Estimation of parameters of the Kumaraswamy distribution based on general progressive type II censoring

Transcription

1 Aeican Jounal of Theoetical and Alied Statistics 4; 3(6: Published online Decebe 2, 4 (htt:// doi: 0.648/j.ajtas ISSN: (Pint; ISSN: (Online Estiation of aaetes of the Kuaasway distibution based on geneal ogessive tye II censoing Mostafa Mohie Eldin, Noa Khalil 2, Montase Aein 3 Pofesso in deatent of Matheatics, faculty of Science, El Azha Univesity, Caio, Egyt 2 Lectue deatent of Matheatics, faculty of Science, Helwan Univesit, Caio, Egyt 3 Lectue deatent of Matheatics, faculty of Science, El Azha Univesity, Caio, Egyt Eail addess: Noa.hassan4@gail.co (N. Khalil To cite this aticle: Mostafa Mohie Eldin, Noa Khalil, Montase Aein. Estiation of Paaetes of the Kuaasway Distibution Based on Geneal Pogessive Tye II Censoing. Aeican Jounal of Theoetical and Alied Statistics. Vol. 3, No. 6, 4, doi: 0.648/j.ajtas Abstact: In this ae, we oduced a study in Estiation fo aaetes of the Kuaasway distibution based on geneal ogessive tye II censoing. These estiates ae deived using the axiu likelihood and Bayesian aoaches. In the Bayesian aoach, the two aaetes ae assued to be ando vaiables and estiatos fo the aaetes ae obtained using the well known suaed eo loss (SEL function. The findings ae illustated with actual and coute geneated data. Keywods: Kuaasway s Distibution, Bayes Estiation, Bayes Pediction, Geneal Pogessive Tye II Censoing. Intoduction The Kuaasway distibution is vey siila to the Beta distibution, but has the iotant advantage of an invetible closed fo cuulative distibution function. Kuaasway (6, 8 has showed that the well known obability distibution functions such as the noal, log-noal, beta and eiical distibutions such as Johnson s and olynoial-tansfoed-noal, etc., do not fit well hydological data, such as daily ainfall, daily stea flow, etc. and develoed a new obability density function known as the sine owe obability density function. Futheoe, Kuaasway (80 develoed a oe geneal obability density function fo double bounded ando ocesses, which is known as Kuaasway s distibution. This distibution is alicable to any natual henoena whose outcoes have lowe and ue bounds, such as the heights of individuals, scoes obtained on a test, atosheic teeatues, hydological data, etc. Also, this distibution could be aoiate in situations whee scientists use obability distibutions which have infinite lowe and/o ue bounds to fit data, when in eality the bounds ae finite. Guta and Kiani (88 discused the connection between non-hoogeneous Poisson ocess (NHPP and ecod values. So, the esults in this ae ay be used fo soe alied situation such as eventive aintenance. The ae by Kuaasway (80 oosed a new obability distibution fo double bounded ando ocesses with hydological alications. The continuous at of Kuaasway s distibution has the obability density function (df and the cuulative distibution function (cdf secified by and ( ( f x = x x ( ( ( x F x = (2 esectively, fo 0 x, > 0 and > 0. Kuaasway (,, whee and ae two ositive shae aaetes. It has any of the sae oeties as the beta distibution but has soe advantages in tes of tactability. This distibution aeas to have eceived consideable inteest in hydology and elated aeas, see Sunda and Subbiah (8, Fletche and Ponnabala (6, Seifi et al. (00, Ponnabala et al. (0, and Ganji et al. (06. Kuaasway distibutions ae secial cases of the thee aaete distibution with density B ( γ, ( γ x x,0 x, and > 0.

