MR(2000) ½ 62N01 / Î O213 A Ê (2012)

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1 Á41ÁÁ6 Vol.41, No ½ ADVANCES IN MATHEMATICS Dec., 2012 º È ² Âͺ ÑÜÖº ÄÅ «² 1,2,, «Ý 1, Đ 3 (1. ÉÐ «Ì È«É Æ ; 2. Ç «¼ «Ì «¼ Ç ; 3. È ««Ð ) 0 Ã É Ø Ù± ß Õ Ö ÜÛ ³ Ò Õ ÑÆÅ Ú EM Û Ð Ñ ³ Ð Ô ÞÛ Ú ³ ÕÓ Û Ø ³ Ò Õ EM MR(2000) ½ 62N01 / Î O213 A Ê (2012) Gupta Kundu(1999) [1] Ľ ß Á ÅÁ Gamma Weibull Î Â Æ Æ ÁÔ ± Þ Ë Æ ß Á ± Ú ÉÆ Ô¹È²Ë ÅÎ ß µý Ú Üµ «Ñß Á à ² [2 8]. Ú Æ È ² ½Ö ³¾ Æ [9 11]. EM ÎÅ Ê Ñ ß Á Ù ß Á Û ÕÌ Ô ½ f(x;α,λ) = αλ(1 e λx ) α 1 e λx, x 0, (1) S(x;α,λ) = 1 (1 e λx ) α, x 0, (2) h(x;α,λ) = f(x;α,λ) S(x;α,λ) = αλ(1 e λx ) α 1 e λx 1 (1 e λx ) α, x 0, (3) ÅÈ α,λ > 0 ¼Û»Ì α = 1 Û «Á Û Û (1) Ô h(x;α,λ) ϵ λ, ϵ α. α > 1 h(x;α,λ) ½Â α < 1 h(x;α,λ) ½Â³ α = 1 h(x;α,λ) ½² [4]. À 1 ² Ù À 2 EM Å Û Ù À 3 ÀÛ À 4 ¾ ß Á Û Î Ûß»º È«ÌÄÛß (No ) ÉÐ «±È«Ûß (ÛßÙ Ò ÊÇ Ü Ì No. 12XNH161). pole1999@163.com

2 756 «41Á 1 Đ ¹ «Ç ± (1) Ûݹ»³ ¹ Þ± Ú ß Á (1), Å F(x;α,λ) = (1 e λx ) α, x 0. (4) Ö n ± Ú¾ Ñ [0,+ ) N +1 ± N ± [T i 1,T i ), ÅÈ i = 1,2,,N; 0 = T 0 < T 1 < < T N < +. c i ¼ [T i 1,T i ) È µý± d i ¼ T i ˾ ± ² n = N (c i +d i ). L(α,λ) = Î logl(α,λ) = = N [F(T i ;α,λ) F(T i 1 ;α,λ)] ci [1 F(T i ;α,λ)] di N [( 1 e λt i ) α ( 1 e λt i 1 ) α ] ci ( [ 1 1 e λt i ) α ] di. { ci log [( 1 e λti) α ( 1 e λt i 1 ) α ] +di log [ 1 ( 1 e λti) α]}. Î Î logl(α,λ) α = = 0, { (1 e λti ) α log(1 e λti ) (1 e λti 1 ) α log(1 e λti 1 ) c i (1 e λti ) α (1 e λti 1 ) α d i (1 e λti ) α log(1 e λti ) (1 e λti ) α } logl(α,λ) λ = α = 0. { (1 e λti ) α 1 e λti T i (1 e λti 1 ) α 1 e λti 1 T i 1 c i (1 e λti ) α (1 e λti 1 ) α d i (1 e λti ) α 1 e λti T i 1 (1 e λti ) α } ¹ É α,λ Ù ½±µ Ù ¹ Ù Ó» ¹¾ À ( Ï ) Ù ½Ú ÑÖ EM Ó Ý Ù 2 Õ¹ Ðß 2.1 EM ¼Þ Dempster µ 1977 Ľ EM ½ Ê Î ² µ (missing data) Û MLE Æ ÁÏ Þ (observed data) Î MLE. Ç

