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Statistics colloquia series

Engaging with universities and industry bodies around the world, the Department of Mathematics and Statistics presents a series of regular colloquiums and seminars during the year. The programs enhance the learning environment, linking students and staff with academic and industry experts.

For information on the statistics colloquium or seminar series at Bundoora, please contact Dr Andriy Olenko on 03 9479 2609 or by sending an email to A.Olenko@latrobe.edu.au.

Future programs

Title: Estimation for Non-negative Le´vy-driven CARMA Processes

• Speaker: Prof. Peter Brockwell, Colorado State University and The University of Melbourne.
  
• Time & date: 11.00am Friday 23 October.
• Venue: Room 310 (Access Grid Room), Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: Continuous-time autoregressive moving average (CARMA) processes with a non-negative kernel and driven by a non-decreasing Le´vy process constitute a useful and very general class of stationary, non-negative continuous-time processes which have been used, in particular, for the modeling of stochastic volatility. Brockwell, Davis and Yang (J. Appl. Prob., 2007) derived efficient estimates of the parameters of a non-negative Le´vy -driven CAR(1) (or stationary Ornstein-Uhlenbeck) process and showed how the realization of the underlying Le´vy process can be estimated from closely-spaced observations of the process itself. In this talk we show how the ideas of that paper can be generalized to higher order CARMA processes with non-negative kernel, the key idea being the decomposition of the CARMA process into a sum of dependent Ornstein-Uhlenbeck processes. (Joint work with Richard Davis and Yu Yang.)

Title: Volatility in the Black-Scholes formula

• Speaker: Dr KaisHamza, Monash University.
  
• Time & date: 11.00am Friday 6 November 2009.
• Venue: Room 310 (Access Grid Room), Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: The Black-Scholes formula has been derived under the assumption of constant volatility in stocks. In spite of evidence that this parameter is not constant, this formula is widely used by financial markets. This talk addresses the question of whether a model for stock price exists that is consistent with the Black-Scholes formula. The results will also be extended to Bachelier and similar formulae.

Past programs

Title: Distributions of quadratic functionals of the ordinary and fractional Brownian motions

• Speaker: Prof. Katsuto Tanaka, Hitotsubashi University, Japan.
  
• Time & date: 10.30am Wednesday 16 September 2009.
• Venue: Room 310 (Access Grid Room), Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: I discuss distributions of quadratic functionals of the ordinary Brownian motion (Bm) and fractional Bm. As far as the ordinary Bm is concerned, quadratic functionals were earlier suggested by several authors as test statistics for goodness of fit tests. Recently, such functionals were used in time-series based econometrics. In this talk I demonstrate how to derive and compute the distributions of such functionals by using various examples. I also discuss ratios of such functionals. Moreover I discuss quadratic functionals of the fractional Bm, but it turns out that the case of the fractional Bm is difficult and there remain some problems to be solved.

Title: A new and practical influence measure for subsets of covariance matrix sample principal components with applications to high dimensional datasets

• Speaker: Connie Li Wai Suen, La Trobe University.
  
• Time & date: 11.00am Friday 18 September 2009.
• Venue: Room 310 (Access Grid Room), Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: Principal Component Analysis (PCA) is a widely-used tool in multivariate analysis. Although much has been done in regards to sensitivity analysis and the development of influence diagnostics for the eigenvector estimators that define sample principal components, little, if any, has been done in this setting regarding sample principal components themselves. We develop a sensitivity measure for principal components associated with the covariance matrix that is very much related to the influence function (Hampel, 1974). This influence measure is based on the average squared canonical correlation and differs from existing measures in that it assesses influence of certain observational types on the sample principal components. We use this measure to derive an influence diagnostic that satisfies two key criteria being (i) it detects influential observations with respect to subsets of sample principal components and (ii) is efficient to calculate even in high dimensions. We use microarray datasets to show that our measure satisfies both criteria.

Title: Applications of Meta-regression using Maximum Likelihood

• Speaker: Michael Malloy, La Trobe University.
  
