Luís M. Grilo

Financial and risk modelling with semicontinuous covariances

Stochastic processes with semicontious covariances are introduced in a flexible way.Useful optimal designs are computed.Groundbreaking financial applications for probabilities of ruin, Famas hypothesis on EMH and IS/LM model. We are encountering deep financial crisis. One of the basic problems we face is how to predict financial developments. Current modelling is clearly insufficient since continuity of autocorrelation oversmoothes the empirical financial data. We prove that geometric probability to chose positive definite continuous covariance by financial data is zero. We also have the empirical evidence for this fact from real stock data. In order to overcome this gap we introduce the novel abc class of semicontinuous covariance functions, which integrates both continuous and discontinuous covariances. The abc class of covariances is much more flexible, powerful, and realistic than its continuous counterpart. It gives answer to the question why financial markets face negative interest rates, cyclic downbreaks, and other difficulties nowadays. As it is clearly illustrated in this paper, the experimenter (e.g.the financial institution) can choose only up to some extent whether abc class member will be positive definite everywhere. To the best knowledge of the authors this fits well to the own Kolmogorovs opinions on axiomatic theory of probability. The Kolmogorovs axiomatic theory of probability was developed as one possible alternative in order to utilize the flexibility of Lebesgue integral in probability theory, but in many experiments e.g.in physics and finance, one should go beyond this axiomatics. That means one should accept negative probabilities (and consequently negative definite covariance functions). In this paper we clarify that the modelers choice of positive/negative definite functions should be considered within broader Information Theory setup. The introduced abc class is purely topologically defined, with soft regularity conditions on covariances, which are still applicable for increasing and infill domain asymptotics for regression problems, kriging, and finance. These conditions are related to semicontinuous maps of Ornstein Uhlenbeck (OU) processes. We provide several novel results for optimal design of random fields with abc covariances, random walks in finance, and probabilities of ruins related to shocks (e.g.by earthquakes). In particular, we construct a random walk model with semicontinuous covariance. The novel abc class provides a proper environment for unifying several contradictions on IS/ML model in the economic literature. All mentioned novel economical applications are hardly visible from currently used continuous covariance framework.

A mixture of Generalized Integer Gamma distributions as the exact distribution of the product of an odd number of independent Beta random variables: applications ⁄

Abstract In this paper we show first how the distribution of the logarithm of a random variable with a Beta distribution may be expressed either as a mixture of Gamma distributions or as a mixture of Generalized Integer Gamma (GIG) distributions and then how the exact distribution of the product of an odd number of independent Beta random variables whose first parameter evolves by 1/2 and whose second parameter is the half of an odd integer may be expressed as a mixture of GIG distributions. Some particularities of these mixtures are analysed. The results are then used to obtain the exact distribution of the logarithm of the Wilks Λ statistic to test the independence of two sets of variables, both with an odd number of variables, and the exact distribution of the logarithm of the generalized Wilks Λ statistic to test the independence of several sets of variables, in the case where two or three of them have an odd number of variables. A discussion of relative advantages and disadvantages of the use of the exact versus near-exact distributions is carried out.

A Family of Near-Exact Distributions Based on Truncations of the Exact Distribution for the Generalized Wilks Lambda Statistic

For the case where at least two sets have an odd number of variables we do not have the exact distribution of the generalized Wilks Lambda statistic in a manageable form, adequate for manipulation. In this article, we develop a family of very accurate near-exact distributions for this statistic for the case where two or three sets have an odd number of variables. We first express the exact characteristic function of the logarithm of the statistic in the form of the characteristic function of an infinite mixture of Generalized Integer Gamma distributions. Then, based on truncations of this exact characteristic function, we obtain a family of near-exact distributions, which, by construction, match the first two exact moments. These near-exact distributions display an asymptotic behaviour for increasing number of variables involved. The corresponding cumulative distribution functions are obtained in a concise and manageable form, relatively easy to implement computationally, allowing for the computation of virtually exact quantiles. We undertake a comparative study for small sample sizes, using two proximity measures based on the Berry-Esseen bounds, to assess the performance of the near-exact distributions for different numbers of sets of variables and different numbers of variables in each set.

The Exact and Near-Exact Distributions for the Wilks Lambda Statistic Used in the Test of Independence of Two Sets of Variables

Synoptic Abstract We develop the exact distribution of the Wilks Lambda statistic to test the independence of two sets of variables, both with an odd number of variables, under the form of an infinite mixture of Generalized Integer Gamma distributions. Based on truncations of the exact characteristic function, for the product of independent Beta random variables, we obtain near-exact distributions for such product and then by direct application of these results, and once again based on truncations, we develop near-exact distributions for the Wilks Lambda statistic. These near-exact distributions are finite mixtures of Generalized Integer Gamma and Generalized Near-Integer Gamma distributions. By construction, the two first moments of these approximations are equal to the exact moments. These distributions are manageable and relatively easy to implement computationally, allowing for the computation of near-exact quantiles which may indeed be regarded as virtually exact, given the good convergence properties of the series involved, mainly when the difference between the sample size and the overall number of variables involved is rather small. We assess the proximity between these near-exact di stributions and the exact distribution by using two measures based on the Berry-Esseen bounds.