3 6 ȹ ÈÒ Ò Â Ú 757 Z = (Y,X) ½±Þ Y µ X f(y,x η) ½ Z Õ Ì f(x Y,η) ½ÑÇÞ Y = y ÉµÑ µ X ɵÕÌ ÅÈ η ½ Ù η MLE ½ÊĐ Þ Y Î L(η Y) À Å L(η Y), Šѽ Î L(η Z) = log[f(y η) f(x Y,η)]. EM Í Í E M E ÑǼÀ η (0), À t 1 Æ η Ù η (t 1), Ç Î À Q Q(η η (t 1) ) = L(η Z) f(x Y,η)dX = E η (t 1){L(η Z)}. η (t 1) M Å Q(η η (t 1) ) η Ó η (t). ÍË E M ¹ Ù ÀÎ η (t) η (t 1) ÜµÞ ÜÀ ε Ã É Ø Å L(η Y) Ú µ Å Q È Ë η (0) ¼ à 2.2 Ó ÆÒÏ Þ n ± Ú X 1,X 2,,X n Ë Ë µ ß Á (1), Î n ± Ú¾ ÁÔ ± [T i 1,T i ) T i Ë ¾ ÔÄĐÞ ± Ú ± [T i 1,T i ) È X j c i T i Ë ¾ X j d i, ÅÈ i = 1,2,,N; j = 1,2,,n; 0 = T 0 < T 1 < < T N < +. ± Ú Å X = (X 1,X 2,,X n ), ½ X ½ ÉÞ EM ȵ µ ĐÞ Y = (c 1,c 2,,c N,d 1,d 2,, d N ), ÁÔ «Ø Z = (X,Y). EM Ô X ih,x il, ÁÔ ¼ ± [T i 1,T i ) T i Ë ¾ ± Ú ÑÖ ÔÒ EM È E M Î Ù Ù ±µ X Þ Y À² µ½ Ô² f(α,λ X,Y) = f(α,λ X). ± ß Á (1) ÕÌ Î ½ logf(α,λ X) = log = n log(αλ)+ N [{ ( ) αλ 1 e λx ih α 1e } λx ci ( ) ih { αλ 1 e λx il α 1e } λx di ] il { ci log [( 1 e λx ) ih α 1e ] λx ih +di log [( 1 e λx ) il α 1e ]} λx il. ÑÇ ¼À α (0),λ (0), EM Í E ÑÇ À t 1 Ù α (t 1),λ (t 1), À t Q Q(α,λ α (t 1),λ (t 1),Y) = E [ logf(α,λ X) α (t 1),λ (t 1),Y ] = n log(αλ)+ + c i E { log [( 1 e λx ) ih α 1e ] λx ih α (t 1),λ (t 1),Y } d i E { log [( 1 e λx il) α 1e ] λx il α (t 1),λ (t 1),Y }.

