A life test experiment based on  progressively group censored sample with periodic change in the failure  rate of an exponential distribution is considered. Estimates of the two  failure rates under different conditions of usage of an item together  with their asymptotic standard error are obtained by the method of  maximum likelihood. A numerical example is given using the data  available in the form of grouped observations under two conditions of  usage during alternate time intervals of fixed lengths T/Sub1 and  T/Sub2

Gajjar, A. V.

Patel, I. D.

English

Defence Science Journal

LIFE TEST BASED ON PROGRESSIVELY GROUP-CENSORED SAMPLES FROM EXPO- NENTIAL DISTRIBUTION WITH PERIODIC CHANGE IN FAILURE RATE I.D. PATEL & A.V. GAJJAR Gujarat University, Ahmedabad-380009 (Received '0 .J~l,me 1976; revi~ed 20 klcirch 1979) A life test experiment based on progressively group-censored sample with periodic change in the failure rate of an exponential distribution is considered. Estimates of the two failure rates under two different conditions of usage of an iten1 together with their asymptotic standard errors are obtained by the method of niaximiirn likelihood. .4 numerical example is given using the data available in the for111 of grouped observations tinder two conditions of wage during alternate tirnc intervals cf  fixed lengths T, and T2. K E Y  W O R D S  Life test Failure rate Periodic change . . * Progressively Parameters Censored Exponential distribution Grouped observations Estimation In life testing;% is a common practice to terminate the experiment when a certain number of items have failed or'a certain stipulated time has elapscd. Cohenl i!~:roduced progressively censored samples in life testing and recommended their use when at various stages of an experiment some of the surviving items are withdrawn (censored) from further observation and the experiment is continued with the remaining items. Several writers have investigated the maximum likelihood,estimation of the parameters of the normal, log-normal, exponential and Weibull distributions when samples are progressively censored. . There are situations in which life-test is carried out periodically and the items that have failed during each time interval (Ti -,, Ti), are counted and some fixed number of items selected at random from the surviving ones is withdrawn (censored) at fixed points of censoring T ,  (i= 1,2,. . . , k). This kind of experi- mentation stems from economic or practical considerations, where it may not be appropriate to collect the exact failure times of items on test and to continue testing until all the items have failed. Srivastavaz has considered a life-test based on grouped observations from an exponential distribution under the assump- tion that the failure rate is constant though different under two different conditions of usage. A trivariate and a multivariate extensions of this life test have recently been studied by Patel & Gajjar3. According to Srivastavaz, we have the following scheme Cycle Phase Duration of time Failure rate j 7 { ( j  1 ) ( T 1 + T 2 ) ,  ( j - - - I )  ( T l - F T 2 )  - ! - T I }  81 2 { ( j - - - 1  ) ( T I  + T , )  + Ti7 TI + TJ 1 8, where J = 1, 2,. . . We note that each cycle is of time duration (T,+T2), the failure rate being 8, during the first phase of duration TI and 8 ,  during the second phase of duration T,. The maximum likelihood estimates of 8, and 8, have been obtained by Srivastava when the sample is singly or doubly truncated/censored and the sample observations are grouped in each phase of the cycle. In this paper, the maximum likklihood estimates of the parameters 0, and 8, along with their asymptotic standard errors are obtained when the sample is pro- gressively censored. It is interesting to note that the maximum likelihood estimating equations obtained by Srivastava2 an be deduced as particular cases of those in section under "Maximum ~ikelihood Estimation" --- of this pnpcr. T H E  M O D E L  A N D  L I K E L I H O O D  Let X, the life span of an item, follow an exponential distribution with density / ( . I : ,  0) 7 - -  0 c-oa:, r01. 