BAYESIAN PREDICTION UNDER A CLASS OF MULTIVARIATE DISTRIBUTIONS

Faculty of Science, Alexandria University, Alexandria, Egypt Mathematics & Statistics Department, Taif University, Hawia, Taif, Saudi Arabia Mathematics Department, Faculty of Science, Assiut University, Egypt Abstract. In this paper the prediction problem is studied under members of a class =∗ of multivariate distributions, constructed by AL-Hussaini and Ateya [7−8]. More attention is paid to bivariate compound Rayleigh (BV CR) distribution, which is a member of this class, as illustrative example.


Introduction
Suppose that a class of distribution functions is of the form where a and b are non-negative real numbers such that a may assume the value zero and b the value infinity, λ η (x) is a continuous, monotone increasing and differentiable function of x such that λ η (x) → 0 as x → a + , λ η (x) → ∞ as x → b − and η is a parameter (could be a vector), (θ, δ, η) belongs to a parameter space Ω . This class covers some important distributions such as the Weibull, exponential, Rayleigh, compound Weibull, compound exponential (Lomax), compound Rayleigh, Pareto, power function, beta, Gompertz and compound Gompertz distributions, among others. The failure rate and survival functions corresponding to F ∈ are, respectively, δθλ η (x) and e −θδλη(x) , so that the probability density function (pdf ) is given, for 0 ≤ a < x < b ≤ ∞, by (2) f X|Θ (x|θ) = δθλ η (x)exp[−θδλ η (x)], ≡ d dx .
It was assumed that Θ is a positive random variable following the gamma(α, β) distribution with pdf g Θ (θ) given by .
Maximum likelihood and Bayes estimation of the parameters of members of the class * were obtained by AL-Hussaini and Ateya [7 − 8] and particularly when the underlying population distribution is bivariate compound Weibull or bivariate compound Gompertz.
In this paper, the prediction problem is studied under members of class * . More attention is paid to bivariate compound Rayleigh (BV CR) distribution as illustrative example.

One-sample prediction
Suppose that X 1 < X 2 < ... < X r is the informative sample, representing the first r ordered lifetimes of a random sample of size n drawn from a population with probability density function (pdf ) f X (x), cumulative distribution function (cdf ) F X (x) and reliability function (rf ) R(x). In one-sample scheme the Bayesian prediction intervals (BP I) for the remaining unobserved future (n − r) lifetimes are sought based on the first r observed ordered lifetimes.
For the remaining (n − r) components, let Y s = X r+s denote the future lifetime of the s th component to fail, 1 ≤ s ≤ (n − r). The conditional density function of Y s given that the r components had already failed is θ is the vector of parameters.
The predictive density function is given by π * (θ|x) is the posterior density function of θ given x and x = (x 1 , ..., x r ).
A (1 − τ ) % BP I for y s is an interval (L, U ) such that By solving equations (7) and (8), we get the interval (L, U ).

Two-sample prediction
Let X 1 < X 2 < ... < X r and Z 1 < Z 2 < ... < Z m represent informative (type II censored) sample from a random sample of size n and a future ordered sample of size m, respectively. It is assumed that the two samples are independent and drawn from a population with (pdf )f X (x), (cdf )F X (x) and (rf )R(x).
Our aim is to obtain the BP I for Z s , s = 1, 2, ..., m. The conditional density function of Z s , given the vector of parameters θ, is θ is the vector of parameters.
The predictive density function is given by π * (θ|x) is the posterior density function of θ given x and x = (x 1 , ..., x r ).
A (1 − τ ) % BP I for z s is an interval (L, U ) such that By solving equations (11) and (12), we get the interval (L, U ).

