Monitoring of crystallization processes by using infrared spectroscopy and multivariate methods

Väitöstutkimuksessa on tarkasteltuinfrapunaspektroskopian ja monimuuttujaisten aineistonkäsittelymenetelmien soveltamista kiteytysprosessin monitoroinnissa ja kidemäisen tuotteen analysoinnissa. Parhaillaan kiteytysprosessitutkimuksessa maailmanlaajuisesti tutkitaan intensiivisesti erilaisten mittausmenetelmien soveltamista kiteytysprosessin ilmiöidenjatkuvaan mittaamiseen niin nestefaasista kuin syntyvistä kiteistäkin. Lisäksi tuotteen karakterisointi on välttämätöntä tuotteen laadun varmistamiseksi. Erityisesti lääkeaineiden valmistuksessa kiinnostusta tämäntyyppiseen tutkimukseen edistää Yhdysvaltain elintarvike- ja lääkeaineviraston (FDA) prosessianalyyttisiintekniikoihin (PAT) liittyvä ohjeistus, jossa määritellään laajasti vaatimukset lääkeaineiden valmistuksessa ja tuotteen karakterisoinnissa tarvittaville mittauksille turvallisten valmistusprosessien takaamiseksi. Jäähdytyskiteytyson erityisesti lääketeollisuudessa paljon käytetty erotusmenetelmä kiinteän raakatuotteen puhdistuksessa. Menetelmässä puhdistettava kiinteä raaka-aine liuotetaan sopivaan liuottimeen suhteellisen korkeassa lämpötilassa. Puhdistettavan aineen liukoisuus käytettävään liuottimeen laskee lämpötilan laskiessa, joten systeemiä jäähdytettäessä liuenneen aineen konsentraatio prosessissa ylittää liukoisuuskonsentraation. Tällaiseen ylikylläiseen systeemiin pyrkii muodostumaan uusia kiteitä tai olemassa olevat kiteet kasvavat. Ylikylläisyys on yksi tärkeimmistä kidetuotteen laatuun vaikuttavista tekijöistä. Jäähdytyskiteytyksessä syntyvän tuotteen ominaisuuksiin voidaan vaikuttaa mm. liuottimen valinnalla, jäähdytyprofiililla ja sekoituksella. Lisäksi kiteytysprosessin käynnistymisvaihe eli ensimmäisten kiteiden muodostumishetki vaikuttaa tuotteen ominaisuuksiin. Kidemäisen tuotteen laatu määritellään kiteiden keskimääräisen koon, koko- ja muotojakaumansekä puhtauden perusteella. Lääketeollisuudessa on usein vaatimuksena, että tuote edustaa tiettyä polymorfimuotoa, mikä tarkoittaa molekyylien kykyä järjestäytyä kidehilassa usealla eri tavalla. Edellä mainitut ominaisuudet vaikuttavat tuotteen jatkokäsiteltävyyteen, kuten mm. suodattuvuuteen, jauhautuvuuteen ja tabletoitavuuteen. Lisäksi polymorfiamuodolla on vaikutusta moniin tuotteen käytettävyysominaisuuksiin, kuten esim. lääkeaineen liukenemisnopeuteen elimistössä. Väitöstyössä on tutkittu sulfatiatsolin jäähdytyskiteytystä käyttäen useita eri liuotinseoksia ja jäähdytysprofiileja sekä tarkasteltu näiden tekijöiden vaikutustatuotteen laatuominaisuuksiin. Infrapunaspektroskopia on laajalti kemian alan tutkimuksissa sovellettava menetelmä. Siinä mitataan tutkittavan näytteenmolekyylien värähtelyjen aiheuttamia spektrimuutoksia IR alueella. Tutkimuksessa prosessinaikaiset mittaukset toteutettiin in-situ reaktoriin sijoitettavalla uppoanturilla käyttäen vaimennettuun kokonaisheijastukseen (ATR) perustuvaa Fourier muunnettua infrapuna (FTIR) spektroskopiaa. Jauhemaiset näytteet mitattiin off-line diffuusioheijastukseen (DRIFT) perustuvalla FTIR spektroskopialla. Monimuuttujamenetelmillä (kemometria) voidaan useita satoja, jopa tuhansia muuttujia käsittävä spektridata jalostaa kvalitatiiviseksi (laadulliseksi) tai kvantitatiiviseksi (määrälliseksi) prosessia kuvaavaksi informaatioksi. Väitöstyössä tarkasteltiin laajasti erilaisten monimuuttujamenetelmien soveltamista mahdollisimman monipuolisen prosessia kuvaavan informaation saamiseksi mitatusta spektriaineistosta. Väitöstyön tuloksena on ehdotettu kalibrointirutiini liuenneen aineen konsentraation ja edelleen ylikylläisyystason mittaamiseksi kiteytysprosessin aikana. Kalibrointirutiinin kehittämiseen kuuluivat aineiston hyvyyden tarkastelumenetelmät, aineiston esikäsittelymenetelmät, varsinainen kalibrointimallinnus sekä mallin validointi. Näin saadaan reaaliaikaista informaatiota kiteytysprosessin ajavasta voimasta, mikä edelleen parantaa kyseisen prosessin tuntemusta ja hallittavuutta. Ylikylläisyystason vaikutuksia syntyvän kidetuotteen laatuun seurattiin usein kiteytyskokein. Työssä on esitetty myös monimuuttujaiseen tilastolliseen prosessinseurantaan perustuva menetelmä, jolla voidaan ennustaa spontaania primääristä ytimenmuodostumishetkeä mitatusta spektriaineistosta sekä mahdollisesti päätellä ydintymisessä syntyvä polymorfimuoto. Ehdotettua menetelmää hyödyntäen voidaan paitsi ennakoida kideytimien muodostumista myös havaita mahdolliset häiriötilanteet kiteytysprosessin alkuhetkillä. Syntyvää polymorfimuotoa ennustamalla voidaan havaita ei-toivotun polymorfin ydintyminen,ja mahdollisesti muuttaa kiteytyksen ohjausta halutun polymorfimuodon saavuttamiseksi. Monimuuttujamenetelmiä sovellettiin myös kiteytyspanosten välisen vaihtelun määrittämiseen mitatusta spektriaineistosta. Tämäntyyppisestä analyysistä saatua informaatiota voidaan hyödyntää kiteytysprosessien suunnittelussa ja optimoinnissa. Väitöstyössä testattiin IR spektroskopian ja erilaisten monimuuttujamenetelmien soveltuvuutta kidetuotteen polymorfikoostumuksen nopeaan määritykseen. Jauhemaisten näytteiden luokittelu eri polymorfeja sisältäviin näytteisiin voitiin tehdä käyttäen tarkoitukseen soveltuvia monimuuttujaisia luokittelumenetelmiä. Tämä tarjoaa nopean menetelmän jauhemaisen näytteen polymorfikoostumuksen karkeaan arviointiin, eli siihen mitä yksittäistä polymorfia kyseinen näyte pääasiassa sisältää. Varsinainen kvantitatiivinen analyysi, eli sen selvittäminen paljonko esim. painoprosentteina näyte sisältää eri polymorfeja, vaatii kaikki polymorfit kattavan fysikaalisen kalibrointisarjan, mikä voi olla puhtaiden polymorfien huonon saatavuuden takia hankalaa.

