Normal distribution and confidence intervals

lecture for COMP6235

DISTRIBUTION FUNCTIONS, MEAN SQUARED ERRORS, AND CONFIDENCE LIMITS FOR RIDGE REGRESSION ESTIMATORS

A large number of ridge regression estimators have been proposed and used with little knowledge of their true distributions. Because of this lack of knowledge, these estimators cannot be used to test hypotheses or to form confidence intervals.^ This paper presents a basic technique for deriving the exact distribution functions for a class of generalized ridge estimators. The technique is applied to five prominent generalized ridge estimators. Graphs of the resulting distribution functions are presented. The actual behavior of these estimators is found to be considerably different than the behavior which is generally assumed for ridge estimators.^ This paper also uses the derived distributions to examine the mean squared error properties of the estimators. A technique for developing confidence intervals based on the generalized ridge estimators is also presented. ^

Conditional Confidence Interval for the Scale Parameter of a Weibull Distribution

2000 Mathematics Subject Classification: 62F25, 62F03.

Small sample confidence intervals for the mean of a positively skewed distribution

This thesis proposes some confidence intervals for the mean of a positively skewed distribution. The following confidence intervals are considered: Student-t, Johnson-t, median-t, mad-t, bootstrap-t, BCA, T1 , T3 and six new confidence intervals, the median bootstrap-t, mad bootstrap-t, median T1, mad T1 , median T3 and the mad T3. A simulation study has been conducted and average widths, coefficient of variation of widths, and coverage probabilities were recorded and compared across confidence intervals. To compare confidence intervals, the width and coverage probabilities were compared so that smaller widths indicated a better confidence interval when coverage probabilities were the same. Results showed that the median T1 and median T3 outperformed other confidence intervals in terms of coverage probability and the mad bootstrap-t, mad-t, and mad T3 outperformed others in terms of width. Some real life data are considered to illustrate the findings of the thesis.

Haplotype distribution of five nuclear genes based on network genealogies and Bayesian inference indicates that Trypanosoma cruzi hybrid strains are polyphyletic

Chagas disease is still a major public health problem in Latin America. Its causative agent, Trypanosoma cruzi, can be typed into three major groups, T. cruzi I, T. cruzi II and hybrids. These groups each have specific genetic characteristics and epidemiological distributions. Several highly virulent strains are found in the hybrid group; their origin is still a matter of debate. The null hypothesis is that the hybrids are of polyphyletic origin, evolving independently from various hybridization events. The alternative hypothesis is that all extant hybrid strains originated from a single hybridization event. We sequenced both alleles of genes encoding EF-1 alpha, actin and SSU rDNA of 26 T. cruzi strains and DHFR-TS and TR of 12 strains. This information was used for network genealogy analysis and Bayesian phylogenies. We found T. cruzi I and T. cruzi II to be monophyletic and that all hybrids had different combinations of T. cruzi I and T. cruzi II haplotypes plus hybrid-specific haplotypes. Bootstrap values (networks) and posterior probabilities (Bayesian phylogenies) of clades supporting the monophyly of hybrids were far below the 95% confidence interval, indicating that the hybrid group is polyphyletic. We hypothesize that T. cruzi I and T. cruzi II are two different species and that the hybrids are extant representatives of independent events of genome hybridization, which sporadically have sufficient fitness to impact on the epidemiology of Chagas disease.

Distribution of Chlamydia pneumoniae DNA in atherosclerotic carotid arteries: significance for sampling procedures

Despite extensive efforts to confirm a direct association between Chlamydia pneumoniae and atherosclerosis, different laboratories continue to report a large variability in detection rates. In this study, we analyzed multiple sections from atherosclerotic carotid arteries from 10 endartectomy patients to determine the location of C. pneumoniae DNA and the number of sections of the plaque required for analysis to obtain a 95% confidence of detecting the bacterium. A sensitive nested PCR assay detected C. pneumoniae DNA in all patients at one or more locations within the plaque. On average, 42% (ranging from 5 to 91%) of the sections from any single patient had C. pneumoniae DNA present. A patchy distribution of C. pneumoniae in the atherosclerotic lesions was observed, with no area of the carotid having significantly more C. pneumoniae DNA present. If a single random 30-mum-thick section was tested, there was only a 35.6 to 41.6% (95% confidence interval) chance of detecting C. pneumoniae DNA in a patient with carotid artery disease. A minimum of 15 sections would therefore be required to obtain a 95% chance of detecting all true positives. The low concentration and patchy distribution of C. pneumoniae DNA in atherosclerotic plaque appear to be among the reasons for inconsistency between laboratories in the results reported.

