Aitchison Geometry for Probability and Likelihood as a new approach to mathematical statistics

The Aitchison vector space structure for the simplex is generalized to a Hilbert space structure A2(P) for distributions and likelihoods on arbitrary spaces. Centralnotations of statistics, such as Information or Likelihood, can be identified in the algebraical structure of A2(P) and their corresponding notions in compositional data analysis, such as Aitchison distance or centered log ratio transform.In this way very elaborated aspects of mathematical statistics can be understoodeasily in the light of a simple vector space structure and of compositional data analysis. E.g. combination of statistical information such as Bayesian updating,combination of likelihood and robust M-estimation functions are simple additions/perturbations in A2(Pprior). Weighting observations corresponds to a weightedaddition of the corresponding evidence.Likelihood based statistics for general exponential families turns out to have aparticularly easy interpretation in terms of A2(P). Regular exponential families formfinite dimensional linear subspaces of A2(P) and they correspond to finite dimensionalsubspaces formed by their posterior in the dual information space A2(Pprior).The Aitchison norm can identified with mean Fisher information. The closing constant itself is identified with a generalization of the cummulant function and shown to be Kullback Leiblers directed information. Fisher information is the local geometry of the manifold induced by the A2(P) derivative of the Kullback Leibler information and the space A2(P) can therefore be seen as the tangential geometry of statistical inference at the distribution P.The discussion of A2(P) valued random variables, such as estimation functionsor likelihoods, give a further interpretation of Fisher information as the expected squared norm of evidence and a scale free understanding of unbiased reasoning

Brain lactate metabolism in humans with subarachnoid hemorrhage.

BACKGROUND AND PURPOSE: Lactate is central for the regulation of brain metabolism and is an alternative substrate to glucose after injury. Brain lactate metabolism in patients with subarachnoid hemorrhage has not been fully elucidated. METHODS: Thirty-one subarachnoid hemorrhage patients monitored with cerebral microdialysis (CMD) and brain oxygen (PbtO(2)) were studied. Samples with elevated CMD lactate (>4 mmol/L) were matched to PbtO(2) and CMD pyruvate and categorized as hypoxic (PbtO(2) <20 mm Hg) versus nonhypoxic and hyperglycolytic (CMD pyruvate >119 μmol/L) versus nonhyperglycolytic. RESULTS: Median per patient samples with elevated CMD lactate was 54% (interquartile range, 11%-80%). Lactate elevations were more often attributable to cerebral hyperglycolysis (78%; interquartile range, 5%-98%) than brain hypoxia (11%; interquartile range, 4%-75%). Mortality was associated with increased percentage of samples with elevated lactate and brain hypoxia (28% [interquartile range 9%-95%] in nonsurvivors versus 9% [interquartile range 3%-17%] in survivors; P=0.02) and lower percentage of elevated lactate and cerebral hyperglycolysis (13% [interquartile range, 1%-87%] versus 88% [interquartile range, 27%-99%]; P=0.07). Cerebral hyperglycolytic lactate production predicted good 6-month outcome (odds ratio for modified Rankin Scale score, 0-3 1.49; CI, 1.08-2.05; P=0.016), whereas increased lactate with brain hypoxia was associated with a reduced likelihood of good outcome (OR, 0.78; CI, 0.59-1.03; P=0.08). CONCLUSIONS: Brain lactate is frequently elevated in subarachnoid hemorrhage patients, predominantly because of hyperglycolysis rather than hypoxia. A pattern of increased cerebral hyperglycolytic lactate was associated with good long-term recovery. Our data suggest that lactate may be used as an aerobic substrate by the injured human brain.

No use for waist-for-height ratio in addition to body mass index to identify children with elevated blood pressure.

