Bayes Prediction Bounds for Right Ordered Pareto Type - II Data

J. Stat. Appl. Pro. 3, No. 3, 335-343 014 335 Journal of Statistics Applications & Probability An International Journal http://dx.doi.org/10.1785/jsap/030304 Bayes Prediction Bounds for Right Ordered Pareto Type - II Data Gyan Praash Department of Community Medicine, S. N. Medical College Agra, U. P., India. Received: May 014, Revised: 13 Jun. 014, Accepted: 17 Jun. 014 Published online: 1 Nov. 014 Abstract: The Pareto Type-II model is considered in present article as the underlying model from which observables are to be predicted under Bayesian approach. The predictions are to be made on the ordered failure items of the remaining businesses by using conditional probability function. A right item failure - censoring criterion has been used for prediction. Keywords: Pareto Distribution, Right Item Censoring, Predictive distribution, Central Coverage Bayes Prediction Bound Length. 1 Introduction Pareto distribution has played a major role in the investigations of previous phenomena providing a satisfactory model at the extremities. Pareto distribution their close relatives provide a very flexible family of fat - tailed distributions, which may be used as a model for income distribution of higher income group. Freiling 1966 applied the Pareto law to study the distributions of nuclear particles. Harris 1968 used this distribution determining times of maintenance service while Davis & Feldstein 1979 employed it in studying time to failure of equipment components. This distribution has established its important role in variety of other problems such as size of cities firms Steindle 1965, business mortality Lomax 1954. It is often used as a model for analyzing areas including city population distribution, stoc price fluctuation, oil field locations military areas. It has also been found to be suitable for approximating the right tails of distribution with positive sewness. It has a decreasing failure rate, so it is also useful for modeling survival after some medical procedures the ability to survive for a longer time appears to increase, the longer one survives after certain medical procedures. Two-parameter Pareto distributios transformational equivalent to the two-parameter exponential distribution Dyer 1981, thus one could analyze Pareto data using nown techniques for exponential distributions. Arnold 1983 gave an extensive historical survey of its use in the context of income distribution. Arnold & Press 1989, Ouyang & Wu 1994, Ali-Mousa 001, Soliman 001, Wu et al. 004 others those who have studied predictive inference for the future observations under the Pareto model. Recently, Praash & Singh 013 present some Bayes prediction length of interval for Pareto model. Al-Hussaini et al. 001 were obtaining Bayes prediction bounds for Type - I censored data from a finite mixture of Lomax Pareto Type - II components. Some Bayes prediction bounds based on one - sample technique for Pareto model have obtained by Nigm et al. 003. The objective of the present paper is to obtain the central coverage Bayes prediction length of bounds for the future observation from Pareto Type - II distribution. We present the Bayesian statistical analysis to predict the future statistic of the considered model based on the right censored item failure data. Based on the first ordered failure items in a sample of size n from Pareto Type - II distribution, Bayesian prediction bounds for the remaining n items are derived in two cases, the first is when the scale parameter is nown Section 3 the second is when both parameters are considered as the rom variables Section 4. A numerical study has been carried out for the illustration of the procedures in next section. Corresponding author e-mail: ggyanji@yahoo.com

