.01 0.05 0.1 40 ˆ δ ( F , E ) 0.0104 0.0518 0.1 044 55 ˆ δ ( F , E ) 0.0108 0.0500 0.1 003 J 0 . 9 0.0637 0.1 243 0.1709 J 0 . 9 0.0550 0.1 101 0.1674 T n 0.0396 0.1 815 0.3157 T n 0.0346 0.1 565 0.2753 Γ 0.0181 0.0 612 0.1110 Γ 0.0151 0.0 569 0.1063 ˆ ∆ F 0.0311 0.0 772 0.1196 ˆ ∆ F 0.0324 0.0 783 0.1237 45 ˆ δ ( F , E ) 0.0106 0.0505 0.1 015 60 ˆ δ ( F , E ) 0.0101 0.0517 0.1 042 J 0 . 9 0.0661 0.1 163 0.1769 J 0 . 9 0.0494 0.1 128 0.1680 T n 0.0380 0.1 704 0.2938 T n 0.0349 0.1 548 0.2666 Γ 0.0167 0.0 584 0.1089 Γ 0.0157 0.0 581 0.1095 ˆ ∆ F 0.0338 0.0 796 0.1232 ˆ ∆ F 0.0304 0.0 742 0.1209 50 ˆ δ ( F , E ) 0.0112 0.0494 0.1 009 70 ˆ δ ( F , E ) 0.0090 0.0489 0.1 024 J 0 . 9 0.0601 0.1 220 0.1736 J 0 . 9 0.0469 0.1 048 0.1587 T n 0.0371 0.1 654 0.2852 T n 0.0303 0.1 410 0.2534 Γ 0.0166 0.0 563 0.1055 Γ 0.0147 0.0 580 0.1072 ˆ ∆ F 0.0312 0.0 810 0.1232 ˆ ∆ F 0.0310 0.0 783 0.1240 In the following, to assess how our prop osed test p erforms relativ ely , w e first con- sider the lar ge sample sizes and us e the measure of Pitman’s a symptotic relativ e effic iency (P ARE) (cf. Nikitin, 1995, Section 1.4). Consider testing H 0 that F is an exp onen tial distribution against H 1 that F = F θ n where θ n = θ 0 + k n − 1 2 , k is an arbitrary p ositiv e constan t and F θ 0 is exp onen tial. Then, Pitman’s asymptotic efficiency (P AE) o f a test based on statistic T n is P AE ( T n ) = lim n →∞ h ∂ E θ ( T n ) ∂ θ | θ = θ 0 i 2 V ar θ 0 [ √ nT n ] . (4.14) Using (4.14), the P AE of our test is give n by P AE ( ˆ δ ( F , E λ )) = ( ∂ δ F θ ∂ θ | θ = θ 0 ) 2 σ 2 F θ 0 . W e conside r three families o f W eibull, Linear fa ilure rate and Make ham distributions with the follo wing densit y functions. (1) W eibull Distribution: f θ ( x ) = θ x θ − 1 e − x θ , x > 0 , θ ≥ 1 . 11 (2) Linear F ailure Rate Distribution: f θ ( x ) = (1 + θ x ) e − x − θx 2 2 , x > 0 , θ ≥ 0 . (3) Mak eham Distribution: f θ ( x ) = (1 + θ (1 − e − x )) e − x − θ ( x + e − x − 1) , x > 0 , θ ≥ 0 . P AE of our test ( ˆ δ ( F , E λ )), Deshpande’s test ( J b ), Ko char’s t est ( T n ), Link’s test (Γ) a nd Ahmad’s test ( ˆ ∆ F ) are presen ted in T able 3 . In T able 4, P ARE of our test with respect to the others has b een obtained. It is observ ed that o ur test dominated the others exc ept Ko c har’s test for the LFR alternativ e case. T able 3: P AE of ˆ δ ( F , E λ ), J 0 . 9 , T n , Γ, ˆ ∆ F . T est  H 1 W eibull LFR Mak eham ˆ δ ( F , E λ ) 1.4414 0.75 0.0833 J 0 . 9 1.35 0.3369 0.0666 T n 1.247 0.8933 0.0784 Γ 1.3867 0.2681 0.0563 ˆ ∆ F 1.35 0.3375 0.0667 T able 4: P ARE( ˆ δ ( F , E ) , T ) = P AE ( ˆ δ ( F ,E λ )) P AE ( T ) ; T = J 0 . 9 , T n , Γ, ˆ ∆ F . T est \ H 1 W eibull LFR Mak eham J 0 . 9 1.0677 2.222 1.2489 T n 1.1558 0.8396 1.0625 Γ 1.0394 2.7974 1.479 ˆ ∆ F 1.0677 2.222 1.2489 In practice, the av ailable samples are small. So, it is imp ortan t to inv estigate the p o w er of the tests and compare them for small sample sizes. Prop ortio n o f 10000 samples (with small sizes 5 (3)15) that reject exponentialit y in fav or of IFRA-ness is considered 12 for estimating the p o wer of the tests. In the alternative, w e cons ider W eibull, LFR and Mak eham distributions. The critical p o ints of J 0 . 