Concomitants of Upper Record Statistics for Bivariate Pseudo Weibull Distribution

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1 Available at Appl Appl Math ISSN: Vol 5, Issue (Decembe ), pp 8 9 (Peviously, Vol 5, Issue, pp ) Applications and Applied Mathematics: An Intenational Jounal (AAM) Concomitants of Uppe Recod Statistics fo Bivaiate Pseudo Weibull Distibution Muhammad Ahsanullah Depatment of Management Sciences Ride Univesity Lawenceville, New Jesey, USA ahsan@ideedu Saman Shahbaz, Muhammad Qaise Shahbaz and Muhammad Mohsin Depatment of Mathematics COMSATS Institute of Infomation Technology Lahoe, Paistan samans@ciitlahoeedup; qshahbaz@gmailcom; mohsinshahid@yahoocom Received: Octobe 7, 9; Accepted: Septembe 9, Abstact In this pape the Bivaiate Pseudo Weibull distibution has been defined as a compound distibution of two andom vaiables to model the failue ate of component eliability The distibution of th concomitant and joint distibution of th and s th concomitant of ecod statistics of the esulting distibution have been deived Single and poduct moments alongside the coelation coefficient have also been obtained Recuence elation fo the single moments has also been obtained fo the esulting distibutions Keywods: Concomitants; Recod values; Pseudo Weibull distibution MSC () No: 6E5, 6E5 8

2 AAM: Inten J, Vol 5, Issue (Decembe ) [Peviously, Vol 5, Issue, pp ] 83 Intoduction The eliability of a component in multicomponent system plays vey impotant ole fo efficient woing of the system Distibutions having longe ight tail can be used to nicely model the failue time of components Seveal pobability distibutions have been in use to model this sot of scenaio Some popula pobability distibutions include Exponential, Weibull and Lognomal Diffeent vesions of Bivaiate Exponential and Bivaiate Weibull distibution have also been available in liteatue which can be used fo modeling of two component system Filus and Filus () intoduced pseudo distibutions as altenate models fo modeling component eliability Chandle (95) intoduced ecods as sequence of andom vaiables such that andom vaiable at th place is lage (smalle) than vaiable at ( ) th place He called andom vaiables as uppe (lowe) ecods in a andom sample of size n fom some pobability distibution f x Afte the intoduction of the field, numbe of people jumped into this aea of statistics Shooc (973) has thooughly discussed about ecod values and ecod time in a sequence of andom vaiables Ahsanullah (979) has chaacteized the exponential distibution by using the ecod values Ahsanullah (99) has also deived the distibutional popeties of ecods by using the Lomax distibution Balaishnan et al (995) have obtained the ecuence elations fo moments of ecod values fo Gumbel distibution Some moment popeties of the ecods have been given by Ahsanullah (99) Sufficient mateial is now available in the field of ecods that can be found in Anold et al (998) Some moe chaacteizations of the pobability distibutions via ecods have been obtained by Balaishnan and Balasubamanian (995) Ahsanullah (995) and Nevzoov () have given a compehensive eview of the mathematical foundation of the ecods Ahsanullah (995) has given the distibution of th uppe ecod, X U, as: fn : x f xrx, () whee Rxln Fx and F x is distibution function of X The joint distibution of th and m th uppe ecods, the following expession given by Ahsanullah (995): X Uand X Um, can be obtained by using with m fmn, : x, xm x f xmrx Rxm Rx, / x R x m () The concomitants in case of ecod statistics have not been extensively studied as compaed with the concomitants of ode statistics This banch is elatively new one in the field of odeed andom vaiables The distibution of concomitant of ecod values can be obtained by using the following expession given in Ahsanullah (995):

