Indian statistical institute

Gopal K. Basak
Title: Adaptive MCMC for general target and proposal distribution through diffusion approximation

(Jointly With Arunangshu Biswas (Stats, Presidency Univ))

Earlier we have worked on the case when the target and the proposal distribution both are standard Normal. We have used moment matching (or the matching of mgfs) to recover the target distribution. Therefore it inherently assumed that the target distribution was light tailed. In many situations the standard Normal is the choice as a proposal, since generating samples from it is easy (for example, using the Box Muller technique). Also in the proof of the diffusion approximation it requires the existence of the first two moments of the proposal. A natural question is how can the results can be extended for general target and proposal distributions. We try to address these issues here. We classify the target

densities depending their existence of m.g.f.s or moments and give criteria that correspond to the existence of the m.g.f of the target density in the whole of R or in a neighbourhood of zero. We further classify the remaining densities with no m.g.f.s with all moments finite or only a few moments finite. We show that the limiting distribution of the diffusion corresponding to the AMCMC also share the same property. We further show that the diffusion approximation method cannot be used for simulation when

the standard Cauchy is the target or density with a finitely moments is the target. We then obtain the diffusion approximation for light tailed proposals. We also explain why heavy tail choices (such as the Cauchy distribution) as the proposal distribution will not work. Specifically, we investigate what goes wrong when we look at the localized infinitesimal drift and diffusion coefficient when the proposal distribution is Cauchy. Work is continuing for further development.

Gopal K. Basak
Title: The dynamics of foreign capital inflow and financial crisis

(Jointly with Pranab K. Das (CSSSC, Kolkata) and Allena Rohit (ICICI, a former M.Stat. Student))

In this work we model foreign capital inflow in a multi-period framework from the developed to the developing countries. The market for foreign loan together with the foreign exchange market simultaneously determines interest rate in the international loan market and the exchange rate. We also derive the conditions for existence of meaningful equilibrium solutions. Because of non- linearity of the functions we adopt a numerical solution method. A number of comparative dynamic analyses explore the impact of parameters of the model on the endogenous variables. The model is then used to explain the possibility of financial crisis originating either in the developed country or in the developing country. The explanation of crisis in this structure is based on trade theoretic terms in a dynamic terms of trade framework rather than in terms of informational imperfections. Work is continuing.