4 758 «41Á Q È X ih X il ɵ ÕÌ p ih (x) = f ih (x α (t 1),λ (t 1),Y) = α(t 1) λ (t 1) (1 e λ(t 1)x ) α(t 1) 1 e λ(t 1) x (1 e λ(t 1) T i) α (t 1) (1 e λ(t 1) T i 1) α (t 1), x [T i 1,T i ) µ½ p il (x) = f il (x α (t 1),λ (t 1),Y) = α(t 1) λ (t 1) (1 e λ(t 1)x ) α(t 1) 1 e λ(t 1) x, x [T 1 (1 e λ(t 1) T i) α (t 1) i,+ ). Q(α,λ α (t 1),λ (t 1),Y) Ti = n log(αλ)+ c i p ih (x) log[(1 e λx ) α 1 e λx ]dx T i d i p il (x) log[(1 e λx ) α 1 e λx ]dx. T i M Å Q α,λ À t Ù α (t),λ (t), Q(α,λ α (t 1),λ (t 1),Y) Î α,λ Q(α,λ α (t 1),λ (t 1),Y) Àà α (t),λ (t). Î α,λ Q α = n N α + Ti c i T i 1 p ih (x) log(1 e λx )dx+ + d i p il (x) log(1 e λx )dx, Q λ = n N λ + Ti c i p ih (x) x(αe λx 1) + T i 1 1 e λx dx+ d i p il (x) x(αe λx 1) T i 1 e λx dx. Q α = 0, Q λ = 0, É α = n Ti c i T i 1 p ih (x) log(1 e λx )dx+ N T i, (5) + d i T i p il (x) log(1 e λx )dx n λ =. (6) Ti c i T i 1 p ih (x) x(αe λx 1) 1 e dx+ N + d λx i T i p il (x) x(αe λx 1) 1 e dx λx Ö ½ (5), (6) ½À (α (t),λ (t) ), Æ (α (t 1),λ (t 1) ) (α (t),λ (t) ), Ë Ö (5), (6)» Å α,λ à 3 X i, i = 1,2,,n ½Î ß Á Û (1) Ë Ë ÔÅ [8] ȹ Û Ç α = 1.50, λ = 0.06, N = 9 10 T 0 = 0, T 1 = 5.5, T 2 = 10.5, T 3 = 15.5, T 4 = 20.5, T 5 = 25.5, T 6 = 30.5, T 7 = 40.5, T 8 = 50.5, T 9 =

5 6 ȹ ÈÒ Ò Â Ú , T 10 = +, Ì 0.001, Î j 8, ± T j ¾ j 9, T 9 À² µý ± Ê ¾ Å n = 60, 120, 200, 500, 1000 ÐË ¾ s = 100, 200, 500 Ù Ý À k ¾ Ù η k = (α k,λ k )(k = 1,2,,s), Ù À Ù Â mean j = 1 s s k=1 η k j, mse j = 1 s 1 s ( ) η k 2, j mean j ÅÈ η j ¼ η À j Ù Ú 1 2. À² ± Matlab2009b µ Ö 1 2 É Ä EM ε ³¾ Ñß Á ² Ù Ý Û Ë ÅÙ Ý Ó 1 ÅÆ ÁÌ ÐÚÔ Ë (mean) k=1 η = (α,λ) n s ˆα ˆλ E E E E E E 2 (1.5, 0.06) E E E E E E E E E 2 2 ÅÆ ÁÌ ÐÚÔ (mse) η = (α,λ) n s ˆα ˆλ E E E E E E E E E E E E 5 (1.5, 0.06) E E E E E E E E E E E E E E E E E E 6

6 760 «41Á 4 À ¹ Đ ÎØ Ð¹Í½Ý ÉÐÕ Ð À À «¾ µð Ñ 2418 Ð¹Í Ð [12] È Parker(1946) Ô ±½ Í ±«² 16 ± 15 ±³Ì I j = (j 1,j], j = 1,2,,15, I 16 = (15, ). Ó ±È µ 3 À¼ 3 ¾ ÚµÙ Interval Death numbers Outfollowed numbers I j D j W j [12] Ƚ ØÙ Ù Ô Í Ï À Ý À 10 ¼ º ± É Ô 10 «¼ Å Ê ÆĐ Í½» ÆÌ 5 ½ Å ß Á Û (1) EM α»ì λ Ù ˆα = 0.769, ˆλ = 0.106, Ô ½ Ŝ(x) = 1 (1 e 0.769x ) 0.106, x 0, (7) ĥ(x) = (1 e 0.769x ) e 0.769x 1 (1 e 0.769x ) 0.106, x 0. (8) ±µ Ù ˆα = < 1, Û Ô ½Â³ Ô 1. 1 Ä Ô ½ ų 2 Úμ À , À , À 3 ú À 10 ³¼Ñ À 10 ú À 30 ³ ²«Ñ º ³ Ò ÚÛ (7) (8),  ڽ ( ), 5 ½ ( [12] È ¼½ ). Ë ÚÁ ½ ³ Ú Ë t ³