3 84 M Ahsanullah et al f y f y x f x dx (3) whee : n f x is the distibution of th uppe ecod given in () n : Ahsanullah (995) has also given joint distibution of concomitants of th and m th ecod values as:, x m, f y y f y x f y x f x x dx dx, (4) m m m, mn : m m whee f x x is given in () mn, :, m In this pape we have obtained the distibution of the concomitants of uppe ecod statistics fo Bivaiate Pseudo Weibull distibution The oganization of the pape is as follows: The Bivaiate Pseudo Weibull distibution is defined in section, distibution of -th concomitant fo Bivaiate Pseudo Weibull distibution with popeties has been studied in section 3, joint distibution of two concomitants is given in section 4 and the ecuence elation fo single moments is pesented in section 5 of the pape Bivaiate Pseudo Weibull Distibution The pseudo distibutions ae elatively new addition to pobability distibutions Fillus and Fillus ( and 6) have intoduced the Pseudo Nomal and Pseudo Gamma distibutions as linea combinations of the nomal and gamma andom vaiables We define the Bivaiate Pseudo Weibull distibution following the lines of Shahbaz and Ahmad (9) as unde: Let andom vaiable X has a Weibull distibution with paamete and The density function of X is: f x;, x exp x,,, x (5) Futhe, let anothe andom vaiable Y has the Weibull distibution with paamete x and, whee x is some function of andom vaiable X The density function of Y is: f y; x, x x y exp x y ; x,, y (6) The Bivaiate Pseudo Weibull distibution is defined as the compound distibution of (5) and (6) The density function of the Bivaiate Pseudo Weibull distibution is given as:

4 AAM: Inten J, Vol 5, Issue (Decembe ) [Peviously, Vol 5, Issue, pp ] 85 f x y x x y exp x x y,,, x, x, y, ; (7) The density (7) can be used to geneate seveal distibutions depending upon vaious choices of function x Shahbaz and Ahmad (9) studied distibution (7) by using x fo which (7) is simply equal to poduct to two maginal densities We have used x x in (7) to obtain following Bivaiate Pseudo Weibull distibution: f x, y x y exp x y ;,,, x, y (8) In following section, distibution of concomitant of ecod statistics has been deived fo (8) 3 Distibution of th Concomitant and Its Popeties The Bivaiate Pseudo Weibull distibution has been given in (7) and (8) In this section the distibution of th concomitants of ecod statistics fo Bivaiate Pseudo Weibull distibution, given in (8), has been obtained The distibution of the th concomitants of ecod statistics can be obtained by using (3) The distibution given in (3) equies the distibution of th ecod statistics fo andom vaiable X, given below: Fo the andom vaiable X we have R x ln F x x ecod statistics fo andom vaiable X is: and so the distibution of th fn : x x exp x ;,, x (9) The conditional distibution, f y x, fom (6) is: f y x x y exp x y,,, x, y ; () Using (9) and () in (3), the distibution of th concomitant of ecod statistic fo Bivaiate Pseudo Weibull distibution is: h y y x exp x y dx ; () Integating (), the pobability distibution of th concomitant of ecod statistics is given as:

5 86 M Ahsanullah et al y hy y y,,,, () The distibution function fo () is: Fy,,,, y y (3) By using (3), the p th pecentile of the distibution is: p p (4) The median can be easily obtained by using p = 5 in (4) The mode can be obtained fom () as: y (5) Also by using () and (3), the hazad ate function of the distibution is: y y,,,, y y (6) The th moment of the distibution given in () is obtained as: y E y y h y dy y dy (7) y / Simplifying (7), the th moment of the concomitant is: / E y, (8) The mean and the vaiance of the concomitants of ecod statistics fo Pseudo Weibull distibution can be obtained by using (8) Table and Table shows numeical values of mean and the vaiance of the concomitants of ecod statistics fo Pseudo Weibull distibution fo = 5 and fo diffeent values of and

6 AAM: Inten J, Vol 5, Issue (Decembe ) [Peviously, Vol 5, Issue, pp ] 87 Table Mean of the concomitants of ecod statistics fo Pseudo Weibull distibution fo Table Vaiance of the concomitants of ecod statistics fo Pseudo Weibull distibution fo Joint Distibution of the Concomitants and Moments In this section we have deived the joint distibution of concomitants of ecod statistics fo Pseudo Weibull distibution given in (8) The expession (4) can be used to obtain the joint distibution of two concomitant of ecod statistics Fom (4) we can see that the distibution given in (4) equies the joint distibution of two ecod statistics given in () The joint distibution () fo andom vaiables X XUand X XUm fo paent Weibull distibution is given as: f x, x x x x x x exp x m m Afte simplification, the joint distibution of two ecod statistics fo Weibull distibution is: m m m f x, x x x x x exp x (9) Using (9) and conditional distibution of Y given X in (4), the joint distibution of th and m th concomitant is obtained as unde:

7 88 M Ahsanullah et al o whee m x g y, y y y x x x x m exp x y exp x y dxdx gy y y y x exp x y m, m x m x x x exp x y dx dx m m m gy, y y y x exp x y I x dx, () x m I x x x x exp x y dx () Simplifying (), we have: m m I x x F, m, x y, m () whee F a, b; x is Kumme confluent Hypegeometic function, defined as: F a, b; x j a b j j j x j! Using () in () and simplifying we get: m m, g y y y y y y y m y y y, (3) whee Y = Y and Y = Y m The poduct moments fo distibution (3) ae obtained below:

8 AAM: Inten J, Vol 5, Issue (Decembe ) [Peviously, Vol 5, Issue, pp ] 89 The q th and th moments of two concomitants is defined as: q, q, ; (4) q E y y y y g y y dy dy /, Using (3) in (4) we have: q, q m m / q, E y y y y y y y y y m y y y dy dy Simplifying above equation, the poduct moments ae: mqm m q m q q / q, q q m q mq (5) The covaiance and coelation can be obtained by using (5) Table 3 shows some numeical values of covaiance fo = 5, m = 6 and fo diffeent values of and Table 3: Covaiance of the concomitants of ecod statistics fo Pseudo Weibull distibution fo 5, m Recuence Relation fo Single Moments It is easy to see fom () and (3) that y H (6) ( y) H ( y) h ( y) y

9 9 M Ahsanullah et al The elation in (6) will be used to establish ecuence elations fo moments of the concomitant of the th ecod statistic fo Pseudo- Weibull distibution Theoem: Fo and intege Poof: E( Y ) y h ( y) dy y H ( y) dy (7) Substituting (6) in (7) and afte simplification, we get Thus, ; as asseted REFERENCES Ahsanullah, M (979) Chaacteization of exponential distibution by ecod values, Sanhya, Vol 4, B, pp 6 Ahsanullah, M (99) Recod values of Lomax distibution, Statisti Nedelandica, Vol 4, No, pp 9 Ahsanullah, M (99) Recod values of independent and identically distibuted continuous andom vaiables, Pa J Statist Vol 8, No, pp 9 34 Ahsanullah, M (995) Recod Statistics, Nova Science Publishes, USA Anold, B C, Balaishnan, N and Nagaaja, H N (99) A Fist Couse in Ode Statistics, John Wiley, New Yo Balaishnan, N and Balasubamanian, K (995) A chaacteization of geometic distibution based on ecod values, J Appl Statist Science, Vol, No, pp 73 87

10 AAM: Inten J, Vol 5, Issue (Decembe ) [Peviously, Vol 5, Issue, pp ] 9 Balaishnan, N, Ahsanullah, M and Chan, P S (995) Relations fo single and poduct moments of ecod values fom Gumbel distibution, Stat and Pob Lett, Vol 5, No 3, pp 3 7 Candle, K N (95) The distibution and fequency of ecod values, J Roy Statist Soc, Vol4, B, pp 8 Filus, JK and Filus, LZ () On some Bivaiate Pseudo nomal densities Pa J Statist Vol 7, No, pp -9 Filus, JK and Filus, LZ (6) On some new classes of Multivaiate Pobability Distibutions Pa J Statist Vol, No, pp 4 Nevzoov, V B, () Recod: Mathematical Theoy, Tanslations of Mathematical Monogaphs, Vol 94, Ameican Mathematical Society Shahbaz, S and Ahmad (9) Concomitants of ode statistics fo Bivaiate Pseudo Weibull distibution, Wold App Sci J, Vol 6, No, pp 49 4 Shooc, R W (973) Recod values and inte ecod times, J Appl Pob, Vol, No 3, pp