9 , T n , Γ and ˆ ∆ F at significance lev el α = 0 . 0 5 for small sample sizes ha v e b een deriv ed from their corresp onding pap ers. T able 5 sho ws the sim ula t ed p o wers of the tests for differen t alt ernat ives . It is observ ed tha t in W eibull and Makeham alternativ es our new test is more p ow erful than the others in all sample sizes. In the LF R alternativ e, Ko c har’s test dominates the other tests while our proposed test is compara ble. Als o, Ko c har’s test and Link’s test are comparable in W eibull and Mak eham alternativ es and a r e more p ow erful than the tests of Deshpande and Ahmad. T able 5: Sim ulated p ow er of the tests at lev el of significance 0.05 for small sample sizes. n W eibu ll( θ ) LFR( θ ) Mak eham( θ ) 1.2 2 3 0.2 1 2.5 0.2 1 2.5 5 ˆ δ ( F , E λ ) 0.0902 0.3886 0.7496 0 .0645 0.1029 0.142 2 0.057 9 0.082 1 0.1111 J 0 . 9 0.0658 0.17 82 0.3157 0.0601 0.0757 0.0944 0.0 520 0.0685 0.0851 T n 0.0897 0.36 77 0.7290 0.0645 0.1027 0.1410 0.0 583 0.0833 0.1112 Γ 0.0826 0.36 18 0.7118 0.0585 0.0961 0.1333 0.0 541 0.0766 0.1037 ˆ ∆ F 0.0781 0.32 58 0.6809 0.059 0.0637 0.0358 0.0 509 0.0519 0.0391 7 ˆ δ ( F , E λ ) 0.1117 0.556 0.9309 0 .0686 0.1292 0.185 4 0.061 7 0.092 7 0.1366 J 0 . 9 0.0784 0.21 53 0.4461 0.0549 0.0812 0.1032 0.0 529 0.0666 0.0881 T n 0.1084 0.52 55 0.9181 0.0664 0.1266 0.1846 0.0 608 0.0906 0.1314 Γ 0.1100 0.54 49 0.9176 0.0706 0.1260 0.1796 0.063 0.0974 0.1 395 ˆ ∆ F 0.0988 0.46 00 0.8758 0.0602 0.0806 0.0668 0.0 533 0.0612 0.0556 9 ˆ δ ( F , E λ ) 0.1294 0.7008 0.9818 0 .0732 0.1497 0.213 1 0.064 5 0.108 7 0.1562 J 0 . 9 0.0857 0.311 0.6371 0.0622 0.0895 0.1222 0.0 569 0.0797 0.1043 T n 0.1227 0.67 05 0.9763 0.0741 0.1469 0.2087 0.0 658 0.1064 0.1494 Γ 0.1284 0.67 98 0.9708 0.0697 0.1364 0.1914 0.0 679 0.1072 0.1520 ˆ ∆ F 0.1068 0.58 28 0.9476 0.0630 0.0952 0.1032 0.0 557 0.0710 0.0804 11 ˆ δ ( F , E λ ) 0.1440 0.8038 0.9967 0 .0767 0.1773 0.287 2 0.066 0.1285 0.199 4 J 0 . 9 0.0864 0.37 79 0.7468 0.0632 0.1094 0.1447 0.0 605 0.0795 0.1154 T n 0.1345 0.77 04 0.9953 0.0770 0.1774 0.2830 0.0 638 0.1227 0.1910 Γ 0.1419 0.77 75 0.9931 0.0717 0.1563 0.2494 0.0 657 0.1202 0.1844 ˆ ∆ F 0.1204 0.68 67 0.9857 0.0695 0.1193 0.1452 0.0 571 0.0849 0.1033 13 ˆ δ ( F , E λ ) 0.1629 0.8777 0.9989 0.0882 0.2042 0.3229 0.0 716 0.1303 0.2333 J 0 . 9 0.1011 0.447 0.8511 0.0737 0.1209 0.1619 0.0 589 0.0875 0.1269 T n 0.1526 0.85 12 0.9985 0.0876 0.2026 0.3213 0.0 718 0.1247 0.2274 Γ 0.1557 0.84 94 0.997 0.0787 0.1767 0.2771 0.0 676 0.1215 0.2076 ˆ ∆ F 0.1326 0.76 73 0.9942 0.0727 0.1365 0.1750 0.0 629 0.0903 0.1278 15 ˆ δ ( F , E λ ) 0.1764 0.925 0.9999 0 .0926 0.2246 0.376 0 0.068 6 0.149 3 0.2554 J 0 . 