Rock fracture mean trace length estimation and confidence interval calculation using maximum likelihood methods

There has been a resurgence of interest in the mean trace length estimator of Pahl for window sampling of traces. The estimator has been dealt with by Mauldon and Zhang and Einstein in recent publications. The estimator is a very useful one in that it is non-parametric. However, despite some discussion regarding the statistical distribution of the estimator, none of the recent works or the original work by Pahl provide a rigorous basis for the determination a confidence interval for the estimator or a confidence region for the estimator and the corresponding estimator of trace spatial intensity in the sampling window. This paper shows, by consideration of a simplified version of the problem but without loss of generality, that the estimator is in fact the maximum likelihood estimator (MLE) and that it can be considered essentially unbiased. As the MLE, it possesses the least variance of all estimators and confidence intervals or regions should therefore be available through application of classical ML theory. It is shown that valid confidence intervals can in fact be determined. The results of the work and the calculations of the confidence intervals are illustrated by example. (C) 2003 Elsevier Science Ltd. All rights reserved.

Five (or so) challenges for species distribution modelling

Species distribution modelling is central to both fundamental and applied research in biogeography. Despite widespread use of models, there are still important conceptual ambiguities as well as biotic and algorithmic uncertainties that need to be investigated in order to increase confidence in model results. We identify and discuss five areas of enquiry that are of high importance for species distribution modelling: (1) clarification of the niche concept; (2) improved designs for sampling data for building models; (3) improved parameterization; (4) improved model selection and predictor contribution; and (5) improved model evaluation. The challenges discussed in this essay do not preclude the need for developments of other areas of research in this field. However, they are critical for allowing the science of species distribution modelling to move forward.

Niche, distribution and global change : modeling insights into biogeography and conservation biology Olivier Broennimann