336 G. Praash: Bayes Prediction Bounds for Right Ordered Pareto Type - II Data The Considered Model Probability density function of the considered Pareto model is f x;θ,σθ σ 1+σx θ 1 ; x>0, θ > 0,σ > 0. 1 Here, θ is the shape parameter σ is the scale parameter. The Pareto distribution 1, is the result of the mixture of Exponential distribution with the parameter α, the exponential scale parameter α is distributed as Gamma with parameters θ σ. In life testing, fatigue failures other inds of destructive test situations, the observations usually occurred in ordered manner such a way that weaest items failed first then the second one so on. Let us suppose that, there are n> 0 items are put to test under the considered model without replacement only > 0 items are fully measured, while the remaining n items are censored. These n censored lifetimes will be ordered separately. This is nown as the Right Item Failure - Censoring scheme. The objective of the present article is the predictions are to be made on the ordered failure remainingn items, by using conditional probability function. The lielihood function for rom sample x x 1,x,...,x under the above censoring criterios define as Lx θ,σ f x i ;θ,σ n xi. 1 f x i ;θ,σ dx i i1 i+1 0 where T 1 1 log 1+σx i T0 i1 log 1+σx i. Lx θ,σ σ θ exp θt 1 T 0, 3 Central Coverage Bayes Prediction Bounds Known Scale Parameter From a Bayesian view point; there is clearly no way in which one can say that one prior is better than other. It is more frequently the case that, we select to restrict attention to a given flexible family of priors, we choose one from that family, which seems to match best with our personal beliefs. A natural family of conjugate prior for the shape parameter θ is consider here as a Gamma when scale parameter is nown, having probability density function g 1 θ σ θ c 1 e d θ ; θ > 0,d > 0,c>0. 3 Posterior density function of the parameter θ is obtained by using 3 as Lx θ,σ.g 1 θ σ πθ x θ Lx θ,σ.g 1θ σ dθ θ x T Γ θ 1 e θt ; T T 1 + d. 4 Nigm & Hy 1987 consider the problem of predicting T r+s ;s 1,,...,n based on the order statistics t 1 < t <... < t r from a sample of size n. In present article, the predictions are to be made on the ordered failure remainingn items, using the conditional probability function for the Right Item failure - censored data. The conditional predictive density of Y s X +s ;s1,,...,n at given θ is defined as where ψ. is the survival function. Solving 5, we have h 1 ys θ,x [ ψ x ψ ys ] [ ψ ys ] n s [ ψ x ] n+r [ f ys ], 5 h 1 ys θ,x θ σ ni θ Bs B s B ; 6

J. Stat. Appl. Pro. 3, No. 3, 335-343 014 / www.naturalspublishing.com/journals.asp 337 where B s 1+σy s, B 1+σx, ni n s+i+1 1 i C i. The joint probability density function of Y s given the data is defined as h y s,θ x h 1 y s θ,x. πθ x h y s,θ x T Γ σ θ e θt Therefore the predictive density of Y s is thus obtained as Using 7 in 8 for y s > x r, we have h y s x σ T Now, the predictive survival functios defined as We now that ni θ Bs B s B. 7 h y s x h y s,θ x dθ. 8 θ B s P [ Y s > y x ] σ T i y 1 T + log y h Y s x dy s x h Y s x dy s σ T 1 i x B T + log s P [ Y s > y x ] Bs Bs B +1 B T + log s B dys Bs +1 B dys B0 i T + log B +1. 9 T ; B 0 1+σy. 10 a { } 1 i a C i i+b 1 a+1 a+b 1. C a+1 11 Hence, the denominator of equation 10 are solving by using 11 as T Using 1 10, the predictive survival functios thus written as P [ Y s > y x ] s n C s T 1 i C i i+n s+1 1 T 1 s n C s T. 1 T + log B0 B. 13 In the context of Bayes prediction, we say thatl 1,l is a 100% prediction limits for the future observation Y s, if they satisfies Pr l 1 < Y s < l. 14

338 G. Praash: Bayes Prediction Bounds for Right Ordered Pareto Type - II Data Here l 1 l are said to be lower upper Bayes prediction bounds for the rom variable Y s, is called the confidence prediction coefficient. The central coverage Bayes prediction lower upper bounds are obtain by solving following equality Pr Y s l 1 Pr Y s l. 15 Using 13 15, the lower upper central coverage Bayes prediction bounds l 1 l are obtained by solving following equalities as s n C s T s n C s T T + log T + log Bl1 B Bl B, 16 where B l1 1+l 1 σ B l 1+l σ. Further, simplification of the above equations 16, do not possible. A numerical technique is applied here for obtaining the values of l 1 l for some ε. For a particular case, substituting s1 in the predictive survival function 13, for predicting the item Y 1 X +1, of the next item to fail, is obtain as P [ Y 1 > y x ] 1+ n log T B0 B. 17 Similarly, for sn, corresponding to predicting the item Y n X n, of the last item to fail, the predictive survival function 13 is written as P [ Y n > y x ] n i1 1 i 1 n C i 1+ i T log B0 Now, the central coverage Bayes prediction lower upper bounds for Y 1 are now obtain as B } l 1 σ {B 1 T 1/ exp 1 1 n. 18 } l σ {B 1 T 1/ exp 1 1. 19 n The central coverage Bayes prediction length of bounds for Y 1 is obtain as Ll l 1. 0 The Central coverage Bayes prediction lower upper bounds do not exist for the item Y n. However, one may obtain lower upper bounds for Y n by equating following equality. n i1 n i1 1 i 1 n C i 1+ i T log 1 i 1 n C i 1+ i T log Bl1 B Bl B. 1