9 0.1074 0.52 71 0.9143 0.0705 0.1259 0.1802 0.0 613 0.0895 0.1430 T n 0.1665 0.90 09 0.9998 0.0942 0.2230 0.3734 0.0 687 0.1422 0.2511 Γ 0.1800 0.90 67 0.9995 0.0862 0.1903 0.3217 0.0 664 0.1362 0.2345 ˆ ∆ F 0.1439 0.82 61 0.9984 0.0767 0.1466 0.2085 0.0 606 0.1005 0.1474 13 4.2 The T w o-Sample As men tioned in the intro duction, Izadi and Khaledi (2012) prop osed and studied a test for the t w o -sample problem based on kerne l densit y estimation for testing H 0 : F = ∗ G against H 1 : F ≤ ∗ G & F 6 = ∗ G . Their test statistic is ˆ ∆( F , G ) = 1 n 2 a n n X i =1 n X j =1 X i k ( X i − X j a n ) − 1 m 2 b m m X i =1 m X j =1 Y i k ( Y i − Y j b m ) where k is a known symmetric and b ounded densit y function and a n and b m are tw o sequence s of p ositiv e real num b ers. k and a n are kno wn as kerne l and bandwidth, resp ec- tiv ely . In this section, w e compare the empirical p ow er o f our new test with the Izadi and Khaledi’s test when the k ernel, k , is the dens ity function of the standar d normal distribu- tion and a n = n − 2 / 5 and b m = m − 2 / 5 . W e kno w that the gamma a nd W eibull family are decreasing with respect to the shap e parameter in the more IFRA order (cf. Marshal and Olkin, 20 0 7, Chapter 9) . Also, Izadi and Khaledi (2 012) sho wed that the b eta family with densit y function f ( x ) = x a − 1 (1 − x ) b − 1 β ( a, b ) , x ∈ [0 , 1] , a, b > 0 . (4.15) is increasing with resp ect t o b in the more IFRA o rder. So, to ev alua te the p ow er of the tests w e use the g amma, W eibull and b eta families denoted b y G ( α, β ), W ( α, β ) and B ( a, b ), respectiv ely , in the alternativ e h yp o t hesis. In T able 6, w e generated 10000 samples with sizes n = m = 20 , 30 , 40 , 50 , 100 from distribution F and G g iven in the table. W e observ e that the e mpirical p ow er of our new test is greater than the empirical p o w er of Izadi and Khaledi’s test when F and G b elong to W eibull family and is smaller when F and G belong to the gamma and beta families. So, our new test is comparable to Iz adi and Khaledi’s test. 5 An a pplic ation In t his section w e apply our test on a data set f rom Nelson (1982, page 529) whic h is a life test to compare tw o differen t (old and new) sn ubb er designs. Let F ( G ) b e the 14 T able 6: The empirical p ow er of the tests ˆ δ ( F , G ) and ˆ ∆( F , G ) Distribution n = m F G test 20 30 40 50 100 G (3 , 1) G (1 . 5 , 1) ˆ δ ( F , G ) 0 . 415 0 . 582 0 . 6764 0 . 7 692 0 . 9589 ˆ ∆( F , G ) 0 . 470 0 . 642 0 . 7554 0 . 8 27 0 . 9712 G (4 , 1) G (2 , 1) ˆ δ ( F , G ) 0 . 435 0 . 570 0 . 681 0 . 785 0 . 966 ˆ ∆( F , G ) 0 . 492 0 . 672 0 . 755 0 . 825 0 . 968 W (3 , 1) W (1 . 5 , 1) ˆ δ ( F , G ) 0.7946 0.9346 0.9804 0 .9940 1 ˆ ∆( F , G ) 0.6714 0.8562 0.9446 0 .9796 1 W (4 , 1) W (2 , 1) ˆ δ ( F , G ) 0 . 811 0.9312 0.9766 0 . 993 1 ˆ ∆( F , G ) 0 . 72 0 . 893 0 . 954 0 . 985 1 B (1 , 1 . 5) B (1 , 3) ˆ δ ( F , G ) 0 . 1292 0 . 1736 0 . 2332 0 . 2692 0 . 4396 ˆ ∆( F , G ) 0 . 1452 0 . 2048 0 . 2806 0 . 3314 0 . 5504 B (1 . 5 , 2) B (1 . 5 , 5 ) ˆ δ ( F , G ) 0 . 166 0 . 209 0 . 278 0 . 364 0 . 597 ˆ ∆( F , G ) 0 . 383 0 . 517 0 . 585 0 . 653 0 . 938 15 distribution of lifetime old ( new) design p opulation. In F ig. 5, Izadi and Khaledi (2012) plotted TTT -plots f or b o th dat a sets of old and new design. The graphs an ticipated IFRA p opulations for b oth p opulat io ns. No w we apply our IFRA test on the t w o data sets. Using o ur one sample test, w e get that √ 12 n ( ˆ δ F − 3 / 2 ) = − 3 . 