Estimation Under Multivariate Inverse Weibull Distribution

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1 Global Journal of Pure and Applied Mathematics. ISSN Volume, Number 8 (07), pp. 4-4 Research India Publications Estimation Under Multivariate Inverse Weibull Distribution Saieed F. Ateya (, ) () Mathematics & Statistics Department Faculty of Science, Taif University, Taif, Saudi Arabia. () Mathematics Department, Faculty of Science, Assiut University, Egypt. Abstract In this paper, a multivariate version of inverse Weibull distribution, denoted by MVIWD, has been constructed and its properties have been studied. Then, the maximum lielihood estimates (MLE s) and Bayes estimates (BE s), under squared error loss (SEL) function, are obtained in case of the trivariate inverse Weibull distribution (TVIWD) as illustrative example. Finally, a simulation study has been carried carrying out to study the goodness of the results and also to compare between the estimation methods using the mean squared errors (MSE s) criterion. Keywords: Inverse Weibull distribution, Continuous multivariate distributions, Maximum lielihood estimation, Bayes estimation, Monte Carlo simulation. AMS Subject Classification 00: 6F0, 6F5, 6N0, 6N0. INTRODUCTION The inverse Weibull distribution (IWD) is used to model degradation of mechanical components such as pistons, cranshafts of diesel engines, as well as breadown of insulating fluid to mention just a few areas. Keller and Kamath[] studied the shapes of the density and failure rate functions for the basic inverse model and Keller et al.[]

2 44 Saieed F. Ateya applied the model for the reliability analysis of commercial vehicle engines. Erto[] introduced further properties and identification of the model. Additional results on the IWD including wor on reliability and tolerance limits, Bayes -sample prediction, maximum lielihood and least squares estimation are given by Calabria and Pulcini[4-6]. Ateya [7] studied the estimation problem under IWD based on Balarishnan s unified hybrid censored scheme. For more details about IWD, some of its generalizations and related distributions with applications, see Oluyede and Yang[8]. Multivariate distributions are important both on theoretical and applied grounds. Their uses in multivariate analysis that have been applied to a variety of disciplines are numerous. Factor, cluster and discrimination analysis and multidimensional scaling are sometimes grouped as multivariate analysis. Regression analysis, variance components, experimental design and generally linear models are examples of the domains of applications. Multivariate normal distributions have probably been studied more than any other multivariate distribution. However, multivariate non-normal distributions are no less important as they may be needed in situations where a multivariate normal distribution is probably not the proper model to use. It is wellnown that multivariate distributions of given marginals are not unique. Some methods of constructing bivariate and multivariate distributions are the multivariate generalization of Pearson system, multivariate linear exponential-type distributions, Sarmanov and Linni multivariate distributions, Fréchet, Placett and Mardia s systems and Farlie-Gumbel-Morgenstern multivariate distributions. For details on such methods of construction and other bivariate and multivariate distributions (see Kotz et al.[9]). AL-Hussaini and Ateya[0-] suggested the compound technique to construct a general class of multivariate distributions, studied the members of this class, estimated the parameters of these members using the maximum lielihood and Bayes methods and studied the one- and two-sample prediction problem. Using the compound technique suggested by AL-Hussaini and Ateya[0-], Ateya and Madhagi[] constructed the multivariate truncated generalized Cauchy distribution and studied its properties and estimated its parameters. A random variable X is said to have an IWD distribution with vector of parameters θ = (α, β) if its PDF is given by f(x; θ) = α β x (α+) exp[ β x α ], x > 0, (α > 0, β > 0). (.) The cumulative distribution function (CDF ) of this random variable can be written as F(x; θ) = exp[ β x α ], x > 0, (α > 0, β > 0). (.) Copula:

3 Estimation under multivariate Inverse Weibull distribution 45 A two -dimensional copula is a function C from properties: I to I, I = [0, ], with the following () For every u and v in I, C(u, 0) = 0 = C(0, v), C(u, ) = u and C(, v) = v, () For every u, u, v and v in I such that u u and v v, C(u, v ) C(u, v ) C(u, v ) + C(u, v ) 0. Let F X,Y (x, y) be a joint distribution function with marginals F X (x) and F Y (y), then there exists a copula C(.,. ) for all x,y in R, such that Remars: F X,Y (x, y) = C(F X (x), F Y (y)). From the definition of the copula we can see that:. If we have the joint distribution function and the marginal distribution functions we can construct the corresponding copula to be c(u, v) = F X,Y (F X (u), F X (v)),. If we have a copula and the marginal distribution functions we can construct the bivariate distribution function. If we have a multivariate copula C(u, u,, u ), u i [0, ], i =,,, and the marginal distribution functions are F X (x ), F X (x ),, F Xn (x ) then the multivariate distribution function will be in the form F X (x) = C (F X (x ), F X (x ),, F X (x )), X = (X, X,, X ), x = (x, x,, x ) 4. If we have a multivariate distribution function F X (x) and the marginal distribution functions are F X (x ), F X (x ),, F X (x ) then the multivariate copula will be C(u, u,, u ) = F X (F X (u ), F X (u ),, F X (u )), u i [0, ], i =,,,. For more details on copulas, see, Nelsen [4].

