Record values of ratio of Weibull random variables

* mkmomamad@kau.edu.sa; Abstract: independently distributed Weibull random variables. The standard distributional properties of the resulting distribution have been studied. We have also obtained the recurrence record values for ratio of Weibull random variables. A real data application has also been given to see the performance of the proposed distribution.


INTRODUCTION
decreasing and constant failure-rates and can be created Weibull probability distribution function of a random variable 'W' is given by: ( ) The h th moment for equation ( 1) is given as: ( ) ( ) The mean and variance of the distribution can be obtained from equation (4).
Order and record statistics have emerged as important variables.A comprehensive treaty of order statistics is given in David and Nagaraja (2003).Record statistics that the density function of k th upper record statistics, n is available from a distribution F(x) is: where ( ) ( ) . Also, the distribution of lower records is . Also, the distribution of ( ) ( ) ( ) ( ) ( ) where ( ) ( ) . Ahsanullah (2004) has discussed record statistics for several distributions including exponential, Weibull and k th record for equation (1) is: The h th moment of the k th record statistics of Weibull distribution is: ) ... (8) In this paper the distribution of record statistics for ratio of Weibull random variables have been derived.Let W 1 and W 2 be independently distributed Weibull and scale parameters 1 and 2 .The probability density and distribution functions of W 1 and W 2 are, respectively: ( ) X W W = be the ratio of two Weibull random variables, then variables, then the density function of X can be obtained by using: and is given as The distribution function of the ratio can be obtained by using: and is given as , 0 The hth moment of ratio of Weibull variates is obtained The h th moment of ratio of Weibull variates is obtained as: ) The mean and variance of the ratio of Weibull variates can be obtained from equation (13).The values in equation (10) ) ...( 14) The distribution function of n th upper record for ratio Again, using the density and distribution function of ratio of Weibull random variables in equation (6), the density function of n th ( ) ( ) ( )( ) , log 1 ...( 16) , log 1 ( 1) In this section, the moments, survival and hazard functions and entropy of the record values of ratio of Weibull random variables are obtained

Moments
The k th moment of the n th record value X(n) is given as: ) Making the transformation ( ) th e e e t dt ( ) ) and hence the k th moment of the n th record of the ratio of Weibull variates is: Now making the transformation ( ) , the k th moment of the n th record value for ratio of Weibull random variables is: ) The mean and variance of the n th upper record for ratio equation (18). .We can also see that for all values of n and , the distribution of upper Again, the k th of Weibull random variables is given as ) x t e e t dx ( ) ; 1 Making the transformation ( ) in Table III for n in the table A.

Survival and hazard functions
The survival and hazard functions of n th , log 1 ( ) ...(20) from equations ( 16) and ( 17) as , log 1 ( ) The hazard rate function is given as The graph of hazard rate function of upper records for various values of parameters are given in Figure II in Appendix.

Entropy
The entropy of ratio of Weibull random variables is given as: The plot of entropy can be made from equation (22).

Estimation using records
of the distribution of ratio of Weibull random variables.
The estimation and application are given based upon the records.

Estimation using upper records
The density and distribution function of the ratio of Weibull random variables are given in equations ( 10) and ( 12) as ( ) ( ) ... (10) and and ( ) , , , , where ( ) . Now using (10) and (12) we have The joint distribution of records for ratio of Weibull random variables is therefore ( ) ( ) | , , , , , The log of likelihood function is w.r.t. the parameters are ( ) the parameters are ( ) ( ) .

..(28)
The maximum likelihood estimates can be obtained by simultaneously solving the equations ( ) ( ) and and ( ) ( ) 0 and ( ) The entries of Fisher information matrix are given ( ) ( ) ( ) ( ) June 2018 Journal of the National Science Foundation of Sri Lanka 46(2) The entries of Fisher information matrix can be computed for any given data.

Recurrence relations for moments of record values
of ratio of Weibull random variables.These recurrence

Recurrence relations for moments of lower record values
the density and distribution function of ratio of Weibull random variables are related as ( ) ( ) The distribution function F(w) and the hazard rate function corresponding to equation (1) are, respectively: of the National Science Foundation of Sri Lanka 46(2) in the context of specialised distributions.Shahbaz and distribution.Some other notable references are Shahbaz et al. (2009), and Shahbaz and Shahbaz (2009; 2010).as discussed by Nadarajah (2010) has been given in the METHODOLOGY The basic methodology contains distribution of ratio of phenomenon, for example stress and strength study of components.In addition the methodology contains sections.Ali et al. (2007) have studied the distribution et al. (2012) have studied the distribution of ratio of several random variables including Rayleigh distribution.The distribution of ratio of Weibull random variables has also been discussed by et al.
n are given in Figure I in Appendix.
...(12) distribution of records.The joint distribution of records is given by Shahbaz et al. (2016) as are given in Table I in Appendix.From Table II contains values of mean, variance,

Table 1 :
Journal of the National Science Foundation of Sri Lanka 46(2) Fitted distribution values ratio of Weibull random variables, ratio of Weibull Weibull and Weibull distribution to above data.The parameter estimates are given in Table