Comparison of Four Methods for Estimating. the Weibull Distribution Parameters

Transcription

1 Appled Mathematal Sees, Vol. 8, 14, o. 83, HIKARI Ltd, Comparso of Four Methods for Estmatg the Webull Dstrbuto Parameters Ivaa Pobočíková ad Zuzaa Sedlačková Departmet of Appled Mathemats Faulty of Mehaal Egeerg, Uversty of Žla Uverztá 1, 1 6 Žla, Slovaka Copyrght 14 Ivaa Pobočíková ad Zuzaa Sedlačková. Ths s a ope aess artle dstrbuted uder the Creatve Commos Attrbuto Lese, whh permts urestrted use, dstrbuto, ad reproduto ay medum, provded the orgal work s properly ted. Abstrat I ths paper we study the performae of the least square method, the weghted least square method, the maxmum lkelhood method ad the method of momets for estmatg the Webull dstrbuto parameters. The omparso s based o the Mote Carlo smulato, the methods are ompared terms of the root mea square error ad sample sze. The omparso shows that the maxmum lkelhood method ad the method of momets provde smlar estmates. We reommed the maxmum lkelhood method to estmate the Webull dstrbuto parameters due to ts good propertes. For very small sample szes we reommed the weghted least square method. Keywords: Webull dstrbuto, parameter estmato, least square method, weghted least square method, maxmum lkelhood method, method of momets, root mea square error 1 Itroduto The Webull dstrbuto s oe of the wdely used dstrbutos tehal prate. It s ofte used weather foreastg, the theory of relablty ad lfetme. Ths dstrbuto was frst trodued by the Swedsh setst Walod Webull ( ), who used t the theory of relablty. We osder the two-parameter Webull dstrbuto. The probablty desty futo of the Webull dstrbuto W(, ) wth parameters ad,

2 4138 Ivaa Pobočíková ad Zuzaa Sedlačková s gve by 1 x f( x) x exp, where x, s the shape parameter, ad s the sale parameter. The umulatve dstrbuto futo of the Webull dstrbuto s gve by x F( x) 1exp, x. The where m k of the Webull dstrbuto s gve by k k 1, th k momet k, 1,,..., m k a s the gamma futo defed by a1 x a x e dx a,. I ths paper we study the performae of the methods for estmatg the Webull dstrbuto parameters ad. Several methods are proposed to estmate the parameters. I ths paper we osder the ommoly used methods: the least square method (LSM), the weghted least square method (), the maxmum lkelhood method () ad the method of momets (). The LSM ad the are ommoly used due to ther smplty. The estmates of the parameters a be alulated easly by the losed-form formula. The ad the are popular methods, but both methods are omputatoally demadg. I the ase of the Webull dstrbuto ether method provdes a explt soluto for the estmates of the parameters the losed-form formula. The estmates of the parameters a be obtaed oly umerally. The performae of the methods s ompared usg the Mote Carlo smulato. The effey of the methods s ompared based o the root mea square error (RMSE) rtero ad the sample sze. Based o the smulato study we reommed the methods, whh have better performae. The smulatos ad the alulato are performed the Matlab. There are some reet works o estmatg the Webull dstrbuto parameters. Trustrum ad Jayatlaka (1979) ompared the LSM, the ad the based o the Mote Carlo smulato. Bergma (1986), Fauher ad Tyso (1988), Hug (1), Lu, Che ad Wu (4) reommed the, ths method outperforms the LSM. Wu, Zhou ad L (6) ompared the LSM, the, the ad the based o the Mote Carlo smulato. Chu ad Ke (1) ompared the LSM ad the based o the umeral smulato study. The rest of ths paper s orgazed as follows. I Seto we trodue the estmato methods. I Seto 3 the smulato study s provded ad fally, Seto 4 we summarze our fdgs. (1) () (3)

3 Comparso of four methods 4139 Estmato methods.1. Least square method Let X1, X,, X be a radom sample of sze from the Webull dstrbuto W(, ) ad let x 1, x,, x be a realzato of a radom sample. The umulatve dstrbuto futo () wll be trasformed to a lear futo. From () by two logarthm alulatos we obta l l 1 F( x) l xl. (4), X l x, 1 ad l. The the equato (4) a be wrtte as Y X (5) Let Y l l 1 F( x) 1. Now let X(1) X() X( ) be the order statsts of X 1, X,, X ad let x(1) x() x( ) be observed ordered observatos. To estmate the values of the umulatve dstrbuto futo F( x ) we use the mea rak Fx() 1, (8) where deotes the th smallest value of x (1), x (),, x ( ), 1,,,. Based o our prevously study Pobočíková ad Sedlačková (1) ths method a be mproved the performae of the estmates. The estmates ad 1 of the regresso parameters ad 1 mmze the futo Q(, 1) ( Y 1 l x( ) ). (7) 1 Therefore, the estmates ad 1 of the parameters ad 1 are gve by l () l l 1 () l () l l 1 () x F x x F x l x() l x() l l 1F x l ( x ) ( ) 1 1 The estmates ĉ ad of the parameters ad are gve by. (9) (8)

