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The purpose of this paper is to show that multivariate t-distribution assumption provides a better description of stock return data than multivariate normality assumption.

The EM algorithm is applied to solve the statistical estimation problem almost analytically, and the asymptotic theory is provided for inference.

The authors find that the multivariate normality assumption is almost always rejected by real stock return data, while the multivariate t-distribution assumption can often be adequate. Conclusions under normality vs under t can be drastically different for estimating expected returns and Jensen’s αs, and for testing asset pricing models.

The results provide improved estimates of cost of capital and asset moment parameters that are useful for corporate project evaluation and portfolio management.

The authors proposed new procedures that makes it easy to use a multivariate t-distribution, which models well the data, as a simple and viable alternative in practice to examine the robustness of many existing results.

The purpose of this study is to propose an effective unsupervised one-class-classifier (OCC) support vector machine (SVM)-based single multivariate control chart (OCC-SVM) to simultaneously monitor “location” and “scale” shifts of a manufacturing process.

The step-by-step approach to developing, implementing and fine-tuning the intrinsic parameters of the OCC-SVM chart is demonstrated based on simulation and two real-life case examples.

A comparative study, considering varied known and unknown response distributions, indicates that the OCC-SVM is highly effective in detecting process shifts of samples with individual observations. OCC-SVM chart also shows promising results for samples with a rational subgroup of observations. In addition, the results also indicate that the performance of OCC-SVM is unaffected by the small reference sample size.

The sample responses are considered identically distributed with no significant multivariate autocorrelation between sample observations.

The proposed easy-to-implement chart shows satisfactory performance to detect an out-of-control signal with known or unknown response distributions.

Various multivariate (e.g. parametric or nonparametric) control chart(s) are recommended to monitor the mean (e.g. location) and variance (e.g. scale) of multiple correlated responses in a manufacturing process. However, real-life implementation of a parametric control chart may be complex due to its restrictive response distribution assumptions. There is no evidence of work in the open literature that demonstrates the suitability of an unsupervised OCC-SVM chart to simultaneously monitor “location” and “scale” shifts of multivariate responses. Thus, a new efficient OCC-SVM single chart approach is proposed to address this gap to monitor a multivariate manufacturing process with unknown response distributions.

In the past few years, several capability indices have been developed for evaluating the performance of multivariate manufacturing processes under the normality assumption. However, this assumption may not be true in most practical situations. Thus, the purpose of this paper is to develop new capability indices for evaluating the performance of multivariate processes subject to non-normal distributions.

In this paper, the authors propose three non-normal multivariate process capability indices (MPCIs) RNMC _p , RNMC _pm and RNMC _pu by relieving the normality assumption. Using the two normal MPCIs proposed by Pan and Lee, a weighted standard deviation method (WSD) is used to modify the NMC _p and NMC _pm indices for the-nominal-the-best case. Then the WSD method is applied to modify the multivariate ND index established by Niverthi and Dey for the-smaller-the-better case.

A simulation study compares the performance of the various multivariate indices. Simulation results show that the actual non-conforming rates can be correctly reflected by the proposed capability indices. The numerical example further demonstrates that the actual quality performance of a non-normal multivariate process can properly reflected by the proposed capability indices.

Process capability index is an important SPC tool for measuring the process performance. If the non-normal process data are mistreated as a normal one, it will result in an improper decision and thereby lead to an unnecessary quality loss. The new indices can provide practicing managers and engineers with a better decision-making tool for correctly measuring the performance for any multivariate process or environmental system.

Once the existing multivariate quality/environmental problems and their Key Performance Indicators are identified, one may apply the new capability indices to evaluate the performance of various multivariate processes subject to non-normal distributions.

The purpose of this paper is to provide an analysis of the dependence structure between returns from real estate investment trusts (REITS) and a stock market index. Further, the aim is to illustrate how copula approaches can be applied to model the complex dependence structure between the assets and for risk measurement of a portfolio containing investments in REIT and equity indices.

The usually suggested multivariate normal or variance‐ covariance approach is applied, as well as various copula models in order to investigate the dependence structure between returns of Australian REITS and the Australian stock market. Different models including the Gaussian, Student t, Clayton and Gumbel copula are estimated and goodness‐of‐fit tests are conducted. For the return series, both the Gaussian and a non‐parametric estimate of the distribution is applied. A risk analysis is provided based on Monte Carlo simulations for the different models. The value‐at‐risk measure is also applied for quantification of the risks for a portfolio combining investments in real estate and stock markets.