Individual and moving range control charts in the production of olive oil

Quality control charts are usually used to monitor peroxideindex of olive oil, which is a variableanalysedin order toassess the quality and tocharacterize the typeof oil. However, the empirical distribution of peroxideindex does not often verify the assumption of normality. In this case study, we obtained the individual and moving range charts for peroxideindex of virgin olive oil, computing the control limits using theaverage and also the median moving range. Afterwards, we compared these charts with the more efficient and robust ones, based on an adequate Box-Cox transformation. The conclusions obtained with these approaches are relatively different.

Comparison of clinical data based on limits of agreement

Summary Two different medical measurement methods, which usually do not produce exactly the same results, are used to analyse the serum levels of folic acid in a blood sample. We assess the (dis)agreement of the available data in order to replace the old method (the reference method, which involves a lot of human intervention) with the new one (which uses mostly machines), without causing problems in clinical interpretation. The 95% limits of agreement are estimated, before and after a logarithmic transformation, and an appropriate use of regression and a nonparametric approach are also considered. The application of these different statistical techniques is very useful and easily interpreted by medical researchers, but the results obtained do not provide confidence that the new method can be used in place of the old one for clinical purposes.

Individual control charts in paperboard industry

The stiffness of paperboard is an important continuous variable measured in the labs of paperboard industry to evaluate the quality of the final product. The variable is approximately normal distributed, in this case study, and individual control charts to monitor the stiffness are obtained based on average moving range. Since the available sample size is small we decide to estimate robust control limits using a non-parametric method based on empirical quantiles (that performs also well under the normality of the observations), with the bootstrap procedure. The comparison of the control limits should be made based on the required accuracy. We also compare the output of the stable process (i.e., in statistical control) with the process specifications and we conclude that it is not a capable process.The stiffness of paperboard is an important continuous variable measured in the labs of paperboard industry to evaluate the quality of the final product. The variable is approximately normal distributed, in this case study, and individual control charts to monitor the stiffness are obtained based on average moving range. Since the available sample size is small we decide to estimate robust control limits using a non-parametric method based on empirical quantiles (that performs also well under the normality of the observations), with the bootstrap procedure. The comparison of the control limits should be made based on the required accuracy. We also compare the output of the stable process (i.e., in statistical control) with the process specifications and we conclude that it is not a capable process.

Attention-deficit/hyperactivity disorder in children. A statistical approach

Attention–deficit/hyperactivity disorder (ADHD) is one of the most common behavioural disorders in school age children, with a high increase of its prevalence rate in the last decade. ADHD has much comorbidity affecting the child’s life in every domain of their quotidian life with a negative influence in the prognosis of this condition. Psychomotor deficits are responsible for many of the learning disabilities of these children.A sample was submitted to a Psychomotor Battery, in order to characterize the population of children with the diagnosis of ADHD, followed at the Centro Hospitalar Cova da Beira (Portugal), under psychomotor therapy. To the data, collected at baseline and at the end of the intervention, we applied a nonparametric testand graphical techniquesbased on limits of (dis)agreement. The main result shows that the psychomotor profile has a statistical significant improvement after psychomotor therapy.

Industrial Production of Gypsum: Quality Control Charts

The production of gypsum (marketed if it accomplishes the required specifications) occurs during the process of flue gas desulphurization in a Portuguese Coal Thermoelectric Central. Important variables in this process are statistically analyzed in the chemical laboratory and quality control charts are implemented to monitor the entire process. In this study individuals and moving range charts of the variable “density of gypsum slurry” are compared with the “more efficient” ones obtained after a Box-Cox transformation. This transformation is used to normalize the data, because its observations come from non-normal models—where classical control charts are considered less appropriate, since they usually exhibit rates of false alarms different from what would be expected. Although it is important to consider different statistical approaches for quality control charts, during the monitoring of an industrial process, in this case study the achieved results lead us, essentially, to similar conclusions.

The Exact and Near-Exact Distributions for the Statistic Used to Test the Reality of Covariance Matrix in a Complex Normal Distribution

The authors start by approximating the exact distribution of the negative logarithm of the likelihood ratio statistic, used to test the reality of the covariance matrix in a certain complex multivariate normal distribution , by an infinite mixture of Generalized Near-Integer Gamma (GNIG) distributions. Based on this representation they develop a family of near-exact distributions for the likelihood ratio statistic, which are finite mixtures of GNIG distributions and match, by construction, some of the first exact moments. Using a proximity measure based on characteristic functions the authors illustrate the excellent properties of the near-exact distributions . They are very close to the exact distribution but far more manageable and have very good asymptotic properties both for increasing sample sizes as well as for increasing number of variables. These near-exact distributions are much more accurate than the asymptotic approximation considered, namely when the sample size is small and the number of variables involved is large. Furthermore, the corresponding cumulative distribution functions allow for an easy computation of very accurate near-exact quantiles.