Vol. 8 o. Joural of Systems Sciece ad Complexity Apr., 5 MOMET-METHOD ESTIMATIO BASED O CESORED SAMPLE I Zhogxi Departmet of Mathematics, East Chia Uiversity of Sciece ad Techology, Shaghai 37, Chia. Email: izhx@sia.com FEI Heliag College of Mathematical Sciece, Shaghai ormal Uiversity, Shaghai 34, Chia Abstract. I reliability theory ad survival aalysis, the problem of poit estimatio based o the cesored sample has bee discussed i may literatures. However, most of them are focused o MLE, BLUE etc; little work has bee doe o the momet-method estimatio i cesorig case. To make the method of momet estimatio systematic ad uifiable, i this paper, the momet-method estimatorsabbr. MEs ad modified mometmethod estimatorsabbr. MMEs of the parameters based o type I ad type II cesored samples are put forward ivolvig mea residual lifetime. The strog cosistecy ad other properties are proved. To be worth metioig, i the expoetial distributio, the proposed momet-method estimators are exactly MLEs. By a simulatio study, i the view poit of bias ad mea square of error, we show that the MEs ad MMEs are better tha MLEs ad the pseudo complete sample techique itroduced i Whitte et al.988. Ad the superiority of the MEs is especially cospicuous, whe the sample is heavily cesored. Key words. Reliability theory, survival aalysis, mea residual lifetime, momet-method estimatio. Itroductio For high reliability ad durability products, it will take much time to do life test to get the estimators of the reliability characteristics. Therefore, cesored samples are usually obtaied i either ormal or accelerated stress life tests. May authors have discussed the poit estimatio i the expoetial distributio, the two-parameter expoetial distributio, the Weibull distributio ad the logormal distributio, ad have got may satisfactory results o MLE, BLUE etc. For the complete sample, the momet-method estimatio was discussed i may literatures, e.g. A. C. Cohe ad R. Helm 4], A. C. Cohe ad B. J. Whitte 5 ], A. C. Cohe, B. J. Whitte ad Y. Dig,3] etc. However, it is a pity that the method of momet estimatio is less discussed for the cesored sample. Oe reaso is that i cesorig case the suitable momet equatios are very difficult to costruct. Mete Sirvaci ad Grace Yag 6] proposed estimators for the shape ad scale parameters i the Weibull distributio uder type I cesorig by a method of momet type argumet. Ad they derived the momet of the estimator for the shape parameter ad established the joit asymptotic ormality of both estimators. Cosiderig that the items still workig regularly at the cesorig time also cotai some iformatio, ad to make the method of momet estimatio systematic ad uifiable, i this Received October 3, 3. *This research is partially supported by atioal Sciece Foudatio of Chia o. 69976.
o. MOMET-METHOD ESTIMATIO BASED O CESORED SAMPLE 55 paper, the momet estimators of the parameters based o type I ad type II cesored samples are put forward accordig to the cocept of mea residual lifetime. As a special case, i the expoetial distributio, the proposed momet estimators are exactly MLEs. I Sectio, we review some cocepts ad give some otatios. Sectio 3 ad Sectio 4 discuss cases of type II cesorig ad type I cesorig respectively. I each case, we obtai the momet estimators ad prove the strog cosistecy ad other good properties. I Sectio 5, the authors compare the momet estimators with MLEs ad the pseudo complete sample techique itroduced i Whitte et al. 8] by simulatio i two-parameter expoetial distributio based o type II cesored sample. Ad the result of the simulatio shows that whe the sample is heavily cesored the momet estimatorsmes ad the modified momet estimatorsmmes proposed i this paper are obviously superior to MLEs, i the light of bias, robustess ad small-sample behavior. Basic Cocepts ad otatios. Basic Coceptscf. Rolski 7] Let X be the lifetime of oe product, with cdfcumulative distributio fuctio F x, i.e. X F x.. Residual Life Distributio Fuctio Suppose a product ca still work regularly after havig worked up to time t. Let F t x deote the probability it fails i the time iterval t, t x. The F t x = P X t x X > t = F x t F t, for x, F t, for x <. It s easy to prove that F t x is a distributio fuctio. Ad, F t x is called residual life distributio fuctio at time t.. Mea Residual Lifetime Let mt deote the time that the product ca keep o workig o coditio that it has worked up to time t. The, mt = = = x df t x F t F t x dx EX t ] F y dy. mt is called mea residual lifetime abbreviated to MRL at time t. The cocepts residual life distributio fuctio ad mea residual lifetime have may applicatios i such fields as reliability, survival aalysis, actuarial sciece, etc.. otatios Throughout this paper, the followig otatios will be used. X i, i =,,, deotes the lifetime of the ith product, ad x, x,, x are the correspodig observed values. X, X,, X are idepedet ad idetically distributed with F x, ad X : X : X : are the correspodig order statistics, with the observed values x : x : x :.
