Notice
For the 2011-2012 academic year, Debanjan Mitra will be organizing the graduate seminars. Good luck!
April 12th 2011: Amidu Raifu
This seminar has been cancelled and will be rescheduled later in the year.
March 15th 2011: Man Ho Ling
EM algorithm for one-shot device testing
The EM algorithm is a powerful technique for determining the maximum likelihood estimates in the presence of binary data since the maximum likelihood estimators of the parameters cannot be expressed in a closed-form. In this talk, we consider one-shot devices that can be used only once and are destroyed after use, and so the actual observation is on the conditions rather than on the real lifetimes of the devices under test. Here, we develop the EM algorithm for this model. Due to the advances in manufacturing design and technology, products have become highly reliable with long lifetime. For this reason, accelerated life tests are performed to collect information on the parameters of the lifetime distributions. For such a test, the Bayesian approach with normal prior was proposed by Fan(2009). Here, through a simulation study for point estimation, we show that the EM algorithm and the mentioned Bayesian approach are both useful for analyzing such binary data arising from one-shot device testing and then make a comparative study of their performance
February 15th 2011: Debanjan Mitra
Likelihood Inferential Procedures for Lognormal Data with Left Truncation and Right Censoring
The lognormal distribution is quite commonly used as a lifetime distribution. Data arising from life-testing and reliability studies are often left truncated and right censored. Here, the EM algorithm is used to estimate the parameters of the lognormal model based on left truncated and right censored data. The maximization step of the algorithm is carried out by two alternative methods, with one involving approximation using Taylor series expansion (leading to approximate maximum likelihood estimate) and the other based on the EM gradient algorithm (Lange, 1995). These two methods are compared based on Monte Carlo simulations. The Fisher scoring method for obtaining the maximum likelihood estimates shows a problem of convergence under this setup, except when the truncation percentage is small. The asymptotic variance-covariance matrix of the MLEs is derived by using the missing information principle (Louis, 1982), and then the asymptotic confidence intervals for scale and shape parameters are obtained and compared with corresponding bootstrap confidence intervals. Finally, some numerical examples are given to illustrate all the methods of inference developed here.
Slides [pdf]
January 11th 2011: William Volterman
Exact Nonparametric Meta-Analysis for Multiple Doubly Type-II Censored Samples
In this talk, we consider the situation where multiple independent doubly type-II censored samples are observed. We present the basic distribution theory for the pooled sample, when the underlying distribution is continuous. We demonstrate how to obtain the necessary mixture weights to represent the pooled order statistics as a mixture of the usual order statistics. Simulation is compared to the exact results for some specific examples. Using this, we are able to construct nonparametric prediction intervals, tolerance intervals for a future sample, and confidence intervals for a population quantile. Since computation can be complex, we do a small simulation to compare analytical results to those obtained from a simulation.
Slides [pdf]
December 14th 2010: Ahmed Roby
This seminar has been cancelled and will be rescheduled later in the year.
November 9th 2010: Fang Xu
The Poisson-Dirichlet distribution and Ewens Sampling Formula
The Poisson-Dirichlet distribution describes the ranked masses of atoms in a Dirichlet process and it appears as an asymptotic distribution in number theory, combinatorics and population genetics. Though it is describe the finite dimensional distribution explicitly, Ewens sampling formula appears a remarkably simple formula related with this distribution. Moreover, the formula has a number of applications in the nonparametric problems of Bayesian inference and exchangeable random partitions. Besides these models, I will also discuss the generalized two-parameter model. Our result shows that the sampling formula determines the underlying population distribution which belongs to the two-parameter family distributions.
Slides [pdf]- Feng, S. (2010): Poisson-Dirichlet distribution and relatedtopics. Springer, Berlin.
- Hoppe, F.M. (1984): Polya-like urns and the Ewens sampling formula. J. Math. Biol., 20, 91-94 [Web Link]
- Kingman, J.F.C. (1975): Random discrete distributions. J. Roy. Statist. Soc. B, 37, 1-22. [Web Link]
- Perman, M., Pitman, J. and Yor, M. (1992): Size-biased sampling of Poisson point processes and excursions. Prob. Thoery Relat. Fields. 92, 21-39. [Web Link]
- Pitman, J. (1995): Exchangeable and partially exchangeable random partitions. Prob. Thoery Relat. Fields. 102, 145-158. [Web Link]