4 46 Saieed F. Ateya - CONSTRUCTION OF MVIWD In this section, a multivariate version of IWD with vector of parameters θ = (α, β) is constructed using the copula introduced by AL-Hussaini and Ateya [] which of the form C(u) = [ + u γ i= i ] γ, 0 u i, i =,,,,, γ > 0, (.) where u = (u, u,, u ). Then, the conditional distribution functions are constructed from the following theorems. Theorem.: Suppose that X = (X, X,, X ) is a random vector of random variables such that Xi~ IWD(α i, ), i=,,,,with PDF shown in (.) after replacing α by α i and β by, respectively. The joint CDF of the random vector X is given by F(x,, x ; α,, α, β,, β, γ) = [ + and the corresponding PDF will be of the form f(x,, x ; α,, α, β,, β, γ) = e γ x α γ i i i= ], (.) (γ + ) (γ) ( α i γ i= x i α i e γ x i α i) [ + e i= γ γ x α i i ], x i > 0, (α,, γ > 0), i =,,,. (.) Theorem.: If X () = (X,, X r ) and X () = (X r+,, X ) are subvectors of X, then the conditional PDF's and CDF's of the MVIWD are given in the following forms: f(x x ) = (γ +r) ( α i α x i (γ ) γ c i e γ x α i r i i= ) [ r + c β i i) γ x γ r i α r i= ] c (e, (.4)

5 Estimation under multivariate Inverse Weibull distribution 47 f(x x ) = (γ + r) (γ ) ( β i i) γ x γ i α +r i=r+ ] c (e α i i=r+ γ c x i α i e ' (.5) γ x α i i ) [ r c + F(x x ) = [ r β i i) + (e γ x i α ] γ r c i=, (.6) c and F(x x ) = [ r c where β i i) + (e γ x i α ] γ i=r+ ' (.7) c c = ( r) + e i=r+ γ x α i i, γ = γ + r, c = r + e r i= γ x α i i and γ = γ + r. - MAXIMUM LIKELIHOOD ESTIMATION In this section, the maximum lielihood estimate of the vector of parameters θ, where θ = (α, α, α, β, β, β, γ) has been obtained. First, the lielihood function of the vector of parameters θ, given the vector of observations (x, y, z) = (x i, y i, z i ), i =,,, n, is given in the form n L(x, y, z θ) = i= f(x i, y i, z i ; α, α, α, β, β, β, γ) (.) where f(x, y, z α, α, α, β, β, β, γ) (γ + ) = ( β β β α α α (γ) γ x α y α z α β x α β y α β z α e γ e γ e γ ) [ + e β γ x α β + e γ y α β + e γ z α ] γ, (.)

6 48 Saieed F. Ateya The MLE s of all parameters are the simultaneous solutions of the following equations ln L α j = 0, ln L β j = 0, j =,, and ln L γ = 0. (.) 4- BAYES ESTIMATION Using the bivariate prior PDF suggested by Ateya[5] for the independent sets of the parameters (, ), (, ), (, ) and which of the forms c c c (, ) exp[ ( c )], 0, > 0, ( c > 0), c 4 c 6 c 6 (, ) exp[ ( c 5 )], 0, > 0, ( c > 0), c 7 c 9 c 9 (, ) exp[ ( c 8 )], 0, > 0, ( c > 0), (4.) (4.) (4.) and c 0 4 ( ) exp[ c ], 0, ( c 0 0, c 0). (4.4) where ci, i,,..., are the prior parameters ( also nown as hyper parameters). Then, the posterior PDF can be written in the form * (,,,,,, data) = A (, ) (, ) (, ) 4( ) L x, y, z θ where A is a normalizing constant. (4.5) Using the previous posterior PDF and using the MCMC technique, the BE s of all parameters can be obtained using SEL function.

7 Estimation under multivariate Inverse Weibull distribution RESULTS AND DISCUSSIONS In the following, the MLE s and BE s have been computed by applying the following steps: ( c, c, c ) the vector of population - For a given vector of prior parameters parameters and have been generated from the joint prior (4.). ( c, c, c ) the vector of population - For a given vector of prior parameters parameters and have been generated from the joint prior (4.). ( c, c, c ) the vector of population - For a given vector of prior parameters parameters and have been generated from the joint prior (4.). ( c, c ) the population parameter 4- For a given vector of prior parameters 0 has been generated from the prior (4.4). 5- Maing use of the generated population parameters, samples from the BVIWD with PDF (.) have been generated by solving the following equations simultaneously: a) F(x) = u, F(y x) = u F(z x, y) = u where u, u and u are random variates from U(0,) and the conditional CDF s can be obtained from (.7). 6- The MLE s of all parameters have been obtained as shown in section using the software Mathematica 8 for solving the resulting nonlinear equations. 7- The BE s for the same parameters under SEL function using MCMC algorithm have been obtained as shown in section The above steps (5-7) are repeated 500 times. 9- If ˆj is an estimate of, based on sample j, j =,,..., m, then the average estimate over the m samples is given by ˆ m = ˆ j. m j = 0- The MSE s of ˆ over the m samples is given by MSE ( ˆ ) = ( ) m ˆ m j = j. - From 0, the MSE s for all parameters will be computed. The computations are shown in Tables 5., 5. and 5..