Abstract Background. In children, waist-for-height ratio (WHtR) has been proposed to identify subjects at higher risk of cardiovascular diseases. The utility of WHtR to identify children with elevated blood pressure (BP) is unclear. Design. Cross-sectional population-based study of schoolchildren. Methods. Weight, height, waist circumference and BP were measured in all sixth-grade schoolchildren of the canton de Vaud (Switzerland) in 2005/06. WHtR was computed as waist [cm]/height [cm]. Elevated BP was defined according to sex-, age- and height-specific US reference data. The area under the receiver operating characteristic curve (AUC) statistic was computed to compare the ability of body mass index (BMI) z-score and WHtR, alone or in combination, to identify children with elevated BP. Results. 5207 children participated (76% response) [2621 boys, 2586 girls; mean (± SD) age, 12.3 ± 0.5 years; range: 10.1-14.9]. The prevalence of elevated BP was 11%. Mean WHtR was 0.44 ± 0.05 (range: 0.29- 0.77) and 11% had high WHtR (> 0.5). BMI z-score and WHtR were strongly correlated (Spearman correlation coefficient r = 0.76). Both indices were positively associated with elevated BP. AUCs for elevated BP was relatively low for BMI z-score (0.62) or for WHtR (0.62), and was not substantially improved when both indices were considered together (0.63). Conclusions. The ability of BMI z-score or WHtR to identify children aged 10-14 with elevated BP was weak. Adding WHtR did not confer additional discriminative power to BMI alone. These findings do not support the measurement of WHtR in addition to BMI to identify children with elevated BP.

Consequences of modified lipid and glucose metabolism on peripheral nervous system structure and function

284 million people worldwide suffered from type 2 diabetes mellitus (T2DM) in 2010, which will, in approximately half of them, lead to the development of diabetic peripheral neuropathy (DPN). Although DPN is the most common complication of diabetes mellitus and the leading cause of non-traumatic amputations its pathophysiology is still poorly understood. To get more insight into the molecular mechanism underlying DPN in T2DM, I used a rodent model of T2DM, the db/db mice.¦ln vivo electrophysiological recordings of diabetic animals indicated that in addition to reduced nerve conduction velocity db/db mice also present increased nerve excitability. Further ex vivo evaluation of the electrophysiological properties of db/db nerves clearly established a presence of the peripheral nerve hyperexcitability (PNH) phenotype in diabetic animals. Using pharmacological inhibitors we demonstrated that PNH is mostly mediated by the decreased activity of Kv1 channels. ln agreement with these data 1 observed that the diabetic condition led to a reduced presence of the Kv1.2 subunits in juxtaparanodal regions of db/db peripheral nerves whereas its mANA and protein expression levels were not affected. Lmportantly, I confirmed a loss of juxtaparanodal Kv1.2 subunits in nerve biopsies from type 2 diabetic patients. Together these observations indicate that the type 2 diabetic condition leads to potassium-channel mediated changes of nerve excitability thus identifying them as potential drug targets to treat sorne of the DPN related symptoms.¦Schwann cells ensheath and isolate peripheral axons by the production of myelin, which consists of lipids and proteins in a ratio of 2:1. Peripheral myelin protein 2 (= P2, Pmp2 or FABP8) was originally described as one of the most abundant myelin proteins in the peripheral nervous system. P2, which is a member of the fatty acid binding protein (FABP) family, is a 14.8 kDa cytosolic protein expressed on the cytoplasmic side of compact myelin membranes. As indicated by their name, the principal role of FABPs is thought to be the binding and transport of fatty acids.¦To study its role in myelinating glial cells I have recently generated a complete P2 knockout mouse model (P2-/-). I confirmed the loss of P2 in the sciatic nerve of P2-/- mice at the mRNA and protein level. Electrophysiological analysis of the adult (P56) mutant mice revealed a mild but significant reduction in the motor nerve conduction velocity. lnterestingly, this functional change was not accompanied by any detectable alterations in general myelin structure. However, I have observed significant alterations in the mRNA expression level of other FABPs, predominantly FABP9, in the PNS of P2-/- mice as compared to age-matched P2+/+ mice indicating a role of P2 in the glial myelin lipid metabolism.¦Le diabète de type 2 touche 284 million de personnes dans le monde en 2010 et son évolution conduit dans la moitié des cas à une neuropathie périphérique diabétique. Bien que la neuropathie périphérique soit la complication la plus courante du diabète pouvant conduire jusqu'à l'amputation, sa physiopathologie est aujourd'hui encore mal comprise. Dans le but d'améliorer les connaissances moléculaires expliquant les mécanismes de la neuropathie liée au diabète de type 2, j'ai utilisé un modèle murin du diabète de type 2, les souris db/db.¦ln vivo, les enregistrements éléctrophysiologiques des animaux diabétiques montrent qu'en plus d'une diminution de la vitesse de conduction nerveuse, les souris db/db présentent également une augmentation de l'excitabilité nerveuse. Des mesures menées Ex­ vivo ont montré l'existence d'un phénotype d'hyperexcitabilité sur les nerfs périphériques isolés d'animaux diabétiques. Grâce à l'utilisation d'inhibiteurs pharmacologiques, nous avons pu démontrer que l'hyperexcitabilité démontrée était due à une réduction d'activité des canaux Kv1. En accord avec ces données, j'ai observé qu'une situation de diabète conduisait à une diminution des canaux Kv1.2 aux régions juxta-paranodales des nerfs périphériques db/db, alors que l'expression du transcrit et de la protéine restait stable. J'ai également confirmé l'absence de canaux Kv1.2 aux juxta-paranoeuds de biopsies de nerfs de patients diabétiques. L'ensemble de ces observations montrent que les nerfs périphériques chez les patients atteints de diabète de type 2 est due à une diminution des canaux potassiques rapides juxtaparanodaux les identifiant ainsi comme des cibles thérapeutiques potentielles.¦Les cellules de Schwann enveloppent et isolent les axones périphériques d'une membrane spécialisée, la myéline, composée de deux fois plus de lipides que de protéines. La protéine P2 (Pmp2 "peripheral myelin protein 2" ou FABP8 "fatty acid binding protein") est l'une des protéines les plus abondantes au système nerveux périphérique. P2 appartient à la famille de protéines FABP liant et transportant les acides gras et est une protéine cytosolique de 14,8 kDa exprimée du côté cytoplasmique de la myéline compacte.¦Afin d'étudier le rôle de P2 dans les cellules de Schwann myélinisantes, j'ai généré une souris knockout (P2-/-). Après avoir validé l'absence de transcrit et de protéine P2 dans les nerfs sciatiques P2-/-, des mesures électrophysiologiques ont montré une réduction modérée mais significative de la vitesse de conduction du nerf moteur périphérique. Il est important de noter que ces changements fonctionnels n'ont pas pu être associés à quelconque changement dans la structure de la myéline. Cependant, j'ai observé dans les nerfs périphériques P2-/-, une altération significative du niveau d'expression d'ARNm d'autres FABPs et en particulier FABP9. Ce dernier résultat démontre l'importance du rôle de la protéine P2 dans le métabolisme lipidique de la myéline.