• Time & date: 11.00am Friday 18 September 2009.
• Venue: Room 310 (Access Grid Room), Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: An analysis of clinical trials using meta-regression is a practice which is not only of great importance to the medical world, but other research fields as well. This talk will discuss the application of meta-regression using maximum likelihood in the statistical software package R. A generalized linear model (GLM) is assumed from which maximum likelihood estimates using an inbuilt minimiser function are obtained. We then describe how to compute the Hessian matrix for a GLM from which approximate variances of the coefficients are achieved. Real life data examples will be provided.

Title: Expansions and variance inequalities of Poisson functionals

• Speaker: Prof. Guenter Last, University of Karlsruhe.
  
• Time & date: 11.00am Friday 11 September 2009.
• Venue: Room 310 (Access Grid Room), Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: The Poisson process is a fundamental object of probability theory and is important for both theory and applications. It can be defined on arbitrary measurable spaces. In the first part of the talk we discuss some basic properties of this process. Then we proceed with an explicit Fock space representation for Poisson processes in terms of iterated difference operators. Some interesting consequences of this representation are the Wiener-Ito chaos expansion and variance inequalities for square integrable functions of the Poisson process. The talk is based on joint work with Mathew Penrose (Bath).

Title: Multifractality of Products of Geometric Ornstein-Uhlenbeck Type Processes

• Speaker: Prof. N.N. Leonenko, Cardiff University, UK.
  
• Time & date: 11am Friday 14 August 2009.
• Venue: Room 310 (Access Grid Room), Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: This is joint work with V.V. Anh (Queensland University of Technology, Brisbane) and N.-R. Shieh (National Taiwan University, Taipei).
  
  We consider multifractal products of stochastic processes as defined in Mannersalo et al. (2002), but we provide a new interpretation of the conditions on the mean, variance and covariance functions of the resulting cumulative processes in terms of the moment generating functions. We show that the logarithms of the corresponding limiting processes have an infinitely divisible distribution such as the gamma and variance gamma distributions (resulting in the log-gamma and log-variance gamma scenarios respectively), the inverse Gaussian and normal inverse Gaussian distributions (yielding the log-inverse Gaussian and log-normal inverse Gaussian scenarios respectively). We describe the behaviour of their q-th order moments and Rényi functions, which are non-linear, hence displaying their multifractality. A property on the dependence structure of the limiting processes, leading to their possible long-range dependence, is also obtained.
  
  We consider very different scenarios such as the log-gamma and log-inverse Gaussian scenarios as typical examples of our approach. We should also note some related results by Barndorff-Nielsen and Schmiegel (2004) who introduced some Lévy-based spatiotemporial models for parametric modelling of turbulence. Log-infinitely divisible scenarios related to independently scattered random measures were introduced in Bacry and Muzy (2003) and others. We should note that Chris Heyde (1999) proposed to use a multifractality into risky asset model with fractal activity time (see also Heyde and Leonenko (2005)).
  
  Similar results can be obtained for the multifractal products of stationary diffusion processes (Anh, Leonenko and N.-R. Shieh (2009b)) and birth-death processes processes (Anh, Leonenko and N.-R. Shieh (2009a)).
  

  
  References
  V.V. Anh, N.N. Leonenko and N.-R. Shieh (2008) Multifractality of products of geomertic Ornstein-Uhlenbeck type processes, Adv. Appl. Prob., 40, 1129-1156.
  V.V. Anh, N.N. Leonenko and N.-R. Shieh (2009a) Multifractal scaling of products of birth-death processes, Bernoulli, 15 (2), 508-531
  V.V. Anh, N.N. Leonenko and N.-R. Shieh (2009b) Multifractal products of stationary diffusion processes, Stochastic Analysis and Applications, 27, 475-499
  E. Bacry and J.F. Muzy (2003) Log-infinitely divisible multifractal processes, Comm. Math. Phys., 236, 449.475

Title: Honours Thesis Presentations

• Time & date: 2pm Friday 7 August 2009.
• Venue: Room 310 (Access Grid Room), Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: Honours students will give presentations detailing work that they covered in their thesis topic.
  