2 28 Mostafa Mohie Eldin et al.: Estiation of Paaetes of the Kuaasway Distibution Based on Geneal Pogessive Tye II Censoing Kuaasway distibutions has secial cases, Kuaasway (, distibution is the owe function distibution, Kuaasway (, distibutions is the distibution of one inus that owe function ando vaiable. Kuaasway (, distibution is the unifo distibution. A futhe secial case of the Kuaasway distibution has also aeaed elsewhee. The Kuaasway (2, distibution is that of the geneating vaiate { } R = x + x when x, x Follow a bivaiate Peason Tye II distibution [K.T. Fang(0], Section Paaete Estiation Suose n indeendent units ae laced on life test with coesonding failue ties (y, y2..., y being indendentically distibuted with cuulative distibution whee, Using e. ( the likelihood function can be exessed as: function F(y, obability density function f(y, and efixed nubes of failues that obseved. The schee (R,..., R is the ogessively tye II censoed, y i; ; n be coletely obseved ties, i=, 2,,. The failue ties of the fist units wee not obseved, at tie (+th failue R + nube of suviving units ae withdawn fo the test andoly, and so on. Finally at the tie th failue, the eaining R = n R + R R ae withdawn fo the test. 2.. ML Estiation The likelihood function to be axiized when geneal ogessive tye II censoed sale based on n indeendent units with F(y is L(θ. * Ri ( = ( +, ( i, ( i,, + < + 2 <... < i= + L θ c F y θ f y θ F y θ y y y (3 n! = +! n! * c n R n R R ( ( ( ( ( ( ( i ( ( ( ( ( ( i * R * R + + i i i + i i i= + i= + L( t;, = c y y y y = c y y y (4 The log-likelihood function l ( y;,, is given by: ( ( ( i ( i ( i + ( Log ( + ( + + y + l( y;, = constant Log y Log ( R + Log y (5 The ML estiates of and can be obtained by axiizing the log-likelihood function, E. 5. By taking ( y + + ( + ( y+ i= + i= + deivatives with esect to and the MLE ae obtained by satisfying the following euations + Log ( yi Log ( yi ( ( Ri yi i= + i= + δl Log y y = = 0 δ ( + ( + ( + δl Log y y = + + δ y i= + ( Ri ( yi = 0 The above syste is nonlinea, but can be easily solved using nueical techniues Bayes Estiation of the Unknown Paaete(s In this subsection we conside the Bayes estiation of the unknown aaete(s. We will assue that the aaetes and of the Kuaasway s distibution ae ando vaiables with a joint bivaiate io density function that was fist suggested by Al-Hussaini and Jaheen (5 as, whee (, g ( g ( = (6 ( α 2 α + α γ g ( = e, α > -, γ > 0 α + Γ + γ is the gaa conjugate io. This io was fist intoduced by Paadooulos (8 and was also used late on by Al- Hussaini and Jaheen (2. The io of is δ β g2 ( = e, δ > 0, β > 0 δ Γ δ β ( ( (8

3 Aeican Jounal of Theoetical and Alied Statistics 4; 3 (6: which is the gaa (δ, β density. Substituting (6 and ( in (5 we obtain the bivaiate io density of and given as (, δ + α α ex ( ( β + γ ( whee α >, β, δ and γ ae ositive eal nubes. Fo (3 and (, the joint osteio distibution is given by ( data 0 0 ( data ( data ln ( y ( ( + + α + δ + α i= + i= + e e (, ( ( L data g g 2 (, ( ( L data g g dd 2 (, ( ( L data g g 2 (, ( ( ln yi + + Ri ln yi + L data g g dd ( ( ( γ β ln ( ln( ( ln y yi Ri y i α δ α γ β i= + i= + e e dd ( ( ( ( data ln yi + + Ri ln yi + ln ( y ( ( + + α + δ + α i= + i= + e e Γ + α + 0,,0,0 ( ( ( ( ( γ β (0 Whee z z d ln ( yi ln ( yi α δ a i= + i= + z e dz = j 0 j = 0 z z z ( Ri ln ( yi j ln ( y f γ i= + ( a, c, d, f ( + α + c If the loss function is the well known suaed eo loss function, the Bayes estiatos fo the aaetes and ae the given by thei aginal osteio exectations as ( y ˆ =E B = (,,0,0 ( 0,,0,0 taken fo the onth of Febuay fo to 0, htt://cdec.wate.ca.gov/esevoi_a.htl. The axiu caacity of the esevoi is AF and the data wee tansfoed to the inteval [0, ]. And ˆ B =E ( y = ( + α + ( 0,2,0,0 ( 0,,0,0 3. Nueical Exales A siulation study with eal data is conducted in ode to coae the efoances between MLE and Bayes estiato. 3.. Illustative Exales Exale (eal data In ode to illustate the findings of Sects. 2 and 3 two exales ae given. The foe is a eal data set obtained fo the Shasta esevoi in Califonia while the latte exale uses siulated data set. In both exales the atheatical ackage was used to obtain the estiates of the aaetes a and b and Bayes edictive estiates. The fist exale deals with the onthly wate caacity data fo the Shasta esevoi in Califonia, USA and wee Fig.. The eiical hazad function. Table, gives the date, actual and tansfoed data.