3; ) 0, 0 > 0, Dm. SCT. J., VOL. 29, OCTOBER 1979 When the item on test is subjected to conditions of usage, the density function of X is f  Jj) ( x )  = 8, e-8, r -1- (j - I )  (0 ,  -. 8,)T7, , ror ( . j  -. 1) (TI -i < 3: < (,i 1 )  (TI j- F ~ )  - 1  T~ f  ( x )  = 1 ,fp(j) ( 2 )  = f 1 2  6 0 ,  z - j  (el - 8,) !PI. where j = 1, 2,. ..... The corresponding distribution function is C - 4 z  + ( j - - l ) ( e ~ - - @ , ) T ,  , I for ( ( j 1 ) (TI  -t- T2) < 2 < ( j - 1) (TI  -+ T ~ J  + p1 1 (') = F~ (j) ($1 = 1 - e-82z - j (8 ,  - 0, ) 1'1 , I 1 for ( (.j 1 )  ( T I  + T2) 4- T I  < z d j  (Tl  + T2)  1 (3) ....... where j = 1, 2,. Let N denote the total number of items put on test. We shall use the following notations : ni j  = number of items which fail during the i-th phase of the j-tli cycle. 1 r 4 j  = fixed number of items which are withdrawn after the i-th phase of the j-tb cycle. 1 (4) Thus, we have i V = f + n , C ~ 1 . ~ + r l + r 2  ( 5 )  where f = number of failed items (not known) before the start of the test, .i = 2 n ; j ,  ri = 5 . I I (6) j = 1  j = 1  J ,.I .s** Since the test is terminated after the k-rlt cycle, Eqn. (5) gives , +- Without any loss of generality, we may assume that the experiment is observed from the (m+l)-th cycle to the (rn+k)-th and it is terminated after the completion of the (m+k)-th cycle. Thus the number f of the items failing before the beginning of the (ti1 f I)-fk cycle is regarded as unknown. Since the sample is censored at fixed points of time, we have the case of type I progressive group-censoring. Using the notations (4) and (6), the likelihood can be written as PATEL & GAJJAR : Life Test Based on l'rogressivzly Group-Censored Samples where M A X l M U M  L I K E L I H O O D  E S T I M A T I O N  Using (2), (3) and (9) the logarithm of the likelihood (8) can be written as Differentiatiqg (10) with respect to 8 ,  and 8, , in turn, we get 3 logqd$, =fm 4 (c," c?) 11, + n, I; 4 (c,) -- mnl T1 - - 1: U - T I - 1 -  1 )  (Il - 1  r - , - l 1  j - 1) , 1 ?by -, vl j  -1- r2j) 1 ( 1  1 )  j - 1  and 2 log L/aOa = ftn 4 ( c i )  c?) T, 4- rb, T_ 4 (c,) - ~ n n ,  11'2 - wrT, ( r e l  $ r2 -k n2) -- where j = l  4 ( t )  = t / ( l  l). Equating ( I  1) and (12) to zero, we get from the maximunl likelihood equations and 8,  T, =. log { 1 -1- ?&,/[ 5 (dl,  0,) --- n, .r,l } where f (dl,  0,) =:- (Z f ~ n  4 (c,*jb c p )  ( I f;) in which s 11 = 71% (aL + 4 -t (m + 1 )  (9-1 -t r,) -1- t z2  + ($ -- 2 )  ( H , ~  + n,, + rlj  + rZj )  .i: j = 1  P ~ t t i n g r , ~  = O f o r j =  1,2 ,......, kandrZj -- 0 f o r f =  1 , 2  ,...., k - - I , a n d ~ = r , ~ =  N - n , -  1 1 ,  -f in (14), (15) and (16), we get the maximum likelihood (22) and (23) obtained by Srivastava2. ],li/iol Approximate Estitnates Sillcc (14) and (15) arc tr:u~sccndcnti~l equations in 0, and 0:,, tllcy arc not  clirectly solvable. Therefore ; I I I I  t f  i u i i i o ~  I N I  I :  I I :  i ~ l y  I I I .  'rile i l t i t i ; I l  h c , l t l t i o n s  8," alld 8," with which the iteration could be slarlcd arc oblair~ccl us tir~tlcr.. DEE. SCI. J., VOI,, 29, OCTOBER 1979 We scc that l t l / r t ,  is tllc ohscrvctl ratio of the ~rt~riihcr of items failed during both tlic phases of the cyclu. 'I'l~is 1.t11ict c.o~~ltl, i l  Ill(* l ivql  : I ~ ~ ~ ~ I ' ~ I ~ ~ I I ~ : I ~ ~ I I I I .  Itc la kc^^ 11%1110 rtilio ol'll~c x(~cclcil t un10cr of litil~crd 0 ,  ' l ' , /U ,  'I',, r;i~lce 0 ,  it11t1 O, iIte ~c.q~~clivo I'i~illllc IXICS 101 tl~l'111io11~ 'i', 1111tl 'I\, I \ I ~ O $ \ ~ I I  i l l  aclvtu~cc ol' I l lo  cxpcrimcnt. Thus wc h:tvc 'It 0 ,  3'1 - . log Cl - - - -- - 982 e2 T, - log c, Now we write zi = 8,. 7'; ( = - log ci ) for C = 1, 2 nsct p = al/np . Thcn fro111 (l4), (16) and (17), wc gcl 6 (el, 8,) == ? A l  ( l?Ot 7', I ) I = gbg / (e% 7'1 -- 1 )  -1- 98, -1 r1 (18) 7 (1 - - / ( ' 1  -4- , 7 - I )  and ZIP 2 Z * .  