4 414 Ivaa Pobočíková ad Zuzaa Sedlačková 1, l l 1F x() l x() (1) 1 1 exp... Weghted least square method The estmates ad 1 of the regresso parameters ad 1 mmze the futo Q(, 1) w( Y 1 l x( ) ), (11) where w s the weght fator, 1,,...,. I ths paper we use the weght fator proposed by Bergma (1986) w 1 F( x() ) l 1 F( x() ), 1,,...,. (1) Therefore, the estmates ad 1 of the parameters ad 1 are gve by 1 w l () l l 1 () l () l l 1 w x F x w x w Fx(), ww l x() wl x() () () wl l 1F x wl x The the estmates ĉ ad of the parameters ad are gve by w (13). (14), exp. wl l 1F x wl x 1 w (15).3. Maxmum lkelhood method The lkelhood futo of the Webull dstrbuto s gve by

5 Comparso of four methods x (, ) (,, ) exp. 1 1 L f x x (16) The maxmum lkelhood estmates ĉ ad of the parameters ad maxmze futo (16) or, equvaletly, the logarthm of the futo (16) 1 l L (, ) l l x 1 l x. (17) 1 1 Dfferetatg (17) wth respet to ad tur ad equatg to zero, we obta the equatos l L (, ) x, 1 1 x l x l x 1 l 1 1 l. (18) l L (, ) x O elmatg from equatos (18) ad smplfyg we solve the equatos 1/ 1 x, (19) 1 x l x 1 x 1 x l. () The estmate ĉ of the parameter we obta by solvg () wth respet to. Ths equato has ot aalytal soluto ad must be solved umerally for. We use the Newto method. As the startg pot we use the result (1) obtaed from the LSM. The estmate of the parameter a be obtaed usg equato (19)..4. Method of momets The estmates ĉ ad of the parameters ad we obta by equatg the th k momets m k of the Webull dstrbuto defed by (3) wth the th orrespodg sample k momets M k. The sample th k momet M, 1,,..., k k s gve by M 1 k x. (1) k 1 It s kow, that the sample mea s x M. 1 The

6 414 Ivaa Pobočíková ad Zuzaa Sedlačková 1 1 x, () 1 1 x. 1 (3) By dvdg (3) by the square () we obta 1 1 x 1. 1 x 1 (4) The estmate ĉ of the parameter we obta by solvg (4) wth respet to. Ths equato has ot aalytal soluto ad must be solved umerally for. We use the Newto method. The startg pot we use aordg to Ramírez ad Carta (5) x, (5) s x 1 1 where x x s the sample mea ad s x x x s the sample varae. The estmate of the parameter a be estmated usg equato () Smulato study I smulato study we geerate the radom samples from the Webull dstrbuto ad ompare the performae of the methods. We osder.5, 1.5,.5, 1 ad sample szes 5, 6,...,1. For eah ombato, ad we geerate by the Mote Carlo smulato N 5 radom samples from the Webull dstrbuto. For eah method uder osderato we obta 5 estmates 1,,..., 5 of the parameter ad 5 estmates 1,,..., 5 of the parameter. The we ompute the sample meas, ad the sample varaes s, s (6),, s, s (7) 1 1

7 Comparso of four methods 4143 To ompare the performae of the methods we ompute the sample root mea square error (RMSE) gve by The estmates wth smaller RMSE are preferred. 5 1 RMSE. 5 (8) 1 4 Results ad dsusso For.5 the omparso shows that geeral the RMSE of the outperforms the other methods. The RMSE of the s slghtly larger tha the RMSE of the. The RMSE of the s the largest. For the small sample sze 5 15 ad for 1.5,.5 the omparso shows that geeral the RMSE of the outperforms the other methods. The RMSE of the LSM, the ad the are larger. The RMSE of the s oly slghtly larger tha the RMSE of the, both methods are omparable. For the sample sze 15 5 ad for 1.5 ad also for the sample sze 15 3 ad for.5 the omparso shows that the RMSE of the outperforms the other methods. The ad the are omparable methods may ases terms of the RMSE. The RMSE of the LSM s larger tha the other methods. For the sample sze 5 ad for 1.5 the omparso shows that the RMSE of the LSM s may ases larger tha the other methods. The ad the are omparable methods may ases terms of the RMSE. The RMSE of the s oly slghtly larger tha the RMSE of the. The RMSE of the s larger tha these two methods. For the sample sze 3 ad for.5 the omparso shows that the RMSE of the LSM s may ases larger tha the other methods. The ad the are omparable methods may ases terms of the RMSE. The RMSE of the s larger tha these two methods. It s evdet that as the sample sze reases the values of the RMSE of all methods derease ad hee the estmato preso of the parameters reases. Fgures 1,, 3 show llustratve plots of the RMSE for 1,11,...,1 ad osdered values of the Webull parameters ad. I Tables 1,, 3 are llustrated the results of the omparso for seleted sample szes ad osdered values of the Webull parameters ad.