4 2 Mostafa Mohie Eldin et al.: Estiation of Paaetes of the Kuaasway Distibution Based on Geneal Pogessive Tye II Censoing Table. Monthly caacity fo August and ootion of total caacity fo Shasta esevoi. Yea caacity Pootion of total caacity Yea caacity Pootion of total caacity The values wee used to veify that the tansfoed data follow Kuaasway s distibution, by exaine the eiical hazad function of the obseved data by alying the scaled Total Tie on Test (TTT lot, see Aaset (8. This ovides a vey good idea about the shae of the hazad function of a distibution. Fo a faily with the suvival function s ( y = F ( y, The scaled TTT with F ( u H ( u H ( u = s( y dy define fo 0 < u < is g ( u =. 0 H ( The coesonding eiical vesion of the scaled TTT tansfo is given by gn ( n H n = = H ( n ( n i= n y( i i= y( + i ( n y(, Failue tie vecto Y= (y 4,, y and censoing schee R= (R 4,, R,=3. whee y ( i denotes the i-th ode statistic of the sale. It has been shown by Aaset (8 that the TTT tansfo is convex (concave if the hazad ate is deceasing (inceasing; and fo bathtub (uniodal hazad ates, the scaled TTT tansfo is fist convex (concave and then concave (convex. The lot of the scaled TTT tansfo of the data, Fig., indicates that the eiical hazad function is inceasing and theefoe, it is easonable to use the Kuaasway s distibution to analyze the data. We also wanted to check by using the Kologoov_Sinov (K_S statistic whethe the Kologoov Sinov test showed that indeed the obsevations follow the Kuaasway s distibution ( value > 0.2. i Y i R i MLE, osteio ean, edian and ode of the aaete Paaete MLE Posteio ean Posteio ode Posteio edian Exale 2 (geneation data The following data is geneal ogessively tye II censoed sale fo Kuaasway distibution ( = 3, = 2 was siulated by using n=60, =, =2 and censoed schee [R i = {4,,,,,3,,0,3,0,2,0,3,2,3,0,0,3,2,}, MLE, osteio ean, edian and ode of the aaete i=+,,]: 0.45, , 0.402, 0.425, , 0.582, , , , 0.580, 0.624, , 0.666, 0.682, 0.056, 0.02, 0.323, 0.566, 0.8, Paaete MLE Posteio ean Posteio ode Posteio edian Monte Calo Siulations In the following, Monte Calo siulation study is conducted in ode to coae the efoance of the Bayes estiato with MLE fo diffeent sale sizes and censoing schees. We geneate ogessively tye II censoed sales fo Kuaasway distibution with aaete =3, =2, diffeent cobinations of the effective sale size = and diffeent ogessive censoing schees ( (R +, R +2,..., R. The diffeent ogessive censoing schees that used wee I ando censoing schees, II

5 Aeican Jounal of Theoetical and Alied Statistics 4; 3 (6: *0, n, *0 and III ( *0, n. Fo 2 2 silicity in notation, we denote these censoing schees, fo exale, by (3*0,,5 which eesents the censoing schee ( R 4 = 0, R 5 = 0, R 6 = 0, R =, R 8 = 5 fo fixed = 3. The ogas wee witten by Matheatica. In couting the estiates we geneated 000 sales fo the Table 2. Aveage estiates of and the associated MSEs when =3. Kuaasway distibution, and we elicated the ocess 000 ties. The aveages and ean suaed eos (MSE in aentheses of estiatos of and ae esented in Tables 2 and 3, esectively. Fo io infoation we have used io with α =, γ = 2, δ =, β =. n Effective sale size (R +, R +2,..., R Censoing sches MLE Bayes Pio (6*0,0 (3*0,0,3*0 (0, 2*,2,,3 (8*0,8 (4*0,8, 4*0 (2,0,2,0,2,0,2,2*0 (6*0, (3*0,, 3*0 (5,0,5,0,5,0,5 (8*0,8 (4*0,8, 4*0 (6*0,0 (8*0,0, 8*0 (2,0,3*,6*0,2*,3*0,3 3.3 ( ( ( ( ( ( ( ( ( ( ( ( ( (0.53 Table 3. Aveage estiates of and the associated MSEs when = ( ( ( ( ( ( ( ( ( ( ( ( ( (0.23 n Effective sale size (R +, R +2,..., R Censoing sches MLE Pio (6*0,0 (3*0,0,3*0 (0, 2*,2,,3 (8*0,8 (4*0,8, 4*0 (2,0,2,0,2,0,2,2*0 (6*0, (3*0,, 3*0 (5,0,5,0,5,0,5 (8*0,8 (4*0,8, 4*0 (4,0,4,0,4,0,4,0,2 (6*0,0 (8*0,0, 8*0 (2,0,3*,6*0,2*,3*0, ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (0.30