We assume that Oi TS < < 1 ( i  = 1, 2), while n,, 1t2, r, and r, are moderate. Expanding the. exponential term in (18) up to the first order and noting zi  = Oi Tc (i = 1,2), we get 5 (91 9 8,) = -f 1 (21 + 22) + 9a1lz1 + 98g/z, + + 9.1 . (20) Hence using (19) and (20), the initial approximate estimate 8," of @a is given by Finally the approximate estimate 0," of 0,  can be obtaincd from (18). t, A S Y M P T O T I C  S T A N D A R D  E R R O R S  O F  T H E  E S T I M A T E S  In this section, we shall obtain the asymptotic standard errors of the estimates of 0, and B 2 .  Differentiating (11) and (12), again with respect to 8, and PI respectively, we get a2 log L 1 jO12 = - f nt2 TI2 4 ( c l m  c2)n) / ( 1 C? cam) - 9tl Tla 4 (el) / (1 - cl )  (22)  * ., 2 2  106 L 1 192 = fi)t2 1'2' 4 (clm c,") / ( 1  -- c,'" ~ 2 ~ )  - 9g2 Ta2 4 (c,) / (I - %)*e,fa - (23)  , f Let n1(m+33 and n2cm+j) denote the numbers of items entering the first and second phase respectively in the (m + j)-th cycle. Then we have nl (rnt.1) = N . f , 9b2 (')'+') = w (q"-l-l) - (,,,+I) - r1 (rn+l) l 12, (m+j) = n2 (7nfj-1) -- ?b2 (t,+j.-1) - r2 (m+j-1) n2 (tn4-j) = nl (m+j) - I l l  (m+j) - r l  (m+j) for .j = 2, 3,. . . . . ., k ~t is easy to verify that B ( al (m+j, ) = E ( nl ('"+j) ) . (I  - cl) , E ( m, (m+j, ) = E { 91, On+j) ) . ( L ca) , f o r j =  I ,  2, ...., k and E (f) = N (1 --clmc2") . P A T ~ L  & GAJJAR : Life Test Based on Progressi\ely Group-Censored Samples Taking expectations in (22), (23), (24) and using (26), we get] .&' ( - : a log L / : 8,2 ) = iVm2 T23 4 (rlm c?) -1- k 1 B ( - aa log L/aOl 8 8 ,  ) = nlms TIT2 4 ( clWL cam ). It may be noted that the expected values of n1 O n  -f- /) and n, (nt 1- 1) can be calculated recursively using the relation (25) and (26). A A The asymptotic standard errors of the estimates 6, and 0, of the parameters 8, and 8, can be obtained from 8 * . *a@ -1 .# A A 7 ~ V ( B : )  - y ~ o v ( ~ s y ~ l , ~ = ~ ~ ( - ~ 2 1 0 g m ~ )  B ( - ~ i ~ g ~ a ~ ~ a ~ )  A J I E ( - a2 log ~ 1 3 8 1  ae2) B ( - 3 2  log L / : o , ~  ) N U M E R I C A L  E X A M P L E  To illustrate the results we assume that the experiment is observed from the second cycle and terminated after the co~npletion of the fifth cycle, so that the number of items failed before the start of the second cycle i s  unknown. Here it is assumed that the observations come from an exponential distribution. The following table gives, for the experiment, the number of tubes failed during the first 16 hours and during the remaining 8 hours of each cycle of 24 hours together with the number of tubes withdrawn from the first and the secoqd phase of each cycle. Thus we have TABLE 1FREQUENCY nlSTRlDUTIO& 01: LIVES Ob VACUUM TIIRFS* Duration in hours Cycle Number of tubes failed Number of tubes withdrawn t\untbcr during from r C \ r- A 7 first phasc second phase first phase second phase "lj "z j  "i , r~ *Total number of tubes put on test is 300. DEF. SCI. J., VOL. 29, OCTOBER 1979 Thc i11iti;ll :~ly~roxi~ii;~tc! csfi~i~;~(cs oht;~i~ictl from (21) colnc out to bc 0",= 0.00678106 and Uop - 0.01 159343 . A A ' 1  llese eslil~~iltcs ar.e usctl I t ,  solvc: (1.4) ;111tl ( I S )  ~IIILI lle e?;l i l~lt~le~ 0 ,  tt lr i l  I),, of lilt! I~ILIIIIIIBICIL~ 1111'( U , ,  are . , A A 8 ,  = 0.0065265 and 8, = 0.0127326. Using (30), the asymptotic standard errors o f  these estimates are given by -1 \ A C K N O W L E D G E M E N T  Authors are grateful to the referee for making helpful suggestions in the inprovement of the paper. R E F E R E N C E S  1. COHEN, A. C., Te~htrornetrics, 5 (1963), 32?-339. 2. SRIVASTAVA, T. N., JASA, 70 (1975), 394-397. 3. PATEL, I. D:& GAJJAR, A. V., VI/II?Y,4. B .Scierrcer. 20 (1977), 55-71. fl 4. GAJJAR, A. V., 'Certain Problems in Life Testings-Thesis (Gijarat University, Ahmedabad), 1972. 

Life Test Based on Progressively Group-Censored Samples From Exponential Distribution With Periodic Change in Failure Rate

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Life Test Based on Progressively Group-Censored Samples From Exponential Distribution With Periodic Change in Failure Rate

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