Title: Testing the equality of error distributions from k independent GARCH models

• Speaker: Ajay Chandra, Department of Mathematics and Statistics, La Trobe University.
  
• Time & date: 11.00am Friday 17 July 2009.
• Venue: Room 310 (Access Grid Room), Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: We study the problem of testing the null hypothesis that errors from k independent parametrically specified generalized autoregressive conditional heteroskedasticity (GARCH) models have the same distribution versus a general alternative. First we establish the asymptotic validity of a class of linear test statistics derived from the k residual-based empirical distribution functions. A distinctive feature is that the asymptotic distribution of the test statistics involves terms depending on the distributions of errors and the parameters of the models, and weight functions providing the flexibility to choose scores for investigating power performance. A Monte Carlo study assesses the asymptotic performance of the Wilcoxon and Van der Waerden tests in terms of empirical size and power in finite samples. The results demonstrate that the proposed tests have overall reasonable size and their power is particularly high when the assumption of Gaussian errors is violated.
  

Title: Kullback-Leibler Information for Model Selection and Comparison in Logistic Linear Models

• Speaker: Guoqi Qian, Department of Mathematics and Statistics, University of Melbourne.
  
• Time & date: 11.30am Friday 5 June 2009.
• Venue: Room 310 (Access Grid Room), Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: We consider two issues involved in model selection:
  (1) Model selection starts with a proposed model class, and it is often unrealistically assumed that the true model generating the data belongs to this model class.
  Then what would happen to model selection if the true model is not in the proposed model class? In other words, how to quantify the model selection bias in the situation of model class mis-specification?
  (2) Model selection often ends up with a selected optimum model minimizing or maximizing a numeric selection criterion function. But it does not or is not able to provide a measure of variability or uncertainty involved in model selection. Such a measure, if available, would be very useful in determining models which are indistinguishable from the optimum model.

  
  We have developed new estimators of Kullback-Leibler information to address these two issues. Our work will be presented in the context of logistic regression model selection but can be extended to other model selection problems.

Title: Variance stabilisation approach to meta-analysis: combining the evidence

• Speaker: Elena Kulinskaya, Director of Statistical Advisory Service, Imperial College, London.
  
• Time & date: 11am Wednesday 26 November 2008.
• Venue: Room 310 (Access Grid Room), Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: In the traditional fixed effects model (FEM) of meta analysis, given the estimated effects from K studies θ1,…, θK , with θi ~N(θ, σi2) , the combined effect θ is estimated as the weighted mean θest =(wi θ1+…+ wK θK)/W ~N(θ ,1/ W ) , where wi=σi-2 and W= (w1+…+ wK ). If the homogeneity of the effects is rejected, the random effects model can be used: θi ~N(θ, σi-2 +τ2). (Sutton et al, 2000).
  
  When the variance stabilizing transformation (vst) is applied to the estimated effects, we deal instead with the transformed standardised effects K(δi). They are estimated by κi =ni-1/2h(Si) ~N(K(δ),1/ni) and can be added with known weights ni in meta-analysis (Kulinskaya, Morgenthaler and Staudte, 2008).
  
  Given variance stabilized statistics from K studies T1,…,TK , with T1 ~N(ni1/2 κ ,1) , the combined effect κest=(n1 κ1+…+nK κK))/ N ~N(K(δ),1/ N ) where N= n1 +…+nK. The back-transformation is used to obtain the inference on the standardised effects δ. If the homogeneity of the transformed effects is rejected, the random transformed effects model can be used: κ i ~N(κ , ni-1 +τ2).
  
  When there are no nuisance parameters (as in the 1-sample Binomial or Poisson case) these two approaches to meta analysis are equivalent. In the general case, the variance stabilization approach can be used even when the inference on the original, non-standardised effects is of primary interest. In this case the optimal weights depend on the nuisance parameters. An example is the variance stabilizing arcsine transformation for the difference in absolute risks, with the average risk as the nuisance parameter.

