# Research

## Frontiers in Statistics

### Friday, December 5, 2008

Title: Growth Mixture Modeling as an Exploratory Analysis Tool in Longitudinal QTL
Speaker: Stephen J. Finch, State University of New York at Stony Brook
Time: 3:30pm‐4:30pm
Place: PHY 118

#### Abstract

Growth mixture modeling (GMM) is a new and important tool for analyzing longitudinal data that may have value in genetic epidemiology. We study the properties of one common GMM statistical package, the SAS TRAJ procedure. We examined three tests: the likelihood ratio test statistic, a direct test of genetic model coefficients, and the chi-square test classifying subjects based on the model posterior Bayesian probability. The null distributions of these tests may be sensitive to departures for Hardy-Weinberg equilibrium in the gene analyzed. Such departures are often due to genotyping error. The power to detect the association with a longitudinal QTL varied by the size of the genetic effect with power essentially one for a gene with large effect. Power was decreased when markers near the gene rather than the gene itself were used indicating that GMM may be useful in genome wide association studies. The Mplus software was much more likely to show failure of convergence and required substantial computing resources.

### Friday, November 21, 2008

Title: Nonlinear Stochastic Modeling and Statistical Analysis
Speaker: Ling Wu
Time: 3:00pm‐4:00pm
Place: PHY 118

#### Abstract

We will first review linear stochastic modeling briefly. Then we will show our algorithms to construct nonlinear stochastic models. We will also present the simulation results using different data partitions. Then we will statistically analyze the jumps, with respect to the length between jumps and the magnitude of jumps.

### Friday, November 14, 2008

Title: Role and Scope of Monte-Carlo Simulation in Stochastic/Statistical Modeling
Speaker: A. Korzeniowski, Department of Mathematics
University of Texas, Arlington TX
Time: 3:00pm‐4:00pm
Place: PHY 118

None given.

### Friday, September 26, 2008

Title: Kernel Density Estimation as an Alternative to the Gumbel Distribution in Modeling Extreme Quantiles and Return Periods for the Purpose of Flood Prevention
Time: 3:00pm‐4:00pm
Place: PHY 118

#### Abstract

The Gumbel probability distribution function has been the function of choice for modeling hydrological extremes since the 1950's. There has been growing evidence in recent years that floods seem to have heavier tails than the Gumbel distribution. Other studies have extended the skepticism for the Gumbel distribution by showing that it underestimates the largest rainfall amounts. In this talk, six models of quantiles and return levels for the annual maxima stream flow of the Hillsborough River, Florida, will be proposed and evaluated. It will be shown that even though the Gumbel distribution provides a good overall fit to the Hillsborough River annual maxima, the non-parametric kernel density provides closer estimates in the tails.

### Friday, September 19, 2008

Title: Classification of Cancers using the Genes selected by Behrens-Fisher Distribution
Speaker: Nabin K. Shrestha
Time: 3:00pm‐4:00pm
Place: PHY 118

#### Abstract

Microarray Data analysis has been one of the most active area of research, because of its small sample size and large number of variables. Although the clasification problem is very old, but it still faces many challenges in the case of microarray data. In my talk, I will (1) introduce the microarray data and related problems (2) Brhrens-Fisher distribution in the Bayesian Settings to select the “marker genes”, and (3) use these genes for calssification of cancers.

### Friday, September 12, 2008

Title: Statistical Analysis of copy number variants data based on the likelihood ratio method
Speaker: Wonkuk Kim
Time: 3:00pm‐4:00pm
Place: PHY 118

#### Abstract

The copy number polymorphisms (CNP) may play an important role in genetic disease. However, the existing statistical methods to analyze the CNP data have not been well developed compared to the single nucleotide polymorphism (SNP) data. My talk covers (1) mixture distribution analysis (2) power analysis of the likelihood ratio test (3) differential error mechanisms, and briefly (4) trend tests.

### Friday, September 5, 2008

Title: A study of present value maximization of monopolist: Continuous and Discrete cases
Speaker: Keshav Pokhrel
Time: 3:00pm‐4:00pm
Place: PHY 118

#### Abstract

The present value of the investment of the monopolist for the continuous case is given by $$PV=\int_0^\infty e^{-rt}q(t)p(t)\,dt$$ with demand conditions $$p(t)=f(t)-q(t)-a_1q'(t)-a_2q''(t), \tag{C}$$ and the present value for the discrete case is $$PV=\sum_{0}^{\infty}\,\beta^t q_tp_t$$ with demand condition $$p_t=f_q-q_t-\alpha q_{t-1}, \tag{D}$$ provided $$0 < \beta\le 1$$.

We will discuss various conditions and possibilities of maximization of present value of a monopolist. Basically we are focused on the Continuous (C) and Discrete (D) cases. In (C), there is an exponential approach o9f growth if and only if $$a_2\ne 0$$. The boundary conditions in (C) generate some mathematical issues. The first derivative of the quantity $$q'(t)$$ has finite jump at $$t=0$$. If $$a_2=0$$ then the jump is similar to the jump of $$q(t)$$ at $$t=0$$. If the sufficient conditions for the problems in (C) are satisfied, then the demand equation is unstable. Finally, in (C) the maximum positive discount rate depends on $$a_1$$ and $$a_2$$ that yields finite maximum present value.

In (D) we do not need to have any adjustment as long as $$\alpha\ne 0$$. The sufficient conditions for maximum present value are satisfied for all $$t\in (0,1)$$. The optimal path is uniquely determined by the boundary condition and the choice of discount factor $$\beta$$. The stability of discount factor is a major player in problem (D). The stable demand condition implies the existence of bounded finite maximum present value for all $$\beta\le 1$$.