Ordinal response models are commonly used to describe how opinions, judgments, evaluations, preferences, and so forth, depend on subjects’ covariates. Since the support of the response variable is discrete and finite, it is often assumed that classical estimation and testing techniques are not affected by deviations from the stochastic assumptions. Nevertheless, outlying covariates as well as anomalous responses can strongly affect the reliability of likelihood based inferential procedures. Consequently alternative robust estimators and tests are proposed, and their properties investigated. Furthermore intrinsic robustness features of the link functions are also considered.

In multivariate analysis, estimating the location vector and dispersion matrix is a fundamental step for many applications. Classical sample mean and covariance estimates are very sensitive to outliers, and therefore their robust counterparts are considered to overcome the problem. For p-dimensional data, the robust mean vector requires estimating p parameters, while the robust covariance matrix requires estimating p(p-1)/2 parameters, and the resulting matrix needs to be positive definite. Therefore, covariance estimation is more challenging than mean estimation. We propose a new robust covariance estimator using the regular vine dependence structure and pairwise robust partial correlation estimators. The resulting positive definite robust covariance estimator delivers high performance for identifying outliers under the Barrow Wheel Benchmark for large high dimensional datasets. Finally, we demonstrate a financial application of active asset allocation using the proposed robust covariance estimator, and the proposed estimator delivers better results compared to many existing asset allocation methods. (Joint work with Zhe Zhu)

We consider the problem of inference for a scalar interest parameter in the presence of a nuisance parameter, using a likelihood-based statistic which is asymptotically normally distributed under the null hypothesis. Two approaches to calculation of an approximate p-value are: analytic methods based on normal approximation to an adjusted form of statistic; simulation (`bootstrap') approximation to the null sampling distribution of the statistic. Higher-order expansions are used to compare the sampling distributions, under a general contiguous alternative hypothesis, of p-values calculated by these different approaches. We establish that comparisons in terms of power under an alternative hypothesis are intrinsically linked to the extent to which testing procedures are conservative or anti-conservative under the null. Empirical examples are discussed which demonstrate that higher-order asymptotic effects may be clearly seen in small sample contexts. This is joint work with Stephen Lee (University of Hong Kong).

We study the sufficient dimension reduction problem in the ultra high dimensional covariate setting. Through introducing latent factors, we avoid the usual sparsity assumption and do not resort to penalization for screening of the covariates. Our treatment does not make the normality assumption on the latent factor distributions. We derive the asymptotic distribution theory of the method. The procedure can be further generalized to sufficient direction analysis in generalized linear latent variable models.

Linear mixed models are widely used in a range of application areas, including ecology and environmental science. We study in detail the effects of fitting the two-level linear mixed model with a single explanatory variable that is misspecified because it incorrectly ignores contextual effects. In particular, we make explicit the effect of (the usually ignored) within-cluster correlation in the explanatory variable. This approach produces a number of unexpected findings. (i) Incorrectly omitting contextual effects affects estimators of both the regression and variance parameters not just, as is currently thought, estimators of the regression parameters and the effects are different for different estimators. (ii) Increasing the within cluster correlation of the explanatory variable introduces a second local maximum into the log-likelihood and REML criterion functions which eventually becomes the global maximum, producing a jump discontinuity (at different values) in the maximum likelihood and REML estimators of the parameters. (iii) Standard statistical software such as SAS, SPSS, STATA, lmer (fromlme4 in R) and GenStat often returns local rather than global maximum likelihood and REML estimates in this very simple problem. (iv) Local maximum likelihood and REML estimators...

Variance swaps are basic contracts to trade volatility. Over the past few decades variance swap markets have experienced an impressive growth, reaching enormous trading volumes. I will review recent approaches to model the term structure of variance swaps, inspired by the term structure literature on interest rates. Then, I will present theoretical and empirical analyses of optimal investment problems in variance swaps, index option, stock index, and risk free bond. This talk is mostly based on joint work with Damir Filipovic and Elise Gourier.