J. Stat. Appl. Pro. 3, No. 3, 335-343 014 / www.naturalspublishing.com/journals.asp 339 4 Central Coverage Bayes Prediction Bounds Both Parameters Unnown When, both scale shape parameters are considered to be rom variable, the lielihood function of the considered model is obtain as Lx θ,σ θ e θt 1 σ e T 0 where T 1 1 log 1+σx i T0 i1 log 1+σx i. It is clear from equation that, both the functions T 0 T 1 are depends only upon scale parameter σ, thus both scale shape parameters are independently distributed. Therefore, in present case when both shape scale parameters are consider being rom variables for the underlying model, the joint prior density for the parameter θ σ is considered as where g 1 θ σ g σ are the gamma densities defined as Now, the joint prior density is g 1 θ σ σ c The joint posterior density is now defined as gθ,σg 1 θ σ. g σ ; 3 Γc θ c 1 e σθ ; θ > 0,σ > 0,c>0 4 g σ ba Γa σ a 1 e bσ ; σ > 0,a>0,b>0. 5 gθ,σ θ c 1 σ a+c 1 e σθ e bσ. 6 Using 6 we have, π θ,σ x σ θ Lx θ,σ. gθ,σ Lx θ,σ. gθ,σ dθdσ. π θ,σ x σ θ 1 e θt1+σ σ a+c+ 1 e T0 bσ ; 7 where σ Γ σ T 1+ σ σ a+c+ 1 e T 1 0 bσ. The conditional predictive density of Y s X r+s ; s1,,...,n at given θ σ is obtain similarly as h ys θ,σ,x Bs θ σ exp θ log. 8 B s The joint probability density function of Y s given the data is now obtain as where T T Bs 1 + σ+ log B. B h ys,θ,σ x h ys θ,σ,x. π θ,σ x h ys,θ,σ x σ θ e θt B s σ a+c+ e T 0+bσ ; 9 Hence, the predictive density of Y s given on the data, for Y s > x r is thus obtained as h y s x h ys,θ,σ x dθ dσ. σ θ h y s x σ σ a+c+ e T0 bσ T B +1 dσ ; 30 s σ

340 G. Praash: Bayes Prediction Bounds for Right Ordered Pareto Type - II Data where σ σγ +1. The predictive survival functios obtain present case as P [ Y s > y x ] y h Y s x dy s x h Y s x dy s P [ Y s > y x ] σ T σ a+c+ 1 e T 0e bσ dσ σ a+c+ 1 e T 0 e bσ dσ ; 31 where T B0, T 1 + σ+ log B T T 1 + σ B 0 1+σy. The Central coverage Bayes prediction lower upper bounds for the rom variable Y s when both parameters are considered to be unnown, are obtained by equating following equality where Ω j Ω P [ Y s > l 1 x ] Ω 1 Ω P [ Y s > l x ] Ω σ T j σ a+c+ 1 e T 0e bσ dσ, B l j 1+l j σ, j 1,, T j σ a+c+ 1 e T 0e bσ dσ. ; 3 Ω Bl T 1 + σ+ log j B Again, closed form of the above equations 3 does not exist. A numerical technique is applied here for obtaining the values of central coverage Bayes prediction lower upper bounds for some ε. Putting s1, for predicting the item Y 1 X +1, of the next item to fail, the predictive survival functios P [ Y s > y x ] where T3 T B0. 1 + σ+n log B σ T 3 σ a+c+ 1 e T 0e bσ dσ σ a+c+ 1 e T 0 e bσ dσ, 33 Similarly, for predicting the last item to fail Y n x n, putting s n, the predictive survival functios rewritten as P [ Y n > y x ] Ω 3 Ω 4, 34 where Ω 3 n i1 1i 1 n C i σ T 4 σ a+c+ 1 e T 0 bσ dσ, Ω 4 n i1 1i 1 n C i σ a+c+ 1 e T0 bσ dσ T4 T 1 + σ+ i log B0 B. The central coverage Bayes prediction lower upper bounds for Y 1 Y n do not exist in closed form. However, one may obtain prediction bounds for Y 1 by equating following equalities. P [ Y s > l 1 x ] where T3 j T Bl 1 + σ+n log j B ; j 1,. σ T 31 σ a+c+ 1 e T 0e bσ dσ σ a+c+ 1 e T 0 e bσ dσ P [ Y s > l x ] σ T 3 σ a+c+ 1 e T 0e bσ dσ σ a+c+ 1 e T 0 e bσ dσ, 35