8 4144 Ivaa Pobočíková ad Zuzaa Sedlačková Table 1. Smulato results of the parameter estmato,.5, 1 Sample sze Method =5 LSM =1 LSM = LSM =3 LSM =4 LSM =5 LSM =6 LSM =7 LSM =8 LSM =9 LSM =1 LSM s s RMSE

9 Comparso of four methods 4145 Table. Smulato results of the parameter estmato, 1.5, 1 Sample sze Method =5 LSM =1 LSM = LSM =3 LSM =4 LSM =5 LSM =6 LSM =7 LSM =8 LSM =9 LSM =1 LSM s s RMSE

10 4146 Ivaa Pobočíková ad Zuzaa Sedlačková Table 3. Smulato results of the parameter estmato,.5, 1 Sample sze Method =5 LSM =1 LSM = LSM =3 LSM =4 LSM =5 LSM =6 LSM =7 LSM =8 LSM =9 LSM =1 LSM s s RMSE

11 Comparso of four methods 4147 RMSE RMSE RMSE, =.5, δ = Fgure 1. Root mea square error,.5, 1 RMSE, =1.5, δ =1 Fgure. Root mea square error, 1.5, 1 LSM LSM RMSE, =.5, δ =1 LSM RMSE Fgure 3. Root mea square error,.5, 1

12 4148 Ivaa Pobočíková ad Zuzaa Sedlačková 5 Colusos Ths paper desrbes the four methods for estmatg the Webull dstrbuto parameters: the least square method (LSM), the weghted least square method (), the maxmum lkelhood method () ad the method of momets (). The performae of these methods s ompared usg the Mote Carlo smulato. The effey of the methods s ompared based o the RMSE rtero ad the sample sze. As the sample sze reases the values of the RMSE of all methods derease ad hee the estmato preso of the parameters reases. It s evdet that the ad the perform better tha the LSM ad the whe the sample sze s mddle or large eough. Oly for very small sample szes the ad the LSM outperform the ad the. The performs better tha the LSM. Both these methods are good methods due to ther smplty. For very small sample szes we reommed the. For mddle or large sample szes the s useful alteratve to the or the the stuato, whe the smple omputg s preferred. The provdes very smlar estmates to the oes obtaed by the. There s oe omplato by usg the. Ths method eeds to use the gamma futo. However the gamma futo a be easly obtaed by usg the software Matlab. The performae of the s more ofte better tha the. The s the most popular for ts effey, good propertes ad t s smpler to ompute tha the. Therefore we reommed the to estmate the Webull dstrbuto parameters. Akowledgemets. The authors gratefully akowledge the Setf Grat Agey (VEGA) of the Mstry of Eduato of Slovak Republ ad the Slovak Aademy of Sees for supportg ths work uder the Grat No. 1/145/1. Referees [1] B. Bergma., Estmato of Webull Parameters Usg a Weght Futo, Joural of Materals See Letters, 5 (1986), [] Y. K. Chu, Ch. J. Ke, Computato approahes for parameter estmato of Webull dstrbuto, Mathematal ad Computatoal Applatos, Vol. 17, No. 1 (1), [3] B. Fauher, W. R. Tyso, O the determato of Webull parameters, Joural of Materals See Letters, 7 (1988), [4] W. L. Hug, Weghted least squares estmato of the shape parameter of the Webull dstrbuto, Qualty ad Relablty Egeerg Iteratoal, 17(6) (1),

13 Comparso of four methods 4149 [5] H. L. Lu, CH. H. Che, J. W. Wu, A Note o Weghted Least-squares Estmato of the Shape Parameter of the Webull Dstrbuto, Qualty ad Relablty Egeerg Iteratoal, (6) (4), [6] I. Pobočíková, Z. Sedlačková, The least square ad the weghted least square methods for estmatg the Webull dstrbuto parameters a omparatve study, Commuatos- Setf Letters of the Uversty of Žla, Vol. 14, No. 4 (1), [7] P. Ramírez, J. A. Carta, Ifluee of the data samplg terval the estmato of parameters of the Webull wd speed probablty desty dstrbuto a ase study, Eergy Coverso ad Maagemet, 46 (5), [8] K. Trustrum, A. S. Jayatlaka, O estmatg the Webull modulus for a brttle materal, Joural of Materal See, 14 (1979), [9] D. Wu, J. Zhou, Y. L, Methods for estmatg Webull parameters for brttle materals, Joural of Materal See, 41 (6), [1] M. Zad, A. M. Sarha, Parameters Estmato of the Modfed Webull Dstrbuto, Appled Mathematal Sees, Vol. 3, No. 11 (9), Reeved: Jue 5, 14