6 222 Mostafa Mohie Eldin et al.: Estiation of Paaetes of the Kuaasway Distibution Based on Geneal Pogessive Tye II Censoing 4. Conclusion In this ae we have consideed Estiation obles fo aaetes of the Kuaasway based on ogessive Tye- II censoed data by using Maxiu Liklehood Estiation and the Bayesian infeence. The io belief of the odel is eesented by the indeendent gaa conjugate ios on the shae aaetes. The suaed eo loss function is used as it is aoiate when lage eos of the estiation ae consideed to be oe seious coaed to othe loss functions. We used Matheatica oga to geneate the sales and bisection ethod to coute the estiated aaetes. The details have been exlained using a eal life exale and coute geneated data. Siulation study was conducted in ode to coae the efoance of the Bayes estiatos with MLE fo diffeent sale sizes and censoing schees, it was seen that the behavio of the MLE and Bayes estiates deends on the kind of censoing schees, and the esults wee suaized as follow: i. Coaing the behavio of MLE and Bayes estiatos: whee when used ando censoing schees the MLE and Bayes estiatos wee alost sae values in tes of MSE; howeve, it was seen that the Bayes estiatos wee bette than thei coesonding MLEs when used the following censoing schees *0, n, *0 and the MLEs wee bette 2 2 than thei coesonding the Bayes estiatos when used the following censoing schees ( *0, n. An iotant oble will be to extend these esults fo othe censoing schees such as Tye-I, hybid censoing schees. The wok is in ogess. ii. behavio of the MLE: the behavio of MLE in censoing schee nube I was bette than nube II, III. iii. behavio of Bayes estiatos: the diffeent kind of censoing schees behavio of does not effect on Bayes estiatos behavio. Refeences [] Aaset, M.V (8 How to identify a bathtub hazad function. IEEE Tansactions on Reliability 36: [2] Al-Hussaini EK, Jaheen ZF (2 Bayesian estiation of the aaetes, eliability and failue ate functions of the Bu Tye XII failue odel. J Stat Cout Siul 4:3 40 [3] Chansoo Ki, Keunhee Han (0 Estiation of the scale aaete of the Rayleigh distibution unde geneal ogessive censoing. Jounal of the Koean Statistical Society 38: 23_246 [4] Fletche SC, Ponnabla K (6 Estiation of esevoi yield and stoage distibution using oents analysis. J Hydol 82:25 25 [5] Guta RC, Kiani SNUA (88 Closue and onotonicity oeties of nonhoogeneous Poisson ocesses and ecod values. Pobab Eng Inf Sci 2: [6] K.T. Fang, S. Kotz, K.W. Ng, Syetic Multivaiate and Related Distibutions, Chaan and Hall, London, 0. [] Kuaasway P (6 Sineowe obability density function. J Hydol 3:8 84 [8] Kuaasway P (8 Extended sineowe obability density function. J Hydol 3:8 8 [] Kuaasway P (80 A genealized obability density function fo double-bounded ando ocesses. J Hydol 46: 88 [0] Ponnabala K, Seifi A, Vlach J (0 Pobabilistic design of systes with geneal distibutions of aaetes. Int J Cicuit Theoy Al 2: [] Seifi A, Ponnabala K, Vlach J (00 Maxiization of anufactuing yield of systes with abitay distibutions of coonent values. Ann Oe Res : [2] Sunda V, Subbiah K (8 Alication of double bounded obability density function fo analysis of ocean waves. Ocean Eng 6:3 0