J. Stat. Appl. Pro. 3, No. 3, 335-343 014 / www.naturalspublishing.com/journals.asp 341 Table 1: Bayes Prediction Length of Bounds for X 16 When σ is Known. σ ε c,d 99% 95% 90% 0.5, 0.50.975 1.6849 1.4879.00 04, 0.1501 1.6160 1.3089 09, 03 1.907 1.4071 1.519 0.5, 0.50.436 1.6454 1.4530 1.00 04, 0.0997 1.5781 1.78 09, 03 1.864 1.3741 1.6 0.5, 0.50.1910 1.6068 1.4189 0.50 04, 0.0504 1.5411 1.48 09, 03 1.8188 1.3418 1.1939 Similarly, the central coverage Bayes prediction lower upper bounds for Y n the last item to fail are obtained by equating following equalities. P [ Y n > l 1 x ] Ω 31 Ω 4 P [ Y n > l x ] Ω 3, 36 Ω 4 where T4 j T Bl 1 + σ+ i log j B Ω3 j n i1 1i 1 n C i σ T 4 j σ a+c+ 1 e T0 bσ dσ j 1,. 5 Numerical Analysis 5.1 When Scale Parameter is Known In this section, we are presenting the findings of the central coverage Bayes prediction lower upper bounds for a particular set of parametric values. We assume here that the failure time follows the considered Pareto Type - II model. The numerical values of the prior parameters c d are selected as c,d 0.5,0.50,04,0 09,03 with scale parameter σ 0.50,1.00,.00. The selections of prior parametric values meet the criterion that the prior variance should be unity. Using above considered parametric values, generate 10, 000 rom samples. A sample of 0 tems is tested their ordered failure times are observed. Suppose that this test is terminated when first 15 of the ordered lifetimes are available. For selected level of significance ε 99%,95%,90%; the central coverage Bayes prediction lower upper bounds for X 16, the next failure item are calculated. The lengths of central coverage Bayes prediction bounds for X 16 are presented in Table 1. We observe from the table that the length of the interval tend to be wider as the value of scale parameter σ increases when other parametric values are consider to be fixed. Opposite trend has been seen when prior parameter c increases. It is also noted further that when confidence level ε or prior parameter d decreases the length of intervals tends to be closer. For the similar set of considered data, the length of central coverage Bayes prediction bounds for X 0, the last failure item are obtained presented in Table. It observes from Table that the magnitude of the length of the interval tends to be wider as compare to failure item X 16. However, the gain the interval is robust. This is a natural, as the prediction of the future order statistic that is far away from the last observation, the value has less accuracy than that of other future order statistic. Other properties are seemed to be similar.

34 G. Praash: Bayes Prediction Bounds for Right Ordered Pareto Type - II Data Table : Bayes Prediction Length of Bounds for X 0 When σ is Known. σ ε c,d 99% 95% 90% 0.5, 0.50 3.5438.5989.951.00 04, 0 3.3165.497.0190 09, 03.9417.1704 1.9311 0.5, 0.50 3.4607.5379.413 1.00 04, 0 3.387.434 1.9716 09, 03.877.1195 1.8858 0.5, 0.50 3.3795.4784.1886 0.50 04, 0 3.167.3771 1.954 09, 03.8054.0697 1.8415 Table 3: Bayes Prediction Length of Bounds for X 16 Both Parameters Unnown. c ε a,b 99% 95% 90% 0.5, 0.50.751.0183 1.783.00 04, 0.5755 1.9358 1.5679 09, 03.846 1.6855 1.4996 0.5, 0.50.6875 1.9710 1.7405 1.00 04, 0.515 1.8904 1.5311 09, 03.309 1.6460 1.4645 0.5, 0.50.645 1.947 1.6997 0.50 04, 0.4561 1.8460 1.495 09, 03.1787 1.6073 1.4301 Table 4: Bayes Prediction Length of Bounds for X 0 Both Parameters Unnown. c ε a,b 99% 95% 90% 0.5, 0.50 4.4538 3.663.8845.00 04, 0 4.1681 3.138.5375 09, 03 3.6971.777.470 0.5, 0.50 4.3494 3.1896.8168 1.00 04, 0 4.0704 3.0593.4779 09, 03 3.6104.6638.3701 0.5, 0.50 4.473 3.1148.7506 0.50 04, 0 3.9748.9875.4198 09, 03 3.558.601.3144 5. When Both Parameters are Unnown When both parameters are treated as the rom variable, study also has been carried out for studying the properties of the length of the central coverage Bayes predictive bounds. We generate the values of the scale parameter σ from 5 by using selected values of prior parameters a,b 0.5,0.50,04,0 09,03. The selections of prior parametric values meet the criterion that the prior variance should be unity. With the help of the generated values of σ c 0.50,1.00,.00, we generated the shape parameter θ. With the help of above generated values, 10, 000 rom samples have been generated from the considered model, a sample of 0 tems is tested their ordered failure times are observed. The length of central coverage Bayes prediction bounds are obtained for X 16, the next failure item, since test is terminated when the first 15 of the ordered lifetimes are available X 0, presented respectively in the Tables 3 4.