Title: On the Asymptotic Variance of Vacancy for Boolean Models with Grain Distortions

• Speaker: Christian Rau, School of Mathematical Sciences, Monash University.
• Time & date: 11am Friday 14 November 2008.
• Venue: Room 310 (Access Grid Room), Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: A Boolean model is a spatial coverage process whose driving point process is homogeneous Poisson, and whose attached random sets, or grains, are independent and identically distributed (i.i.d). Apart from a host of traditional applications, Boolean models have been employed recently in the modelling of sensor networks, which motivated this research. The asymptotic variance of vacancy (AVV) in the Boolean model is defined by letting the intensity of the Poisson process diverge to infinity, and simultaneously scaling the grains to become small. We consider optimality and continuity properties of the AVV when the grains are subject to i.i.d. distortions, which includes rotations and shearings as special cases. An important role in the formulation and derivation of our results is played by notions of symmetry well known from multivariate analysis and stochastic simulation, such as conjugation-invariance and group models. This is joint work with Sung Nok Chiu, Hong Kong Baptist University.

Title: Honours Thesis Presentations

• Time & date: 1pm Friday 31 October 2008.
• Venue: Room 310 (Access Grid Room), Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: Six honours student will give a 20 minute presentation detailing work that they covered in their thesis topic.

Title: Multiscale Simulation of Biochemical Systems

• Speaker: Linda Petzold, University of California Santa Barbara.
  
• Time & date: 2pm Friday 22nd July 2008.
• Venue: SEMS meeting room 221, Physical Sciences 1, La Trobe University, Bundoora Campus.
• Abstract: In microscopic systems formed by living cells, the small numbers of some reactant molecules can result in dynamical behavior that is discrete and stochastic rather than continuous and deterministic. An analysis tool that respects these dynamical characteristics is the stochastic simulation algorithm (SSA). Despite recent improvements, as a procedure that simulates every reaction event, the SSA is necessarily inefficient for most realistic problems. There are two main reasons for this, both arising from the multiscale nature of the underlying problem: (1) the presence of multiple timescales (both fast and slow reactions); and (2) the need to include in the simulation both chemical species that are present in relatively small quantities and should be modeled by a discrete stochastic process, and species that are present in larger quantities and are more efficiently modeled by a deterministic differential equation. We will describe several recently developed techniques for multiscale simulation of biochemical systems, and outline some of the future challenges.

Title: Bayesian Methods in Software Testing and Capability Maturity Model

• Speaker: Salilesh Mukhopadhyay, Feasible Solution Inc, USA.
  
• Time & date: 2pm Friday 4 July 2008.
• Venue: Room 310 (Access Grid Room), Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: Most of the time the testing phase of the application does not allow an end-to-end testing with all the interfaces up and running in QA environment. To remedy the situation the common practice is to have coverage analysis with appropriate risk mitigation in Finacial (large) applications mainly. However the testing scenarios of E-Commerce, Business to Business Models, Electronic Raw Material Acquisition, Auction Engines are not at all different. The present paper provides a statistical analysis of the scenarios and calculates the associated risk for each phase of testing cycle like, Unit, System Integration Testing, End-to-End Testing, UAT.
  
  The purpose of the present paper is to provide an outline of Bayesian Methods (Prior and Posterior analysis) in Software testing with special emphasize on Capability Maturity Model. Different types of testing scenarios will be analysed for each phase of testing like Unit testing, System Integration Testing, End-to-End Testing, User Acceptance Testing and post production maintenance Testing. Manual and Automated testing will be discussed in details for stable QA environment. Finally the benefits of using quantitative analysis to mitigate the associated risk in each phase will be discussed in details.

Title: The Coverage Probability of Confidence Intervals in 2r factorial Experiments after Preliminary Hypothesis Testing

• Speaker: Khageswor Giri, Department of Mathematics and Statistics, La Trobe University.
  