Simulation-Based Finite-and Large-sample Inference Methods in Multivariate Regressions and Seemingly Unrelated Regressions

In the context of multivariate regression (MLR) and seemingly unrelated regressions (SURE) models, it is well known that commonly employed asymptotic test criteria are seriously biased towards overrejection. in this paper, we propose finite-and large-sample likelihood-based test procedures for possibly non-linear hypotheses on the coefficients of MLR and SURE systems.

Multivariate methods for correlated effects sizes

Current statistical methods for estimation of parametric effect sizes from a series of experiments are generally restricted to univariate comparisons of standardized mean differences between two treatments. Multivariate methods are presented for the case in which effect size is a vector of standardized multivariate mean differences and the number of treatment groups is two or more. The proposed methods employ a vector of independent sample means for each response variable that leads to a covariance structure which depends only on correlations among the $p$ responses on each subject. Using weighted least squares theory and the assumption that the observations are from normally distributed populations, multivariate hypotheses analogous to common hypotheses used for testing effect sizes were formulated and tested for treatment effects which are correlated through a common control group, through multiple response variables observed on each subject, or both conditions.^ The asymptotic multivariate distribution for correlated effect sizes is obtained by extending univariate methods for estimating effect sizes which are correlated through common control groups. The joint distribution of vectors of effect sizes (from $p$ responses on each subject) from one treatment and one control group and from several treatment groups sharing a common control group are derived. Methods are given for estimation of linear combinations of effect sizes when certain homogeneity conditions are met, and for estimation of vectors of effect sizes and confidence intervals from $p$ responses on each subject. Computational illustrations are provided using data from studies of effects of electric field exposure on small laboratory animals. ^

Censored Linear Regression Models For Irregularly Observed Longitudinal Data Using The Multivariate-t Distribution.