J. Stat. Appl. Pro. 3, No. 3, 335-343 014 / www.naturalspublishing.com/journals.asp 343 Similar properties have been seen for length of the central coverage Bayes prediction bounds in both cases. It is also noted that the magnitude of the length of bounds are wider when both parameter are unnown than compare to length of bounds when one parameter is unnown. The gain magnitude in the length of the Bayes prediction bounds is nominal. Acnowledgement The author are grateful to the anonymous referee for a careful checing of the details for helpful comments that improved this paper. References [1] E. K. Al - Hussaini, A. M. Nigm & Z. F. Jaheen. Bayesian prediction based on finite mixtures of Lomax components model Type - I censoring. Statistics, 35 3, 59-68, 001. [] M. A. M. Ali - Mousa. Inference prediction for Pareto progressively censored data. Journal of Statistics - Computation Simulation, 71, 163-181, 001. [3] B. C. Arnold. Pareto Distributions. International Co-Operative Publishing House, Maryl, 1983. [4] B. C. Arnold & S. J. Press. Bayesian estimation prediction for Pareto data. Journal of American Statistical Association, 84, 1079-1084, 1989. [5] H. T. Davis & M. L. Feldstein. The generalized Pareto law as a model for progressively censored survival data. Biometria, 66, 99-306,1979. [6] D. Dyer. A Comparison of the fallout mass-size distributions calculated by Lognormal Power law models. Canadian Journal Statistics, 9, 71-77, 1981. [7] E. C. Freiling. A Comparison of the fallout mass-size distributions calculated by Lognormal Power law models. U. S. Naval Radiological Defense Laboratory, San Francisco, 1966. [8] C. M. Harris. The Pareto distribution as a Queue discipline. Operations Research, 16, 307-313, 1968. [9] K. S. Lomax. Business failures. Another example of the analysis of failure data. Journal of the American Statistical Association, 49, 847-85, 1954. [10] A. M. Nigm & H. I. Hamdy. Bayesian two - sample prediction under the Lomax model with fixed rom sample size. Communication Statistics - Theory Methods, 16, 1761-177, 1987. [11] A. M. Nigm & E. K. Al-Hussaini & Z. F. Jaheen. Bayesian prediction bounds for the Pareto lifetime. Statistics, 37 6, 57-536, 003. [1] L. Y. Ouyang & S. J. Wu. Predictiontervals for an ordered observation from a Pareto distribution. IEEE Transactions on Reliability, 43, 64-69, 1994. [13] G. Praash & D. C. Singh. Bayes Prediction Intervals for the Pareto Model. Journal of Probability Statistical Science, 11 1, 109-1, 013. [14] A. A. Soliman. Bayes prediction a Pareto lifetime model with rom sample size. Journal of the Royal Statistical Society - Series D, 49 1, 51-6, 001. [15] J. Steindle. Rom processes the growth of firms, a study of the Pareto law. Heffner, New Yor,1965. [16] J. W. Wu, H. L. Lu, C. H. Chen & C. H. Yang. A note on the predictiontervals for a future ordered observation from a Pareto distribution. Quality Quantity, 38, 17-33, 004.