7 6 ȹ ÈÒ Ò Â Ú 761 ÚÖ» µ(t) = 1 S(t) t S(x)dx Survival function Survival value years diagnosed 0.18 Hazard function 0.16 Hazard value years diagnosed 1 ĐºÎ ² Õ ± 1 ³ Ú½ ( ), 5 ³ Ú½ ( ), 10 ³ Ú½ ( ). ± Ë É ½ Å Ê ÆĐ Í½ ³ Ú ³Þ Ö Å ÍÄ Ç Ð¹Í Ý Î Ð¹Í Æ Â³ Ú Ä Õ [1] Gupta, R.D. and Kundu, D., Generalized exponential distributions, Austr. New Zealand J. Statist., 1999, 41(2): [2] Raqab, M.Z., Inferences for generalized exponential distribution based on record statistics, J. Statist. Plann. Inference, 2002, 104(2): [3] Sarhan, A.M., Analysis of incomplete, censored data in competing risks models with generalized exponential distribution, IEEE Trans. Reliability, 2007, 56(1): [4] Gupta, R.D. and Kundu, D., Generalized exponential distribution: existing results and some recent developments, J. Statist. Plann. Inference., 2007, 137(11): [5] Raqab, M.Z. and Madi, M.T., Bayesian inference for the generalized exponential distribution, J. Statist. Comput. Simul., 2005, 75(10): [6] Gupta, R.D. and Kundu, D., Generalized exponential distribution: Bayesian estimations, Comput. Statist. Data Anal., 2008, 52(4): [7] Kundu, D. and Pradhan, B., Estimating the parameters of the generalized exponential distribution in presence of hybrid censoring, Communications in Statistics Theory and Methods, 2009, 38(12):

8 762 «41Á [8] Chen D.G. and Lio, Y.L., Parameter estimations for generalized exponential distribution under progressive type-i interval censoring, 2010, 54(6): [9] Pettitt, A.N., Re-weighted least squares estimation with censored and grouped data: an application of the EM algorithm, Royal Statistical Society, 1985, 47(2): [10] Liu L.P. Estimation of MLE for Weibull distribution with grouped and censored data, Chinese Journal of Applied Probability and Statistics, 2001, 17(2): [11] Liu X., Chen H. and Fei H.L., Estimation of the parameters in the lognormal distribution with grouped and right-censored data, Chinese Journal of Applied Probability and Statistics, 2008, 24(4): [12] Lee, E.T. and Wang J.W., Statistical Methods for Survival Data Analysis(3rd Edition), New York: John Wiley & Sons, Parameters Estimation and Application of Generalized Exponential Distribution Under Grouped and Right-censored Data TIAN Yuzhu 1,2, TIAN Maozai 1, CHEN Ping 3 (1. Center for Applied Statistics, Renmin University of China, Beijing, , P. R. China; 2. School of Mathematics and Statistics, Tianshui Normal University, Tianshui, Gansu, , P. R. China; 3. Dependment of Mathematics, Southeast University, Nanjing, Jiangsu, , P. R. China) Abstract: Generalized exponential distribution is a class of important distribution in lifedata analysis, especially in some skewed lifedata. The estimation problem for generalized exponential distribution model with grouped and right-censored data is considered. The maximum likelihood estimators by using the EM algorithm are obtained. Some simulations are carried out to illustrate that proposed algorithm is effective to the model. Finally, a set of medicine data is analyzed by use of generalized exponential distribution. Key words: generalized exponential distribution; grouped and right-censored data; EM algorithm

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