56 I ZHOGXI FEI HELIAG Vol. 8 3 Case of Type II Cesorig Suppose the lifetime of oe product is X, with the cdf. F x, ad products of this type are put ito test, which will be termiated whe just products fail. The obtaied sample is X : X : X :. As a special case, whe =, i.e. complete sample, it is well kow that the momet estimator ca be easily obtaied. Geerally, <, i.e. data are cesored o the right. It is importat to otice that these cesored data cosist of some iformatio which we ca ot easily fid. Oe reasoable solutio is to resort to the idea of data augmetatio. Thus, to get the momet estimator i this case, set where U = X i: X : F x: x = P X x : x X > x :, x df X: x ], ad x df x: x is the mea residual lifetime of the product at time x :. We ca easily have the followig theorem: Theorem 3. For geeral life distributios, there holds EU = EX. Proof To see this, let I i =, if X i X : ; ad = otherwise. Defie X i = I ix i I i gx :, where gx = EX X > x. The we have U = X i. It suffices to show EX i = EX i, which is equivalet to EX i I i = = EX i I i =. The latter implies for every x. otice that ad EX i I i =, X : = x = EX i I i =, X : = x, EX i I i =, X : = x = EX i X i > X :, X : = x = gx, EX i I i =, X : = x = gx. The we complete the proof. By Theorem 3., U = EX may be viewed as a ubiased estimatig equatio. Let X i: X : x df X: x ] = EX. 3. From 3. we ca get the momet estimator ˆ of the parameter i oe-parameter distributio. Corollary If X Exp, with cdf exp x, for x, F x = >,, for x <,
o. MOMET-METHOD ESTIMATIO BASED O CESORED SAMPLE 57 the the momet estimator ˆ = X i: X : is the ubiased estimator of, ad furthermore ˆ is MVUE ad MLE. Eq. 3. solves the problem of momet estimatio for the oe-parameter distributio. To obtai the momet estimators based o type II cesored sample i the distributio cotaiig two or three parameters, we eed more equatios. Therefore, set S = def = { Xi: X : X i: X : ] x df X: x ] } x df X: x {P Q}. 3. For the two-parameter expoetial distributio, we have the followig coclusio. Theorem 3. If X Exp, η, with cdf exp x η, for x η, F x = >,, for x < η, the EU = EX, ω = { ES = ω VarX, where, U, S are the same as above, ad i j i j i ] i l j= i= l= i m=i l= i i m i j= j= i j i i i j= i j l j i l= l= i j l= l j l ] j i j j l j= l= i ] l l= i ] i i l i= l= } i. l Proof The first equatio is obvious. We oly eed to give the proof of the secod equatio. By 3. { } ES = E P Q = EP EQ],
58 I ZHOGXI FEI HELIAG Vol. 8 where EP = EXi: X E : EQ = E EX i: X : Usig the above trasformatio, we have EP = x df X: x], E Z i: η E Z : η ] = E Zi: η EZ i: η ] x df X: x. E Z: η EZ: η ], ] EQ = E Z i: η Z : η. Accordig to Balakrisha ad Cohe ], EZ i: = i l= E Z: = EZ :, E Z i: l, i, = E Z i : EZ i:, i =, 3,,, EZ i: Z i: = E Z i: EZ i: Z j: = EZ i: Z j : i EZ i:, i, j EZ i:, i < j, j i. Usig the above equatios to simplify EP ad EQ, the the proof is completed. By Theorem 3. we ca costruct the followig momet equatios: U = EX, 3.3 S = ω VarX. 3.4 Whe X Exp, η, the momet estimators of the parameters based o type II cesored sample ca be obtaied from the followig equatios: η = X i: X :, ω A A X : B A =, 3.5