In acquired immunodeficiency syndrome (AIDS) studies it is quite common to observe viral load measurements collected irregularly over time. Moreover, these measurements can be subjected to some upper and/or lower detection limits depending on the quantification assays. A complication arises when these continuous repeated measures have a heavy-tailed behavior. For such data structures, we propose a robust structure for a censored linear model based on the multivariate Student's t-distribution. To compensate for the autocorrelation existing among irregularly observed measures, a damped exponential correlation structure is employed. An efficient expectation maximization type algorithm is developed for computing the maximum likelihood estimates, obtaining as a by-product the standard errors of the fixed effects and the log-likelihood function. The proposed algorithm uses closed-form expressions at the E-step that rely on formulas for the mean and variance of a truncated multivariate Student's t-distribution. The methodology is illustrated through an application to an Human Immunodeficiency Virus-AIDS (HIV-AIDS) study and several simulation studies.

Evaluation of natural and synthetic compounds according to their antioxidant activity using a multivariate approach

The antioxidant activity of natural and synthetic compounds was evaluated using five in vitro methods: ferric reducing/antioxidant power (FRAP), 2,2-diphenyl-1-picrylhydradzyl (DPPH), oxygen radical absorption capacity (ORAL), oxidation of an aqueous dispersion of linoleic acid accelerated by azo-initiators (LAOX), and oxidation of a meat homogenate submitted to a thermal treatment (TBARS). All results were expressed as Trolox equivalents. The application of multivariate statistical techniques suggested that the phenolic compounds (caffeic acid, carnosic acid, genistein and resveratrol), beyond their high antioxidant activity measured by the DPPH, FRAP and TBARS methods, showed the highest ability to react with the radicals in the ORAC methodology, compared to the other compounds evaluated in this study (ascorbic acid, erythorbate, tocopherol, BHT, Trolox, tryptophan, citric acid, EDTA, glutathione, lecithin, methionine and tyrosine). This property was significantly correlated with the number of phenolic rings and catecholic structure present in the molecule. Based on a multivariate analysis, it is possible to select compounds from different clusters and explore their antioxidant activity interactions in food products.

Exploring the knowledge contained in neuroimages: Statistical discriminant analysis and automatic segmentation of the most significant changes

Objective: The aim of this article is to propose an integrated framework for extracting and describing patterns of disorders from medical images using a combination of linear discriminant analysis and active contour models. Methods: A multivariate statistical methodology was first used to identify the most discriminating hyperplane separating two groups of images (from healthy controls and patients with schizophrenia) contained in the input data. After this, the present work makes explicit the differences found by the multivariate statistical method by subtracting the discriminant models of controls and patients, weighted by the pooled variance between the two groups. A variational level-set technique was used to segment clusters of these differences. We obtain a label of each anatomical change using the Talairach atlas. Results: In this work all the data was analysed simultaneously rather than assuming a priori regions of interest. As a consequence of this, by using active contour models, we were able to obtain regions of interest that were emergent from the data. The results were evaluated using, as gold standard, well-known facts about the neuroanatomical changes related to schizophrenia. Most of the items in the gold standard was covered in our result set. Conclusions: We argue that such investigation provides a suitable framework for characterising the high complexity of magnetic resonance images in schizophrenia as the results obtained indicate a high sensitivity rate with respect to the gold standard. (C) 2010 Elsevier B.V. All rights reserved.

Some reflections on acquisition, processing and analysis of statistical data in forest soils

Beyond the classical statistical approaches (determination of basic statistics, regression analysis, ANOVA, etc.) a new set of applications of different statistical techniques has increasingly gained relevance in the analysis, processing and interpretation of data concerning the characteristics of forest soils. This is possible to be seen in some of the recent publications in the context of Multivariate Statistics. These new methods require additional care that is not always included or refered in some approaches. In the particular case of geostatistical data applications it is necessary, besides to geo-reference all the data acquisition, to collect the samples in regular grids and in sufficient quantity so that the variograms can reflect the spatial distribution of soil properties in a representative manner. In the case of the great majority of Multivariate Statistics techniques (Principal Component Analysis, Correspondence Analysis, Cluster Analysis, etc.) despite the fact they do not require in most cases the assumption of normal distribution, they however need a proper and rigorous strategy for its utilization. In this work, some reflections about these methodologies and, in particular, about the main constraints that often occur during the information collecting process and about the various linking possibilities of these different techniques will be presented. At the end, illustrations of some particular cases of the applications of these statistical methods will also be presented.

