New in the present paper is a differential-geometrical framework for analyzing statistical problems related to the Weibull distribution using a one-parameter family of affine connections. The geometry of the statistical manifold is given. A development of the relation between the J-divergence and geodesic distance is obtained. Finally, the curvatures of the affine connections for each of the curved J-space and S-space (constant curvature) on Weibull manifolds are derived.

New in the present paper is a differential-geometrical framework for analyzing statistical problems related to the Weibull distribution using a one-parameter family of affine connections. The geometry of the statistical manifold is given. A development of the relation between the J-divergence and geodesic distance is obtained. Finally, the curvatures of the affine connections for each of the curved J-space and S-space (constant curvature) on Weibull manifolds are derived.

A nonlinear discriminant rule may be estimated by maximum likelihood estimation using unclassified observations. The performance of a nonlinear discriminant function based on a sample from a mixture of two Weibull distributions, with parameters λ, θ1, θ2 and p, is examined. Asymptotic expansion and asymptotic expected values of probabilities of misclassification are presented. The asymptotic relative efficiencies (AREs) of mixture and classified discrimination procedures are evaluated and discussed for selected parameters. Computations show that for fixed λ and p, as Δ = | θ1 − θ2| increases the ARE increases. Furthermore, for fixed λ and Δ, as p varies from 0.2 to 0.8 the values of ARE decrease. On the other hand, for fixed p and Δ, the ARE in case of λ = 0.5 are close to the ARE in the case of λ = 2.

Information geometry (Geometry and Nature) has emerged from the study of invariant properties of the manifold of probability distributions. It is regarded as mathematical sciences having vast developing areas of applications as well as giving new trends in geometrical and topological methods. Information geometry has many applications which are treated in many different branches, for instance, statistical inference, linear systems and time series, neural networks and nonlinear systems, linear programing, convex analysis and completely integrable dynamical systems, quantum information geometry and geometric modelling. Here, we give a brief account of information geometry and the deep relationship between the differential geometry and the statistics[1,4,5,10,11].The parameter space of the random walk distribution using its Fisher's matrix is defined. The Riemannian and scalar curvatues of the parameter space are calculated. The differential equations of the geodesic are obtained and solved. The relations between the J-divergence and the geodesic distance in that space are found.

Information geometry (Geometry and Nature) has emerged from the study of invariant properties of the manifold of probability distributions. It is regarded as mathematical sciences having vast developing areas of applications as well as giving new trends in geometrical and topological methods. Information geometry has many applications which are treated in many different branches, for instance, statistical inference, linear systems and time series, neural networks and nonlinear systems, linear programing, convex analysis and completely integrable dynamical systems, quantum information geometry and geometric modelling. Here, we give a brief account of information geometry and the deep relationship between the differential geometry and the statistics[1,4,5,10,11].The parameter space of the random walk distribution using its Fisher's matrix is defined. The Riemannian and scalar curvatues of the parameter space are calculated. The differential equations of the geodesic are obtained and solved. The relations between the J-divergence and the geodesic distance in that space are found.

Information geometry (Geometry and Nature) has emerged from the study of invariant properties of the manifold of probability distributions. It is regarded as mathematical sciences having vast developing areas of applications as well as giving new trends in geometrical and topological methods. Information geometry has many applications which are treated in many different branches, for instance, statistical inference, linear systems and time series, neural networks and nonlinear systems, linear programing, convex analysis and completely integrable dynamical systems, quantum information geometry and geometric modelling. Here, we give a brief account of information geometry and the deep relationship between the differential geometry and the statistics[1,4,5,10,11].The parameter space of the random walk distribution using its Fisher's matrix is defined. The Riemannian and scalar curvatues of the parameter space are calculated. The differential equations of the geodesic are obtained and solved. The relations between the J-divergence and the geodesic distance in that space are found.

The main object of this paper is the approximate Bayes estimation of the five dimensional vector of parameters and the reliability function of a mixture of two Weibull distributions under Type-2 censoring. Under Type-2 censoring, the posterior distribution is complicated, and the integrals involved cannot be obtained in a simple closed form. In this work, Lindley's (1980) approximate form of Bayes estimation is used in the case of a mixture of two Weibull distributions under Type-2 censoring. Through Monte Carlo simulation, the root mean squared errors (RMSE's) of the Bayes estimates are computed and compared with the corresponding estimated RMSE's of the maximum likelihood estimates.