o. MOMET-METHOD ESTIMATIO BASED O CESORED SAMPLE 59 where A = X i: X :, B = ad ω is the same as before. Whe ad >, there holds that ω >, Xi: X :, ad also for B A >, >, the solutio to the above equatios is uique. The MEs of, η are ˆ = ˆη = A A X: A A X: ω X i: X : ˆ. B A, ω 3.6 Remark Whe =, it is easy to verify that the estimators give by 3.6 are just equal to the momet estimators derived from the complete sample by the geeral method. Remark ote that the above estimators are developed from Eq. 3.3 ad Eq. 3.4. We ca also use Eq. 3.3 ad the followig equatio 3.7 to get the estimators. X : = EX :. 3.7 We ame them the Modified Momet EstimatorsMMEs, ad they are give i the followig: ˆ = ] X i: X : X :, 3.8 ˆη = X : ˆ. The MMEs proposed above are exactly MVUEsSee ]. Cosiderig that i the expoetial distributio Exp, η, η is a threshold parameter, ad η ca be estimated by X : directly. Combiig with Eq. 3., we have ˆ = ] X i: X i: X :, 3.9 ˆη = X :. Remark 3 The estimators of ad η, ˆ, ˆη i 3.9, are exactly MLEs. Remark 4 ω ca be viewed as the oe coefficiet to elimiate the bias. ote that the umber ω i 3.6 is very complicated. I practical applicatios, ω ca be replaced by approximately. I fact, ω is very approximate to, which ca be oticed clearly i the case of type I cesorig. Ad the simulatio study shows that the replacemet is satisfactory. Theorem 3.3 If X Exp, η, ad is the predetermied umber of failures i type II cesorig life test, such that p, as, where p is a costat ad p,
6 I ZHOGXI FEI HELIAG Vol. 8 the there holds that ˆ, ˆη η as, with probability oe, i.e. ˆ, ˆη are the strog cosistet estimators of ad η respectively. The proof of Theorem 3.3 is give i Appedix. If X Wei, β, ad with the cdf { x } β exp, for x, F x = >,, for x <, its residual lifetime distributio is very complicated. Of course, we ca use Eq.3. ad Eq.3. directly to fid the MEs of its parameters. But, cosiderig that the computatio is very hard, the authors suggest the trasformatio Y = X β; ad as a result Y Exp. ext, usig the above method i oe-parameter expoetial distributio, the the estimators ca be obtaied from the followig equatios: β = X i: β X : β, β β X : β X : β X : β =. It is obvious that this idea ca be used i other distributio. 4 The Case of Type I Cesorig Suppose products are put ito type I cesorig life test, with their lifetimes X i idepedet ad idetically distributed with the distributio fuctio F x, i =,,,, ad the cesorig time T is predetermied. The obtaied sample is X : X : X r: T, where r is the observed umber of failures before time T. X i:, i =,,, r are real failure datas, while X i:, i = r, r,, are oly kow ot less tha T, i.e. X i: T, i = r, r,,. Aalogously to the case of type II cesored sample, set W = X i I,T ] X i I T, X i T x df T x ], where x df T x is the MRL of the product at time T ad I E x is the idicator fuctio of the set E, satisfyig {, if x E, I E x =, if x / E. Similarly, we have the followig theorem. Theorem 4. For geeral life distributios, there holds EW = EX. The proof is similar to that of Theorem 3., so omitted. Let X i I,T ] X i X i I T, X i T x df T x ] = EX. 4.