DESARROLLO DE APLICACIONES ESTADÍSTICAS PARA LA AGRICULTURA DE PRECISIÓN. STATISTICAL APPLICATIONS FOR PRECISION AGRICULTURE

A partir de las últimas décadas se ha impulsado el desarrollo y la utilización de los Sistemas de Información Geográficos (SIG) y los Sistemas de Posicionamiento Satelital (GPS) orientados a mejorar la eficiencia productiva de distintos sistemas de cultivos extensivos en términos agronómicos, económicos y ambientales. Estas nuevas tecnologías permiten medir variabilidad espacial de propiedades del sitio como conductividad eléctrica aparente y otros atributos del terreno así como el efecto de las mismas sobre la distribución espacial de los rendimientos. Luego, es posible aplicar el manejo sitio-específico en los lotes para mejorar la eficiencia en el uso de los insumos agroquímicos, la protección del medio ambiente y la sustentabilidad de la vida rural. En la actualidad, existe una oferta amplia de recursos tecnológicos propios de la agricultura de precisión para capturar variación espacial a través de los sitios dentro del terreno. El óptimo uso del gran volumen de datos derivado de maquinarias de agricultura de precisión depende fuertemente de las capacidades para explorar la información relativa a las complejas interacciones que subyacen los resultados productivos. La covariación espacial de las propiedades del sitio y el rendimiento de los cultivos ha sido estudiada a través de modelos geoestadísticos clásicos que se basan en la teoría de variables regionalizadas. Nuevos desarrollos de modelos estadísticos contemporáneos, entre los que se destacan los modelos lineales mixtos, constituyen herramientas prometedoras para el tratamiento de datos correlacionados espacialmente. Más aún, debido a la naturaleza multivariada de las múltiples variables registradas en cada sitio, las técnicas de análisis multivariado podrían aportar valiosa información para la visualización y explotación de datos georreferenciados. La comprensión de las bases agronómicas de las complejas interacciones que se producen a la escala de lotes en producción, es hoy posible con el uso de éstas nuevas tecnologías. Los objetivos del presente proyecto son: (l) desarrollar estrategias metodológicas basadas en la complementación de técnicas de análisis multivariados y geoestadísticas, para la clasificación de sitios intralotes y el estudio de interdependencias entre variables de sitio y rendimiento; (ll) proponer modelos mixtos alternativos, basados en funciones de correlación espacial de los términos de error que permitan explorar patrones de correlación espacial de los rendimientos intralotes y las propiedades del suelo en los sitios delimitados. From the last decades the use and development of Geographical Information Systems (GIS) and Satellite Positioning Systems (GPS) is highly promoted in cropping systems. Such technologies allow measuring spatial variability of site properties including electrical conductivity and others soil features as well as their impact on the spatial variability of yields. Therefore, site-specific management could be applied to improve the efficiency in the use of agrochemicals, the environmental protection, and the sustainability of the rural life. Currently, there is a wide offer of technological resources to capture spatial variation across sites within field. However, the optimum use of data coming from the precision agriculture machineries strongly depends on the capabilities to explore the information about the complex interactions underlying the productive outputs. The covariation between spatial soil properties and yields from georeferenced data has been treated in a graphical manner or with standard geostatistical approaches. New statistical modeling capabilities from the Mixed Linear Model framework are promising to deal with correlated data such those produced by the precision agriculture. Moreover, rescuing the multivariate nature of the multiple data collected at each site, several multivariate statistical approaches could be crucial tools for data analysis with georeferenced data. Understanding the basis of complex interactions at the scale of production field is now within reach the use of these new techniques. Our main objectives are: (1) to develop new statistical strategies, based on the complementarities of geostatistics and multivariate methods, useful to classify sites within field grown with grain crops and analyze the interrelationships of several soil and yield variables, (2) to propose mixed linear models to predict yield according spatial soil variability and to build contour maps to promote a more sustainable agriculture.

