Kai Wang Ng

The Maximum of Randomly Weighted Sums with Long Tails in Insurance and Finance

In risk theory we often encounter stochastic models containing randomly weighted sums. In these sums, each primary real-valued random variable, interpreted as the net loss during a reference period, is associated with a nonnegative random weight, interpreted as the corresponding stochastic discount factor to the origin. Therefore, a weighted sum of m terms, denoted as , represents the stochastic present value of aggregate net losses during the first m periods. Suppose that the primary random variables are independent of each other with long-tailed distributions and are independent of the random weights. We show conditions on the random weights under which the tail probability of —the maximum of the first n weighted sums—is asymptotically equivalent to that of —the last weighted sum.

On improved EM algorithm and confidence interval construction for incomplete rXc tables

Constructing confidence interval (CI) for functions of cell probabilities (e.g., rate difference, rate ratio and odds ratio) is a standard procedure for categorical data analysis in clinical trials and medical studies. In the presence of incomplete data, existing methods could be problematic. For example, the inverse of the observed information matrix may not exist and the asymptotic CIs based on delta methods are hence not available. Even though the inverse of the observed information matrix exists, the large-sample delta methods are generally not reliable in small-sample studies. In addition, existing expectation-maximization (EM) algorithm via the conventional data augmentation (DA) may suffer from slow convergence due to the introduction of too many latent variables. In this article, for rxc tables with incomplete data, we propose a novel DA scheme that requires fewer latent variables and this will consequently lead to a more efficient EM algorithm. We present two bootstrap-type CIs for parameters of interest via the new EM algorithm with and without the normality assumption. For rxc tables with only one incomplete/supplementary margin, the improved EM algorithm converges in only one step and the associated maximum likelihood estimates can hence be obtained in closed form. Theoretical and simulation results showed that the proposed EM algorithm outperforms the existing EM algorithm. Three real data from a neurological study, a rheumatoid arthritis study and a wheeze study are used to illustrate the methodologies.

MODEL RISK IN VaR ESTIMATION: AN EMPIRICAL STUDY

This paper studies the model risk; the risk of selecting a model for estimating the Value-at-Risk (VaR). By considering four GARCH-type volatility processes exponentially weighted moving average (EWMA), generalized autoregressive conditional heteroskedasticity (GARCH), exponential GARCH (EGARCH), and fractionally integrated GARCH (FIGARCH), we evaluate the performance of the estimated VaRs using statistical tests including the Kupiecs likelihood ratio (LR) test, the Christoffersens LR test, the CHI (Christoffersen, Hahn, and Inoue) specification test, and the CHI nonnested test. The empirical study based on Shanghai Stock Exchange A Share Index indicates that both EGARCH and FIGARCH models perform much better than the other two in VaR computation and that the two CHI tests are more suitable for analyzing model risk.

A unified method for checking compatibility and uniqueness for finite discrete conditional distributions

Checking compatibility for two given conditional distributions and identifying the corresponding unique compatible marginal distributions are important problems in mathematical statistics, especially in Bayesian inferences. In this article, we develop a unified method to check the compatibility and uniqueness for two finite discrete conditional distributions. By formulating the compatibility problem into a system of linear equations subject to constraints, it can be reduced to a quadratic optimization problem with box constraints. We also extend the proposed method from two-dimensional cases to higher-dimensional cases. Finally, we show that our method can be easily applied to checking compatibility and uniqueness for a regression function and a conditional distribution. Several numerical examples are used to illustrate the proposed method. Some comparisons with existing methods are also presented.

Hierarchical models for repeated binary data using the IBF sampler

Hierarchical models have emerged as a promising tool for the analysis of repeated binary data. However, the computational complexity in these models have limited their applications in practice. Several approaches have been proposed in the literature to overcome the computational difficulties including maximum likelihood estimation from a frequentist perspective (e.g., J. Amer. Statist. Assoc. 89 (1994) 330-335) and Markov chain Monte Carlo (MCMC) methods from a Bayesian perspective (e.g., Generalized Linear Models: A Bayesian Perspective, Marcel Dekker, New York, pp. 113-131). Although MCMC methods provide the whole posterior of the parameter of interest, the convergence diagnostics problem of the Markov chain and the slow convergence problem owing to the introduction of too many Gaussian latent variables are still unresolved. Recently, Tan et al. (Statist. Sinica 13 (2003) 625-639) proposed a noniterative sampling approach, the inverse Bayes formulae (IBF) sampler, for computing posteriors in the structure of EM algorithm. This article develops the IBF sampler in the structure of Monte Carlo EM (MCEM) for the hierarchical model with repeated binary data for which current methods encounter difficulty. An efficient IBF sampler is implemented by utilizing the estimated posterior modes obtained via MCEM algorithm. The proposed method generates independent and identically distributed (iid) samples approximately from the observed posterior distribution and thus alleviates the convergence problem associated with the MCMC methods. In addition, the slow convergence problem in Gibbs sampler can be bypassed in the noniterative IBF sampler via running some fast EM-type algorithm. Real datasets from six cities childrens wheeze study and childrens ear fluid study illustrate the proposed methods.