o. MOMET-METHOD ESTIMATIO BASED O CESORED SAMPLE 6 Eq.4. ca give the ME of the parameter i the oe-parameter expoetial distributio i type I cesorig case. Accordig to Theorem 4., we ca easily have the followig corollary. Corollary If X Exp ad T is cesorig time, the the ME of is r ˆ = X i: rt, r where r is the observed failure umber before the cesorig time T. It is easy to fid that ˆ is exactly MLE. For the two-parameter ad three-parameter distributios, similarly to Sectio 3, set S = def = { r Xi: T r r X i: r T ] x df T x ] } x df T x {R M}. 4. Theorem 4. If X Exp, η ad T is the cesorig time, the where W, S are the same as before, ad Proof ote that r = I,T ] X i, EW = EX, 4.3 ES = ω VarX, 4.4 ω = exp r X i: = X i I,T ] X i, r X i: = X i I,T ] X i, T η. η T EX i I,T ] X i = η T exp, η T EX i I,T ] X i = η η T T exp Usig the above equatios to simplify the left of 4.3 ad 4.4 respectively, the the proof will be fiished. By Theorem 4., if X Exp, η, we ca costruct the correspodig momet equatios, i which ω is estimated by r, the the MEs of ad η i type II cesorig case ca be obtaied.
6 I ZHOGXI FEI HELIAG Vol. 8 from the followig equatios r η r = X i: rt, r r r C C r rt D C =, where C = r X i: rt, D = r X i: rt. Similarly to the case of type II cesorig, it s easy to verify that whe > ad >, the solutio to the above equatios is uique, ad the MEs of ad η are C r C rt C C r rt r r r D r C ˆ = ˆη = r X i: rt rˆ. r r r r, As a special case, whe r =, the above estimators are exactly equal to the commo momet estimators obtaied from the complete sample. Theorem 4.3 If X Exp, η, ad T is the cesorig time i type I cesorig life test, for the ˆ, ˆη give by 4.6, there also holds that ˆ, ˆη η as, with probability oe. The proof of Theorem 4.3 is give i Appedix. As Sectio 3 describes, the MMEs of the parameters ad η ca be easily obtaied: r ] ˆ = X i: rt X :, r 4.7 ˆη = X : ˆ. As was stated above, we have solved the problem of the momet estimatio i two-parameter expoetial distributio based o type I cesored sample. For other distributios, we suggest the method itroduced i Sectio 3. 5 Mote-Carlo Simulatio ad Compariso I this sectio, we give a compariso of the above four estimators i the two-parameter expoetial distributio uder type II cesorig by simulatio Mote-Carlo rus. I the followig, PME deotes the momet estimator obtaied from pseudo complete sample see ]; SMSEME is a abbreviatio for the square root of the MSE of ME, ad SMSEMLE is a abbreviatio for the square root of the MSE of MLE.. X Exp5,4, = 3, =, the simulatio results are give i the followig: ME MLE MME PME SMSEME SMSEMLE ˆ 49.33 5.5595 5.754 49.55 53.79 53.77 ˆη 4.566 5.5957 3.998 5.33 6.959 6.95. X Exp6,6, = 3, =, the simulatio results are give i the followig: ME MLE MME PME SMSEME SMSEMLE ˆ 59.3396 6.66 6.3 58.58 6.75 6.4 ˆη 6.553 7.98 5.9765 7.956.495.3985 4.5 4.6
o. MOMET-METHOD ESTIMATIO BASED O CESORED SAMPLE 63 3. X Exp5,4, = 3, =, the simulatio results are give i the followig: ME MLE MME PME SMSEME SMSEMLE ˆ 49.8738 56.67 5.679 48.99 7.43.77 ˆη 4.866 5.58 3.835 5.989.4775.4768 4. X Exp6,6, = 3, =, the simulatio results are give i the followig: ME MLE MME PME SMSEME SMSEMLE ˆ 59.594 7.3384 6.368 57.936 5.84 5.88 ˆη 6.493 8.36 5.993 8.439 8.74 8.734 From the above simulatio results, we ca fid that the MEs ad MMEs are obviously superior to MLEs ad PMEs i respect of the bias, especially whe the samples are heavily cesored. Appedix Proof of Theorem 4.3 For X Exp, η, we have: T EX i I,T ] X i = x η exp x η η T dx = η T exp T EX i I,T ] X i = x η exp x η dx η T = η η T T exp. ote that r C = X i: rt = X i I,T ] X i I,T ] X i T, r D = X i: rt = X i I,T ] X i I,T ] X i T. By the classical strog law of large umbers of Kolmogorov, whe, with probability oe, C η T η exp, r r r C C r rt D C η T exp exp η T η T exp ] η T, η T exp, = X i I,T ] X i η T T η exp r C T exp η T,.