Multivariate Analysis for Animal Selection in Experimental Research

Background: Several researchers seek methods for the selection of homogeneous groups of animals in experimental studies, a fact justified because homogeneity is an indispensable prerequisite for casualization of treatments. The lack of robust methods that comply with statistical and biological principles is the reason why researchers use empirical or subjective methods, influencing their results. Objective: To develop a multivariate statistical model for the selection of a homogeneous group of animals for experimental research and to elaborate a computational package to use it. Methods: The set of echocardiographic data of 115 male Wistar rats with supravalvular aortic stenosis (AoS) was used as an example of model development. Initially, the data were standardized, and became dimensionless. Then, the variance matrix of the set was submitted to principal components analysis (PCA), aiming at reducing the parametric space and at retaining the relevant variability. That technique established a new Cartesian system into which the animals were allocated, and finally the confidence region (ellipsoid) was built for the profile of the animals’ homogeneous responses. The animals located inside the ellipsoid were considered as belonging to the homogeneous batch; those outside the ellipsoid were considered spurious. Results: The PCA established eight descriptive axes that represented the accumulated variance of the data set in 88.71%. The allocation of the animals in the new system and the construction of the confidence region revealed six spurious animals as compared to the homogeneous batch of 109 animals. Conclusion: The biometric criterion presented proved to be effective, because it considers the animal as a whole, analyzing jointly all parameters measured, in addition to having a small discard rate.

A limited area model intercomparison on the "Montserrat-2000" flash-flood event using statistical and deterministic methods

In the scope of the European project Hydroptimet, INTERREG IIIB-MEDOCC programme, limited area model (LAM) intercomparison of intense events that produced many damages to people and territory is performed. As the comparison is limited to single case studies, the work is not meant to provide a measure of the different models' skill, but to identify the key model factors useful to give a good forecast on such a kind of meteorological phenomena. This work focuses on the Spanish flash-flood event, also known as "Montserrat-2000" event. The study is performed using forecast data from seven operational LAMs, placed at partners' disposal via the Hydroptimet ftp site, and observed data from Catalonia rain gauge network. To improve the event analysis, satellite rainfall estimates have been also considered. For statistical evaluation of quantitative precipitation forecasts (QPFs), several non-parametric skill scores based on contingency tables have been used. Furthermore, for each model run it has been possible to identify Catalonia regions affected by misses and false alarms using contingency table elements. Moreover, the standard "eyeball" analysis of forecast and observed precipitation fields has been supported by the use of a state-of-the-art diagnostic method, the contiguous rain area (CRA) analysis. This method allows to quantify the spatial shift forecast error and to identify the error sources that affected each model forecasts. High-resolution modelling and domain size seem to have a key role for providing a skillful forecast. Further work is needed to support this statement, including verification using a wider observational data set.

Multivariate study and regression analysis of gluten-free granola

This study developed a gluten-free granola and evaluated it during storage with the application of multivariate and regression analysis of the sensory and instrumental parameters. The physicochemical, sensory, and nutritional characteristics of a product containing quinoa, amaranth and linseed were evaluated. The crude protein and lipid contents ranged from 97.49 and 122.72 g kg-1 of food, respectively. The polyunsaturated/saturated, and n-6:n-3 fatty acid ratios ranged from 2.82 and 2.59:1, respectively. Granola had the best alpha-linolenic acid content, nutritional indices in the lipid fraction, and mineral content. There were good hygienic and sanitary conditions during storage; probably due to the low water activity of the formulation, which contributed to inhibit microbial growth. The sensory attributes ranged from 'like very much' to 'like slightly', and the regression models were highly fitted and correlated during the storage period. A reduction in the sensory attribute levels and in the product physical stabilisation was verified by principal component analysis. The use of the affective test acceptance and instrumental analysis combined with statistical methods allowed us to obtain promising results about the characteristics of gluten-free granola.

Simulation-Based Finite and Large Sample Tests in Multivariate Regressions.

In the context of multivariate linear regression (MLR) models, it is well known that commonly employed asymptotic test criteria are seriously biased towards overrejection. In this paper, we propose a general method for constructing exact tests of possibly nonlinear hypotheses on the coefficients of MLR systems. For the case of uniform linear hypotheses, we present exact distributional invariance results concerning several standard test criteria. These include Wilks' likelihood ratio (LR) criterion as well as trace and maximum root criteria. The normality assumption is not necessary for most of the results to hold. Implications for inference are two-fold. First, invariance to nuisance parameters entails that the technique of Monte Carlo tests can be applied on all these statistics to obtain exact tests of uniform linear hypotheses. Second, the invariance property of the latter statistic is exploited to derive general nuisance-parameter-free bounds on the distribution of the LR statistic for arbitrary hypotheses. Even though it may be difficult to compute these bounds analytically, they can easily be simulated, hence yielding exact bounds Monte Carlo tests. Illustrative simulation experiments show that the bounds are sufficiently tight to provide conclusive results with a high probability. Our findings illustrate the value of the bounds as a tool to be used in conjunction with more traditional simulation-based test methods (e.g., the parametric bootstrap) which may be applied when the bounds are not conclusive.