The findings suggest that the multivariate normal model is not appropriate to measure the complex dependence structure between the returns of the two asset classes. Instead, a model using non‐parametric estimates for the return series in combination with a Student t copula is clearly more suitable. It further illustrates that the usually applied variance‐covariance approach leads to a significant underestimation of the actual risk for a portfolio consisting of investments in REITS and equity indices. The nature of risk is better captured by the suggested copula models.

To the authors', knowledge, this is one of the first studies to apply and test different copula models in real estate markets. Results help international investors and portfolio managers to deepen their understanding of the dependence structure between returns from real estate and equity markets. Additionally, the results should be helpful for implementation of a more adequate risk management for portfolios containing investments in both REITS and equity indices.

Empirical research on European stock markets has shown that they behave differently according to the performance of the leading financial market identified as the US market. A positive sign is viewed as good news in the international financial markets, a negative sign means, conversely, bad news. As a result, we assume that European stock market returns are affected by endogenous and exogenous shocks. The former raise in the market itself, the latter come from the US market, because of its most influential role in the world. Under standard assumptions, the distribution of the European market index returns conditionally on the sign of the one-day lagged US return is skew-normal. The resulting model is denoted Skew-GARCH. We study the properties of this new model and illustrate its application to time-series data from three European financial markets.

Portfolio credit derivatives, such as basket credit default swaps (basket CDS), require for their pricing an estimation of the dependence structure of defaults, which is known to exhibit tail dependence as reflected in observed default contagion. A popular model with this property is the (Student's) t-copula; unfortunately there is no fast method to calibrate the degree of freedom parameter.

In this paper, within the framework of Schönbucher's copula-based trigger-variable model for basket CDS pricing, we propose instead to calibrate the full multivariate t distribution. We describe a version of the expectation-maximization algorithm that provides very fast calibration speeds compared to the current copula-based alternatives.

The algorithm generalizes easily to the more flexible skewed t distributions. To our knowledge, we are the first to use the skewed t distribution in this context.

Purpose – In this chapter, copula theory is used to model dependence structure between hedge fund returns series.

Methodology/approach – Goodness-of-fit tests, based on the Kendall's functions, are applied as selection criteria of the “best” copula. After estimating the parametric copula that best fits the used data, we apply previous results to construct the cumulative distribution functions of the equally weighted portfolios.

Findings – The empirical validation shows that copula clearly allows better estimation of portfolio returns including hedge funds. The three studied portfolios reject the assumption of multivariate normality of returns. The chosen structure is often of Student type when only indices are considered. In the case of portfolios composed by only hedge funds, the dependence structure is of Franck type.

Originality/value of the chapter – Introducing goodness-of-fit bootstrap method to validate the choice of the best structure of dependence is relevant for hedge fund portfolios. Copulas would be introduced to provide better estimations of performance measures.

The purpose of this paper is to propose two control charts to monitor multi-attribute processes and then a maximum likelihood estimator for the change point of the parameter vector (process fraction non-conforming) of multivariate binomial processes.

The performance of the proposed estimator is evaluated for both control charts using some simulation experiments. At the end, the applicability of the proposed method is illustrated using a real case.

The proposed estimator provides accurate and useful estimation of the change point for almost all of the shift magnitudes, regardless of the process dimension. Moreover, based on the results obtained the estimator is robust with regard to different correlation values.

To the best of authors’ knowledge, there are no work available in the literature to estimate the change-point of multivariate binomial processes.

In this paper, we consider the analysis of models for univariate and multivariate ordinal outcomes in the context of the latent variable inferential framework of Albert and Chib (1993). We review several alternative modeling and identification schemes and evaluate how each aids or hampers estimation by Markov chain Monte Carlo simulation methods. For each identification scheme we also discuss the question of model comparison by marginal likelihoods and Bayes factors. In addition, we develop a simulation-based framework for analyzing covariate effects that can provide interpretability of the results despite the nonlinearities in the model and the different identification restrictions that can be implemented. The methods are employed to analyze problems in labor economics (educational attainment), political economy (voter opinions), and health economics (consumers’ reliance on alternative sources of medical information).