64 I ZHOGXI FEI HELIAG Vol. 8 Thus, with probability oe C C r rt r r η T exp r η T exp D C ] η T. Furthermore, by Slutsky theorem, we have whe, ˆ, with probability oe, i.e. ˆ is the strog cosistet estimator of. The proof of the cosistecy of ˆη is similar. The proof of Theorem 3.3 is similar to that of Theorem 4.3, so we omit it. Refereces ]. Balakrisha ad A. C. Cohe, Order Statistic ad Iferece, Academic Press, Ic, 99. ] J. L. Bai, Statistical Alysis of Reliability ad Life-testig Models, Marcel Dekker, Ic., ew York ad Basel, 978. 3] R. E. Barlow ad F. Proscha, Mathematical Theory of Reliability, Wiley, ew York, 965. 4] A. C. Cohe ad R. Helm, Estimatio i the Expoetial Distributio, Techometrics, 973, 4: 84 846. 5] A. C. Cohe ad B. J. Whitte, Estimatio i the three-parameter logormal distributio, J. Amer. Statist. Assoc., 98, 75: 399 44. 6] A. C. Cohe ad B. J. Whitte, Estimatio of logormal distributio, Amer. J. Math. Mgt. Sci., 98, : 39 53. 7] A. C. Cohe ad B. J. Whitte, Modified momet ad maximum likelihood estimators for parameters of the three-parameter gamma distributio, Commu. Statiat. Simul. Comput., 98, : 97 6. 8] A. C. Cohe ad B. J. Whitte, Modified maximum likelihood ad modified momet estimators for the three-parameter Weibull distributio, Commu. Statiat. Theor. Meth., 98, : 63 656. 9] A. C. Cohe ad B. J. Whitte, Mdified momet estimatio for the three-parameter iverse Gussia distributio, J. Qual. Tech., 985, 7: 47 54. ] A. C. Cohe ad B. J. Whitte, Modified momet estimatio for the threeparameter Gamma distributio, J. Qual. Tech., 986, 8: 53 6. ] A. C. Cohe ad B. J. Whitte, Parameter Estimatio i Reliability ad Life Spa Models, Marcel Dekker, ew York, 988. ] A. C. Cohe, B. J. Whitte ad Y. Dig, Modified momet estimatio for the three-parameter Weibull distributio, J. Qual. Tech., 984, 6: 59 67. 3] A. C. Cohe, B. J. Whitte ad Y. Dig, Modified momet estimatio for the three-parameter Logormal distributio, J. Qual. Tech., 985, 7: 9 99. 4] J. F. Lawless, Statistical Models ad Methods for Lifetime Data, Joh Wiley, 98. 5]. R. Ma, R. E. Schafer ad. D. Sigpurwalla, Methods for Statistical Aalysis of Reliability ad Life Data, Joh Wiley ad Sos, Ic., 974. 6] Mete Sirvaci ad Grace Yag, Estimatio of the Weibull parameters uder type I cesorig, J. Amer. Statist. Assoc., 984, 79: 83 87. 7] T. Rolski, Mea residual life, Bulleti of the Iteratioal Statistical Istitute, 975, 4: 66 7. 8] B. J. Whitte, A. C. Cohe ad V. Sudaraiyer, A pseudo-complete sample techique for estimatio from cesored samples, Commu. Stat. Theory. Meth., 988, 7: 39 58. 9] Y. S. Chow ad H. Teicher, Probability Theory: Idepedece, Iterchageability, Martigales, Spriger-Verlag, 978.