EM-type algorithms for computing restricted MLEs in multivariate normal distributions and multivariate t-distributions

Constrained parameter problems arise in a variety of statistical applications but they have been most resistant to solution. This paper proposes methodology for estimating restricted parameters in multivariate normal distributions with known or unknown covariance matrix. The proposed method thus provides a solution to an open problem to find penalized estimation for linear inverse problem with positivity restrictions [Vardi, Y., Lee, D. 1993. From image deblurring to optimal investments: Maximum likelihood solutions for positive linear inverse problems (with discussion). Journal of the Royal Statistical Society, Series B 55, 569-612]. By first considering the simplest bound constraints and then generalizing them to linear inequality constraints, we propose a unified EM-type algorithm for estimating constrained parameters via data augmentation. The key idea is to introduce a sequence of latent variables such that the complete-data model belongs to the exponential family, hence, resulting in a simple E-step and an explicit M-step. Furthermore, we extend restricted multivariate normal distribution to multivariate t-distribution with constrained parameters to obtain robust estimation. With the proposed algorithms, standard errors can be calculated by bootstrapping. The proposed method is appealing for its simplicity and ease of implementation and its applicability to a wide class of parameter restrictions. Three real data sets are analyzed to illustrate different aspects of the proposed methods. Finally, the proposed algorithm is applied to linear inverse problems with possible negativity restrictions and is evaluated numerically.

Further properties and new applications of the nested Dirichlet distribution

Recently, Ng et al. (2009) studied a new family of distributions, namely the nested Dirichlet distributions. This family includes the traditional Dirichlet distribution as a special member and can be adopted to analyze incomplete categorical data. However, other important aspects of the family, such as marginal and conditional distributions and related properties are not yet available in the literature. Moreover, diverse applications of the family to the real world need to be further explored. In this paper, we first obtain the marginal and conditional distributions and other related properties of the nested Dirichlet distribution. We then present new applications of the family in fitting competing-risks model, analyzing incomplete categorical data and evaluating cancer diagnosis tests. Three real data involving failure times of radio transmitter receivers, attitude toward the death penalty and ultrasound ratings for breast cancer metastasis are provided.

Non-iterative sampling-based Bayesian methods for identifying changepoints in the sequence of cases of Haemolytic uraemic syndrome

Diarrhoea-associated Haemolytic uraemic syndrome (HUS) is a disease that affects the kidneys and other organs. Motivated by the annual number of cases of HUS collected in Birmingham and Newcastle of England, respectively, from 1970 to 1989, we consider Bayesian changepoint analysis with specific attention to Poisson changepoint models. For changepoint models with unknown number of changepoints, we propose a new non-iterative Bayesian sampling approach (called exact IBF sampling), which completely avoids the problem of convergence and slow convergence associated with iterative Markov chain Monte Carlo (MCMC) methods. The idea is to first utilize the sampling inverse Bayes formula (IBF) to derive the conditional distribution of the latent data given the observed data, and then to draw iid samples from the complete-data posterior distribution. For the purpose of selecting the appropriate model (or determining the number of changepoints), we develop two alternative formulae to exactly calculate marginal likelihood (or Bayes factor) by using the exact IBF output and the point-wise IBF, respectively. The HUS data are re-analyzed using the proposed methods. Simulations are implemented to validate the performance of the proposed methods.

OPTIMAL CONSTANT-REBALANCED PORTFOLIO INVESTMENT STRATEGIES FOR DYNAMIC PORTFOLIO SELECTION

In this paper we propose a variant of the continuous-time Markowitz mean-variance model by incorporating the Earnings-at-Risk measure in the portfolio optimization problem. Under the Black-Scholes framework, we obtain closed-form expressions for the optimal constant-rebalanced portfolio (CRP) investment strategy. We also derive explicitly the corresponding mean-EaR efficient portfolio frontier, which is a generalization of the Markowitz mean-variance efficient frontier.

Inversion of Bayes' formula for events

In standard notation, let {H1,H2, · · · ,Hm} and {A1, A2, · · · , An} be two distinct partitions of the sample space, or equivalently two sets of events satisfying three properties: (i) each event is non-void, (ii) events in the same set are mutually exclusive (i.e. P (Hj ∪Hk) = P (Hj)+P (Hk) for j 6= k), and (iii) each set is collectively exhaustive, (i.e. P (∪i=1Hj) = 1). The Bayes formula in general form is, for j = 1, · · · ,m, i = 1, · · · , n,