579222 and √ 12 m ( ˆ δ G − 3 / 2 ) = − 3 . 085525 whic h are less than − 2 . 376 441 (the critical v alue at lev el of significance α = 0 . 01 from T able 1). So, our tes t reject exponentialit y of b oth p opulation in fa v or of IFRA-ness. T o compare t w o p opulations with resp ect to more IFRA order, the test statistic v alue of the tw o sample problem is √ N ˆ δ ( F , G ) / ˆ σ F ,G = − 0 . 3762 ≮ − 2 . 326348 = z 0 . 01 . So, at lev el of significance α = 0 . 0 1, the equalit y of tw o p o pula t io ns in more IFRA order is not rejected. 6 Summary and Con clusion In o rder t o ev aluate the p erformance of an appliance, w e need to compare its aging b e- ha vior with some distributions suc h as exponential, W eibull, gamma, linear failure rate distributions. The notion of the star or der ( denoted b y ≤ ∗ ) is one of the useful to ols for this comparison b et w een tw o distributions. In this paper, w e ha v e intro duced a new s imple test for the pro blem of testing H 0 : F = ∗ G against H 1 : F ≤ ∗ G and F 6 = ∗ G . In t he one-sample problem, let X 1 , . . . , X n b e a random sample from F and G = G 0 where G 0 is a kno wn distribution. H 0 is rejected at lev el o f significance α , fo r large sample size, if n 1 / 2 ( ˆ δ F − δ G 0 ) /σ G 0 < z α , where δ G 0 = E G 0 [max { X 1 , X 2 } ] µ G 0 , ˆ δ F = X X i 6 = j max { X i , X j } n ( n − 1) ¯ X and σ 2 G 0 = 4 µ 2 G 0 × V ar G 0  X G 0 ( X ) + Z ∞ X tdG 0 ( t ) − δ G 0 2 X  . In particular, when G 0 is an exponential distribution, the null h yp othesis in fa v or of IFR A- ness is rejected, if √ 12 n ( ˆ δ F − 3 / 2) < z α . The exact n ull distribution of the tes t statistic has b een obta ined and, for small sample sizes 2(1)40 , the exact critical p o in ts of the 16 test statistics ha ve b een computed. Based on P itman’s asymp totic relativ e efficiency and sim ulat ed p ow er, we hav e compared our test with the tests given by Deshpande (1983), Ko c har ( 1 985), Link (1989) and Ahmad (2000 ) . The results sho w ed that our test relativ ely dominates the other tests. In t wo-sample pro blem, let X 1 , . . . , X n and Y 1 , . . . , Y m b e t wo ra ndom samples fro m F and G respective ly . F or large sample sizes, w e reject H 0 in fa v or of H 1 if √ N ( ˆ δ F − ˆ δ G ) / ˆ σ F ,G < z α where N = n + m and ˆ σ F ,G has been give n in (3.10). Using sim ulation study , w e ha v e sho wn that our test in this case is comparable with the test of Izadi and Khaledi (2012). Ac kno wledgemen t : The authors w ould lik e to thank the anon ymous asso ciate editor and the referee for their v a luable comme n ts leading to the impro v emen t of our man uscript. The researc h of Baha-Eldin Khaledi is partially supp orted from Ordered and Spatial Data Cen ter of Ex cellence of F erdowsi Univers ity of Mashhad. References Ahmad IA (2000 ) T esting exponen tiality against p ositiv e ageing using k ernel metho ds. Sankhy¯ a Series A 62:244–257. Barlo w RE, Prosc han F (19 81) Statistical theory of reliability and life testing. T o Be gin With, S ilver Sprin g. 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