Baysian prediction intervals for future bivariate observations
The main goal in this section is to study the one-sample and two-sample prediction problems in case of bivariate informative observations.
While ordering a set of univariate random variables is a clear and straight-forward matter as it can be done by simply ordering the set of random variables, such ordering is not as clear if we are dealing with a set of random vectors.
Barnett [10] classified the principles used for ordering multivariate date into four principles : marginal, reduced (aggregate), partial and conditional (sequential) ordering. An interesting detailed discussion of such principles with illustrative examples are given in Barnett's paper.
In our paper, we wish to predict bivariate random vectors. The first components of the predicted random vectors are based on the ordered first components of the informative sample, as is done in the univariate case. To predict the second components, we compute the norms of each vector of the informative sample, order the norms and then predict the future norms as is done in the univariate case. The relation between the components of vectors and norms enables us to obtain the second components of the predicted vectors.
In other words, we obtain the second component of a predicted vector from the knowledge of the values of the first component and the norm of the vector.

One-sample prediction
Let (X 1 , Y 1 ), ..., (X r , Y r ) be the first r bivariate informative observations from a random sample of size n of bivariate observations. Suppose that the first components of such informative vectors are ordered, that is X 1 < X 2 < ... < X r and that their norms are given by Z 1 , Z 2 , ..., Z r .
Step 1 The norm Z of the vector (X, Y ) is given by Z = (X 2 + Y 2 ) 1/2 . In APPENDIX A the pdf and hence cdf and rf are derived. Such functions are given by From (16) and (18), the conditional density of Z * s given (c, α) is obtained ( see APPENDIX B ), as Suppose that the prior belief of the experimenter is given by the pdf π(c, α) = π 1 (c|α) π 2 (α), c|α ∼ Gamma(c 1 , α) and α ∼ Gamma(c 2 , c 3 ).
Step 2 By using the pdf (14) and its cdf , the predictive density function of X * s can be written as follows where where A 1 is a normalizing constant and B i,s = (−1) i To obtain (1 − τ ) % BP I for X * s , say (L 2s , U 2s ),we solve the following two nonlinear equations, numerically, Step 3 From steps 2 and 3, a In this case we apply the steps in Subsection 2.2 as follows Step 1 Substituting from (16) and (18) in (9) and then using (20) and (21) we can write where * * It then follows that the predictive density function of Z * s is given by g * 2 (z * s | z 1:r , ..., z r:r ) = ∞ 0 ∞ 0 g 1 (z * s | c, α)π * (c, α| z 1:r , ..., z r:r )dc dα.
Step 2 Using the pdf (14), its cdf and the same prior as in (20) the predictive density function of X * s is given by where where A 1 is a normalizing constant and To obtain (1 − τ ) % BP I for X * s , say (L 2s , U 2s ),we solve the following two nonlinear equations, numerically, Step 3 From steps 2 and 3, a

Numerical example
In this section we follow the steps (1) given the set of prior parameters, generate the parameters (c, α), (2) using the generated population parameters, generate a bivariate random sample of size n, say (X 1 , Y 1 ), ..., (X n , Y n ) as shown in subsection 1.2 (3) follow steps in Subsections 2.1 and 2.2.
In Tables (1) and (2) 95% BP I s are computed in case of the one-and two-sample predictions, respectively, with the same parameters c, α, hyperparameters c 1 , c 2 , c 3 and using informative samples of different sizes, r.

Concluding remarks
In Tables (1) and (2) we take different sizes for the informative sample, 10, 20 and 45 and predict the first three future observations .
In these tables, we observe that (1) The length of the BP I s and the number of samples which cover these intervals increase by increasing s and decrease by increasing the informative sample size.
(2) The results become better as the informative sample size r gets larger.
(3) In all cases, the simulated percentage coverages are at least 95%.
(5) If the hyperparameters are unknown, they can be estimated by using the empirical Bayes method [see Maritz and Lwin [13]] or the hierarchical method [see Bernardo and Smith [11]].

Proof of equations (16)-(18)
From the joint density function of the random variables X and Y which is given by (13) and using the transforms X = Z cos Θ and Y = Z sin Θ we get the joint density function of the random variables Z and Θ in the form The cdf (17) is obtained by integrating by parts the integral in (A.2). The rf is then obtained as in (18), since R(z) = 1 − F Z (z).