• Time & date: 2pm Friday 23 May 2008.
• Venue: Room 310 (Access Grid Room), Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: We consider a 2r factorial experiment with at least two replicates. Our aim is to find a confidence interval for θ, a specified linear combination of regression parameters. We suppose that preliminary hypothesis tests are carried out sequentially beginning with the rth order interaction. After these preliminary hypothesis tests, a confidence interval for θ with nominal coverage 1-α is constructed under the assumption that the selected model is given to us a priori. We call this the naïve 1-α confidence interval for θ. We describe a new efficient Monte Carlo method, which employs conditioning for variance reduction, for estimating the minimum coverage probability of the naive confidence interval. The application of this method is demonstrated in the context of a 23 factorial experiment with 2 replicates and a particular contrast θ of interest. The naive confidence interval, with nominal coverage probability 0.95, has minimum coverage probability that is, to a good approximation, 0.464. This shows that the naive confidence interval is completely inadequate.

Title: Variance stabilizing the risk difference to obtain confidence intervals for effects and effect sizes

• Speaker: Associate Professor, Dr Robert Staudte, Department of Mathematics and Statistics, La Trobe University.
  
• Time & date: 2pm Friday 16 May 2008.
• Venue: Room 310 (Access Grid Room), Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: I will be discussing the study of dimension reduction of high dimensional data for binomial response variable data sets when the number of individuals sampled is less than the number of measurement variables. The application of principle component analysis as a prestep to Sliced Inverse Regression and Sliced Average Variance Estimation will be presented. In this setting in order to produce an efficient algorithm, an approximation technique using first and second order perturbations is suggested as a one step eigen-analysis in the cross-validation step of the classification of the binomial response variable data sets.

Title: Approximating cross-validation results for binary classification methods preceded by principal component dimension reduction

• Speaker: Mitra Jazayeri, Department of Mathematics and Statistics, La Trobe University.
  
• Time & date: 2pm Friday 9 May 2008.
• Venue: Room 310 (Access Grid Room), Physical Sciences 2, La Trobe University, Bundoora Campus.
  • Abstract: The usual estimator of the risk difference is variance stabilized, conditionally on an estimated weighted average of the unknown risks. This leads to conditional confidence intervals for the standardized risk difference, and hence for a correlation effect size. In addition, it leads to confidence intervals for the risk difference itself, with more accurate unconditional coverage than those obtained by standard asymptotic methods, as shown by simulations studies. Methods for combining the results of several studies are presented, and illustrated on nine independent randomized clinical trials of the effect of diuretics on pre-eclampsia.

Title: On Survival Equivalence Function

• Speaker: Prof. T. K. Pogany, University of Rijeka, Croatia .
  
• Time & date: 2pm Friday 29 February 2008.
• Venue: Room 310 (Access Grid Room), Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: The reliability of composite systems is improved by decreasing the argument of the associated survival function in the case of general positive i.i.d. random life components connected in series (paralell). The gamma-Weibull distribution has been recently introduced by Leipnik and Pearce. The density function, the probability distribution function and the characteristic function are of is expressed in terms of the confluent Fox-Wright generalization of the hypergeometric function, and its incomplete variant. The composite series (paralell) systems reliability is presented taking for the case study system components having gamma-Weibull life distribution.

Title: Parameter Estimation and Bias Correction for Diffusion Processes

• Speaker: Prof. Song Xi Chen, Department of Statistics, Iowa State University.
  
• Time & date: 2pm Friday 7 March 2008.
• Venue: Room 310 (Access Grid Room), Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: This lecture considers parameter estimation for continuous-time diffusion processes which are commonly used to model dynamics of financial securities including interest rates. To understand why the drift parameters are more difficult to estimate than the diffusion parameter as observed in many empirical studies, we develop expansions for the bias and variance of parameter estimators for two mostly employed interest rate processes. A parametric bootstrap procedure is proposed to correct bias in parameter estimation of general diffusion processes with a theoretical justification. Simulation studies confirm the theoretical findings and show that the bootstrap proposal can effectively reduce both the bias and the mean square error of parameter estimates for both univariate and multivariate processes. The advantages of using more accurate parameter estimators when calculating various option prices in finance are demonstrated by an empirical study on a Fed fund rate data.