Statistical distributions also known as probability distributions are used to model a random experiment. Probability distributions consist of probability density functions (pdf) and cumulative density functions (cdf). Probability distributions are widely used in the area of engineering, actuarial science, computer science, biological science, physics, and other applicable areas of study. Statistics are used to draw conclusions about the population through probability models. Sample statistics such as the minimum, first quartile, median, third quartile, and maximum, referred to as the five-number summary, are examples of order statistics. The minimum and maximum observations are important in extreme value theory. This paper will focus on the probability distribution of the minimum observation, also known as the first order statistic, when the sample size is random.

In today's competitive business and industrial environment, it is becoming more crucial than ever to assess precisely process losses due to non-compliance to customer specifications. To assess these losses, industry is extensively using Process Capability Indices for performance evaluation of their processes. Determination of the performance capability of a stable process using the standard process capability indices such as and requires that the underlying quality characteristics data follow a normal distribution. However it is an undisputed fact that real processes very often produce non-normal quality characteristics data and also these quality characteristics are very often correlated with each other. For such non-normal and correlated multivariate quality characteristics, application of standard capability measures using conventional methods can lead to erroneous results. The research undertaken in this PhD thesis presents several capability assessment methods to estimate more precisely and accurately process performances based on univariate as well as multivariate quality characteristics. The proposed capability assessment methods also take into account the correlation, variance and covariance as well as non-normality issues of the quality characteristics data. A comprehensive review of the existing univariate and multivariate PCI estimations have been provided. We have proposed fitting Burr XII distributions to continuous positively skewed data. The proportion of nonconformance (PNC) for process measurements is then obtained by using Burr XII distribution, rather than through the traditional practice of fitting different distributions to real data. Maximum likelihood method is deployed to improve the accuracy of PCI based on Burr XII distribution. Different numerical methods such as Evolutionary and Simulated Annealing algorithms are deployed to estimate parameters of the fitted Burr XII distribution. We have also introduced new transformation method called Best Root Transformation approach to transform non-normal data to normal data and then apply the traditional PCI method to estimate the proportion of non-conforming data. Another approach which has been introduced in this thesis is to deploy Burr XII cumulative density function for PCI estimation using Cumulative Density Function technique. The proposed approach is in contrast to the approach adopted in the research literature i.e. use of best-fitting density function from known distributions to non-normal data for PCI estimation. The proposed CDF technique has also been extended to estimate process capability for bivariate non-normal quality characteristics data. A new multivariate capability index based on the Generalized Covariance Distance (GCD) is proposed. This novel approach reduces the dimension of multivariate data by transforming correlated variables into univariate ones through a metric function. This approach evaluates process capability for correlated non-normal multivariate quality characteristics. Unlike the Geometric Distance approach, GCD approach takes into account the scaling effect of the variance-covariance matrix and produces a Covariance Distance variable that is based on the Mahanalobis distance. Another novelty introduced in this research is to approximate the distribution of these distances by a Burr XII distribution and then estimate its parameters using numerical search algorithm. It is demonstrates that the proportion of nonconformance (PNC) using proposed method is very close to the actual PNC value.

Les écosystèmes aquatiques contiennent environ 25% de la biodiversité globale et sont parmi les plus affectés par l’activité humaine. Cela est entre autres causé par la position de « receveur » des rivières, lacs et océans dans leur bassin versant. Les espèces aquatiques, en eau douce particulièrement, sont ainsi hautement à risque d’être affectées par l’activité humaine. La protection de ces espèces peut inclure la protection et la restauration de leurs habitats. Les modèles de qualité d’habitats (MQH) peuvent être utilisés afin de déterminer quels habitats protéger et restaurer. Les MQH définissent la relation entre un indice de qualité d’habitats (IQH, e.g. densité) et des conditions environnementales. Toutefois, la performance des MQH dépend de l’IQH sélectionné. Ici, notre objectif est de comparer des MQH basés sur deux IQH estimés pour des poissons en rivière : 1) la densité instantanée, échantillonnée en transect par plongée en apnée et 2) la densité cumulative, échantillonnée en point fixe par caméra-vidéo en stéréo. Au total, douze modèles ont été construits et nos analyses indiquent que les MQH basés sur la densité instantanée ont des capacités explicatives significativement supérieures. Les variables environnementales retenues pour expliquer la distribution de chaque espèce sont toutefois différentes. Cela semble être causé en partie par des différences inhérentes à l’échantillonnage (e.g. échelle spatiale). Ces résultats démontrent que la densité instantanée en tant qu’IQH produit des MQH aux capacités explicatives supérieures et que les deux IQH semblent donner des informations complémentaires sur les caractéristiques des habitats à protéger et à restaurer. / Aquatic ecosystems contain approximately 25% of the global biodiversity and are among the most affected by human activity. This may be caused by the position of “receivers” rivers, lakes and oceans have in their watershed. Aquatic species, specially in freshwater, are thus at high risk of being affected by human activity. Assuring the survival of these species may include protecting and restoring their habitats. Habitat quality models (HQM) can be used to determine which habitats to protect and how to restore damaged habitats. HQM are relationships between habitat quality indices (HQI, e.g., density) and environmental conditions prevailing in those habitats. However, how well an HQM performs depends on the chosen HQI it is computed with. For this research, we compared HQM based on two HQI estimated for fish in a river : 1) instantaneous density, sampled by transect snorkeling survey and 2) cumulative density, sampled by fixed stereo-video recording. Analyses of twelve HQM show that, contrary to our hypothesis, HQM based on instantaneous density had higher explanatory capacities. However, environmental conditions selected by both types of HQM to explain a species’ distribution were different. This may in part be explained by inherent differences of the sampling methods (e.g., spatial scale). We conclude that instantaneous density as HQI produces HQM of higher explanatory capacities, yet both HQI may provide complementary information on the characteristics of habitats to protect and restore.