RESUME Les évidences montrant que les changements globaux affectent la biodiversité s'accumulent. Les facteurs les plus influant dans ce processus sont les changements et destructions d'habitat, l'expansion des espèces envahissantes et l'impact des changements climatiques. Une évaluation pertinente de la réponse des espèces face à ces changements est essentielle pour proposer des mesures permettant de réduire le déclin actuel de la biodiversité. La modélisation de la répartition d'espèces basée sur la niche (NBM) est l'un des rares outils permettant cette évaluation. Néanmoins, leur application dans le contexte des changements globaux repose sur des hypothèses restrictives et demande une interprétation critique. Ce travail présente une série d'études de cas investiguant les possibilités et limitations de cette approche pour prédire l'impact des changements globaux. Deux études traitant des menaces sur les espèces rares et en danger d'extinction sont présentées. Les caractéristiques éco-géographiques de 118 plantes avec un haut degré de priorité de conservation sont revues. La prévalence des types de rareté sont analysées en relation avec leur risque d'extinction UICN. La revue souligne l'importance de la conservation à l'échelle régionale. Une évaluation de la rareté à échelle globale peut être trompeuse pour certaine espèces car elle ne tient pas en compte des différents degrés de rareté que présente une espèce à différentes échelles spatiales. La deuxième étude test une approche pour améliorer l'échantillonnage d'espèces rares en incluant des phases itératives de modélisation et d'échantillonnage sur le terrain. L'application de l'approche en biologie de la conservation (illustrée ici par le cas du chardon bleu, Eryngium alpinum), permettrait de réduire le temps et les coûts d'échantillonnage. Deux études sur l'impact des changements climatiques sur la faune et la flore africaine sont présentées. La première étude évalue la sensibilité de 227 mammifères africains face aux climatiques d'ici 2050. Elle montre qu'un nombre important d'espèces pourrait être bientôt en danger d'extinction et que les parcs nationaux africains (principalement ceux situé en milieux xériques) pourraient ne pas remplir leur mandat de protection de la biodiversité dans le futur. La seconde étude modélise l'aire de répartition en 2050 de 975 espèces de plantes endémiques du sud de l'Afrique. L'étude propose l'inclusion de méthodes améliorant la prédiction des risques liés aux changements climatiques. Elle propose également une méthode pour estimer a priori la sensibilité d'une espèce aux changements climatiques à partir de ses propriétés écologiques et des caractéristiques de son aire de répartition. Trois études illustrent l'utilisation des modèles dans l'étude des invasions biologiques. Une première étude relate l'expansion de la laitue sáuvage (Lactuca serriola) vers le nord de l'Europe en lien avec les changements du climat depuis 250 ans. La deuxième étude analyse le potentiel d'invasion de la centaurée tachetée (Centaures maculosa), une mauvaise herbe importée en Amérique du nord vers 1890. L'étude apporte la preuve qu'une espèce envahissante peut occuper une niche climatique différente après introduction sur un autre continent. Les modèles basés sur l'aire native prédisent de manière incorrecte l'entier de l'aire envahie mais permettent de prévoir les aires d'introductions potentielles. Une méthode alternative, incluant la calibration du modèle à partir des deux aires où l'espèce est présente, est proposée pour améliorer les prédictions de l'invasion en Amérique du nord. Je présente finalement une revue de la littérature sur la dynamique de la niche écologique dans le temps et l'espace. Elle synthétise les récents développements théoriques concernant le conservatisme de la niche et propose des solutions pour améliorer la pertinence des prédictions d'impact des changements climatiques et des invasions biologiques. SUMMARY Evidences are accumulating that biodiversity is facing the effects of global change. The most influential drivers of change in ecosystems are land-use change, alien species invasions and climate change impacts. Accurate projections of species' responses to these changes are needed to propose mitigation measures to slow down the on-going erosion of biodiversity. Niche-based models (NBM) currently represent one of the only tools for such projections. However, their application in the context of global changes relies on restrictive assumptions, calling for cautious interpretations. In this thesis I aim to assess the effectiveness and shortcomings of niche-based models for the study of global change impacts on biodiversity through the investigation of specific, unsolved limitations and suggestion of new approaches. Two studies investigating threats to rare and endangered plants are presented. I review the ecogeographic characteristic of 118 endangered plants with high conservation priority in Switzerland. The prevalence of rarity types among plant species is analyzed in relation to IUCN extinction risks. The review underlines the importance of regional vs. global conservation and shows that a global assessment of rarity might be misleading for some species because it can fail to account for different degrees of rarity at a variety of spatial scales. The second study tests a modeling framework including iterative steps of modeling and field surveys to improve the sampling of rare species. The approach is illustrated with a rare alpine plant, Eryngium alpinum and shows promise for complementing conservation practices and reducing sampling costs. Two studies illustrate the impacts of climate change on African taxa. The first one assesses the sensitivity of 277 mammals at African scale to climate change by 2050 in terms of species richness and turnover. It shows that a substantial number of species could be critically endangered in the future. National parks situated in xeric ecosystems are not expected to meet their mandate of protecting current species diversity in the future. The second study model the distribution in 2050 of 975 endemic plant species in southern Africa. The study proposes the inclusion of new methodological insights improving the accuracy and ecological realism of predictions of global changes studies. It also investigates the possibility to estimate a priori the sensitivity of a species to climate change from the geographical distribution and ecological proprieties of the species. Three studies illustrate the application of NBM in the study of biological invasions. The first one investigates the Northwards expansion of Lactuca serriola L. in Europe during the last 250 years in relation with climate changes. In the last two decades, the species could not track climate change due to non climatic influences. A second study analyses the potential invasion extent of spotted knapweed, a European weed first introduced into North America in the 1890s. The study provides one of the first empirical evidence that an invasive species can occupy climatically distinct niche spaces following its introduction into a new area. Models fail to predict the current full extent of the invasion, but correctly predict areas of introduction. An alternative approach, involving the calibration of models with pooled data from both ranges, is proposed to improve predictions of the extent of invasion on models based solely on the native range. I finally present a review on the dynamic nature of ecological niches in space and time. It synthesizes the recent theoretical developments to the niche conservatism issues and proposes solutions to improve confidence in NBM predictions of the impacts of climate change and species invasions on species distributions.

Spontaneous viral clearance, viral load, and genotype distribution of hepatitis C virus (HCV) in HIV-infected patients with anti-HCV antibodies in Europe.

BACKGROUND: Variables influencing serum hepatitis C virus (HCV) RNA levels and genotype distribution in individuals with human immunodeficiency virus (HIV) infection are not well known, nor are factors determining spontaneous clearance after exposure to HCV in this population. METHODS: All HCV antibody (Ab)-positive patients with HIV infection in the EuroSIDA cohort who had stored samples were tested for serum HCV RNA, and HCV genotyping was done for subjects with viremia. Logistic regression was used to identify variables associated with spontaneous HCV clearance and HCV genotype 1. RESULTS: Of 1940 HCV Ab-positive patients, 1496 (77%) were serum HCV RNA positive. Injection drug users (IDUs) were less likely to have spontaneously cleared HCV than were homosexual men (20% vs. 39%; adjusted odds ratio [aOR], 0.36 [95% confidence interval {CI}, 0.24-0.53]), whereas patients positive for hepatitis B surface antigen (HBsAg) were more likely to have spontaneously cleared HCV than were those negative for HBsAg (43% vs. 21%; aOR, 2.91 [95% CI, 1.94-4.38]). Of patients with HCV viremia, 786 (53%) carried HCV genotype 1, and 53 (4%), 440 (29%), and 217 (15%) carried HCV genotype 2, 3, and 4, respectively. A greater HCV RNA level was associated with a greater chance of being infected with HCV genotype 1 (aOR, 1.60 per 1 log higher [95% CI, 1.36-1.88]). CONCLUSIONS: More than three-quarters of the HIV- and HCV Ab-positive patients in EuroSIDA showed active HCV replication. Viremia was more frequent in IDUs and, conversely, was less common in HBsAg-positive patients. Of the patients with HCV viremia analyzed, 53% were found to carry HCV genotype 1, and this genotype was associated with greater serum HCV RNA levels.