8 40 Saieed F. Ateya Table 5.: MSE s of the MLE s based on different sample sizes n and m=500 repetitions., ( α =.7440, α = 0.509, α =.99879, β =.086, β = 9.85, β =.85, γ =.65) N MSE(γ ) MSE( α ) MSE( α ) MSE( α ) MSE( β ) MSE(β ) MSE(β ) Table 5.:- MSE's of the BE's under SEL function of α, α, α, β, β, β and α for different sample sizes n and m= 500 repetitions. (c =., c =., c =., c 4 =.5, c 5 =., c 6 = 4., c 7., c 8 =., c 9 =.0, c 0 =.0, c =.0 ), ( α =.7440, α = 0.509, α =.99879, β =.086, β = 9.85, β = n MSE(γ ) MSE(α ) MSE(α ) MSE(α ) MSE(β ) , γ =.65) MSE(β ) MSE(β ) Table 5.: MSE s of the MLE s and BE s under SEL function based on different sample sizes n and m=500 repetitions. (c =., c =., c =., c 4 =.5, c 5 =., c 6 = 4., c 7 =., c 8 =., c 9 =.0, c 0 =.0, c =.0), ( α =.7440, α =

9 Estimation under multivariate Inverse Weibull distribution , α =.99879, β =.086, β = 9.85, β =.85, γ =.65) n MSE MSE(γ ) MSE( α ) MSE( α ) MSE( α ) MSE( β ) MSE(β ) MSE(β ) 5 MLE SEL MLE SEL MLE SEL MLE SEL MLE SEL CONCLUSIONS In this project, MLE s and BE s of the parameters of TVIWD have been obtained. A simulation study is carried out to examine and compare the performance of the proposed methods for different sample sizes. From the results which are summarized in tables 5., 5. and 5., we observe the following. - The MSE s of the BE s based on SEL function are less than that obtained for the MLE s which means that the BE s are better than the MLE s. - The MSE s of the MLE s and BE s decrease by increasing the sample size n. REFERENCES [] Keller, A.Z. and Kamath, A.R., Reliability analysis of CNC machine tools, Reliab. Eng., (98), [] Keller, A.Z., Giblin, M.T. and Farnworth, N.R., Reliability Analysis of Commercial Vehicle Engines, Reliab. Eng., 0(985), [] Erto, P., Genesis, properties and identification of the inverse Weibull lifetime model, Statistica Applicata, (989), 7-8.

10 4 Saieed F. Ateya [4] Calabria, R. and Pulcini, G., Confidence Limits for Reliability and Tolerance Limits in the Inverse Weibull Distribution, Eng. and Sys. Safety,4(989), [5] Calabria, R. and Pulcini, G., Bayes -Sample Prediction for the Inverse Weibull Distribution, Communications in Statistics-Theory and Methods, (994), [6] Calabria, R. and Pulcini, G., On the Maximum Lielihood and Least Squares Estimation in Inverse Weibull Distribution, Statistica Applicata, (990), [7] Ateya, S.F., Estimation under Inverse Weibull Distribution based on Balarishnan s Unified Hybrid Censored Scheme, Communications in Statistics-Simulation and Computation, (05), DOI: 0.080/ [8] Oluyede, B.O. and Yang, T., Generalizations of the Inverse Weibull and Related Distributions with Applications, Elec. J. of Appl. Statist. Analy., 7(04), [9] Kotz, S., Balarishnan, N. and Johnson, N. L. (000). Continuous multivariate distributions: models and applications, vol. Wiley, New Yor [0] AL-Hussaini, E.K. and Ateya, S.F., Parametric estimation under a class of multivariate distributions, Stat. Pap., 46(005), 8. [] AL-Hussaini, E.K. and Ateya, S.F., A class of multivariate distributions and new copulas, J. Egypt. Math. Soc.,4(006), [] AL-Hussaini, E.K. and Ateya, S.F., Bayesian prediction under a class of multivariate distributions, Arabian J. Math. (0), 8-9. [] Ateya, S.F. and Madhagi E.A., On multivariate truncated generalized Cauchy distribution, Stat Papers (0) 54, [4] Nelsen,R. B., An Introduction to Copulas, Springer, New Yor(999). [5] Ateya,S. F., Bayesian Prediction Intervals of Future Nonadjacent Generalized Order Statistics from Generalized Exponential Distribution Using Marov Chain Monte Carlo Method. Applied Mathematical Sciences, 6(0), 7, 5 45.