Finite-Sample Diagnostics for Multivariate Regressions with Applications to Linear Asset Pricing Models

In this paper, we propose several finite-sample specification tests for multivariate linear regressions (MLR) with applications to asset pricing models. We focus on departures from the assumption of i.i.d. errors assumption, at univariate and multivariate levels, with Gaussian and non-Gaussian (including Student t) errors. The univariate tests studied extend existing exact procedures by allowing for unspecified parameters in the error distributions (e.g., the degrees of freedom in the case of the Student t distribution). The multivariate tests are based on properly standardized multivariate residuals to ensure invariance to MLR coefficients and error covariances. We consider tests for serial correlation, tests for multivariate GARCH and sign-type tests against general dependencies and asymmetries. The procedures proposed provide exact versions of those applied in Shanken (1990) which consist in combining univariate specification tests. Specifically, we combine tests across equations using the MC test procedure to avoid Bonferroni-type bounds. Since non-Gaussian based tests are not pivotal, we apply the “maximized MC” (MMC) test method [Dufour (2002)], where the MC p-value for the tested hypothesis (which depends on nuisance parameters) is maximized (with respect to these nuisance parameters) to control the test’s significance level. The tests proposed are applied to an asset pricing model with observable risk-free rates, using monthly returns on New York Stock Exchange (NYSE) portfolios over five-year subperiods from 1926-1995. Our empirical results reveal the following. Whereas univariate exact tests indicate significant serial correlation, asymmetries and GARCH in some equations, such effects are much less prevalent once error cross-equation covariances are accounted for. In addition, significant departures from the i.i.d. hypothesis are less evident once we allow for non-Gaussian errors.

Exact Skewness-Kurtosis Tests for Multivariate Normality and Goodness-of-fit in Multivariate Regressions with Application to Asset Pricing Models

We study the problem of testing the error distribution in a multivariate linear regression (MLR) model. The tests are functions of appropriately standardized multivariate least squares residuals whose distribution is invariant to the unknown cross-equation error covariance matrix. Empirical multivariate skewness and kurtosis criteria are then compared to simulation-based estimate of their expected value under the hypothesized distribution. Special cases considered include testing multivariate normal, Student t; normal mixtures and stable error models. In the Gaussian case, finite-sample versions of the standard multivariate skewness and kurtosis tests are derived. To do this, we exploit simple, double and multi-stage Monte Carlo test methods. For non-Gaussian distribution families involving nuisance parameters, confidence sets are derived for the the nuisance parameters and the error distribution. The procedures considered are evaluated in a small simulation experi-ment. Finally, the tests are applied to an asset pricing model with observable risk-free rates, using monthly returns on New York Stock Exchange (NYSE) portfolios over five-year subperiods from 1926-1995.

Exact Multivariate Tests of Asset Pricing Models with Stable Asymmetric Distributions

In this paper, we propose exact inference procedures for asset pricing models that can be formulated in the framework of a multivariate linear regression (CAPM), allowing for stable error distributions. The normality assumption on the distribution of stock returns is usually rejected in empirical studies, due to excess kurtosis and asymmetry. To model such data, we propose a comprehensive statistical approach which allows for alternative - possibly asymmetric - heavy tailed distributions without the use of large-sample approximations. The methods suggested are based on Monte Carlo test techniques. Goodness-of-fit tests are formally incorporated to ensure that the error distributions considered are empirically sustainable, from which exact confidence sets for the unknown tail area and asymmetry parameters of the stable error distribution are derived. Tests for the efficiency of the market portfolio (zero intercepts) which explicitly allow for the presence of (unknown) nuisance parameter in the stable error distribution are derived. The methods proposed are applied to monthly returns on 12 portfolios of the New York Stock Exchange over the period 1926-1995 (5 year subperiods). We find that stable possibly skewed distributions provide statistically significant improvement in goodness-of-fit and lead to fewer rejections of the efficiency hypothesis.