Asymptotic Distribution of a Simple Linear Estimator for VARMA Models in Echelon Form

In this paper, we study the asymptotic distribution of a simple two-stage (Hannan-Rissanen-type) linear estimator for stationary invertible vector autoregressive moving average (VARMA) models in the echelon form representation. General conditions for consistency and asymptotic normality are given. A consistent estimator of the asymptotic covariance matrix of the estimator is also provided, so that tests and confidence intervals can easily be constructed.

Confidence sets for asset correlations in portfolio credit risk

Asset correlations are of critical importance in quantifying portfolio credit risk and economic capitalin financial institutions. Estimation of asset correlation with rating transition data has focusedon the point estimation of the correlation without giving any consideration to the uncertaintyaround these point estimates. In this article we use Bayesian methods to estimate a dynamicfactor model for default risk using rating data (McNeil et al., 2005; McNeil and Wendin, 2007).Bayesian methods allow us to formally incorporate human judgement in the estimation of assetcorrelation, through the prior distribution and fully characterize a confidence set for the correlations.Results indicate: i) a two factor model rather than the one factor model, as proposed bythe Basel II framework, better represents the historical default data. ii) importance of unobservedfactors in this type of models is reinforced and point out that the levels of the implied asset correlationscritically depend on the latent state variable used to capture the dynamics of default,as well as other assumptions on the statistical model. iii) the posterior distributions of the assetcorrelations show that the Basel recommended bounds, for this parameter, undermine the levelof systemic risk.

Hitting emissions targets with (statistical) confidence in multi-instrument Emissions Trading Schemes

A means of assessing, monitoring and controlling aggregate emissions from multi-instrument Emissions Trading Schemes is proposed. The approach allows contributions from different instruments with different forms of emissions targets to be integrated. Where Emissions Trading Schemes are helping meet specific national targets, the approach allows the entry requirements of new participants to be calculated and set at a level that will achieve these targets. The approach is multi-levelled, and may be extended downwards to support pooling of participants within instruments, or upwards to embed Emissions Trading Schemes within a wider suite of policies and measures with hard and soft targets. Aggregate emissions from each instrument are treated stochastically. Emissions from the scheme as a whole are then the joint probability distribution formed by integrating the emissions from its instruments. Because a Bayesian approach is adopted, qualitative and semi-qualitative data from expert opinion can be used where quantitative data is not currently available, or is incomplete. This approach helps government retain sufficient control over emissions trading scheme targets to allow them to meet their emissions reduction obligations, while minimising the need for retrospectively adjusting existing participants’ conditions of entry. This maintains participant confidence, while providing the necessary policy levers for good governance.

The distribution of solar wind speeds during solar minimum: calibration for numerical solar wind modeling constraints on the source of the slow solar wind

It took the solar polar passage of Ulysses in the early 1990s to establish the global structure of the solar wind speed during solar minimum. However, it remains unclear if the solar wind is composed of two distinct populations of solar wind from different sources (e.g., closed loops which open up to produce the slow solar wind) or if the fast and slow solar wind rely on the superradial expansion of the magnetic field to account for the observed solar wind speed variation. We investigate the solar wind in the inner corona using the Wang-Sheeley-Arge (WSA) coronal model incorporating a new empirical magnetic topology–velocity relationship calibrated for use at 0.1 AU. In this study the empirical solar wind speed relationship was determined by using Helios perihelion observations, along with results from Riley et al. (2003) and Schwadron et al. (2005) as constraints. The new relationship was tested by using it to drive the ENLIL 3-D MHD solar wind model and obtain solar wind parameters at Earth (1.0 AU) and Ulysses (1.4 AU). The improvements in speed, its variability, and the occurrence of high-speed enhancements provide confidence that the new velocity relationship better determines the solar wind speed in the outer corona (0.1 AU). An analysis of this improved velocity field within the WSA model suggests the existence of two distinct mechanisms of the solar wind generation, one for fast and one for slow solar wind, implying that a combination of present theories may be necessary to explain solar wind observations.