A genome-wide screen for interactions reveals a new locus on 4p15 modifying the effect of waist-to-hip ratio on total cholesterol.

Recent genome-wide association (GWA) studies described 95 loci controlling serum lipid levels. These common variants explain ∼25% of the heritability of the phenotypes. To date, no unbiased screen for gene-environment interactions for circulating lipids has been reported. We screened for variants that modify the relationship between known epidemiological risk factors and circulating lipid levels in a meta-analysis of genome-wide association (GWA) data from 18 population-based cohorts with European ancestry (maximum N = 32,225). We collected 8 further cohorts (N = 17,102) for replication, and rs6448771 on 4p15 demonstrated genome-wide significant interaction with waist-to-hip-ratio (WHR) on total cholesterol (TC) with a combined P-value of 4.79×10(-9). There were two potential candidate genes in the region, PCDH7 and CCKAR, with differential expression levels for rs6448771 genotypes in adipose tissue. The effect of WHR on TC was strongest for individuals carrying two copies of G allele, for whom a one standard deviation (sd) difference in WHR corresponds to 0.19 sd difference in TC concentration, while for A allele homozygous the difference was 0.12 sd. Our findings may open up possibilities for targeted intervention strategies for people characterized by specific genomic profiles. However, more refined measures of both body-fat distribution and metabolic measures are needed to understand how their joint dynamics are modified by the newly found locus.

On the inefficiency of the restricted maximum likelihood

The restricted maximum likelihood is preferred by many to the full maximumlikelihood for estimation with variance component and other randomcoefficientmodels, because the variance estimator is unbiased. It is shown that thisunbiasednessis accompanied in some balanced designs by an inflation of the meansquared error.An estimator of the cluster-level variance that is uniformly moreefficient than the fullmaximum likelihood is derived. Estimators of the variance ratio are alsostudied.

Conditional maximum likelihood timing recovery: estimators and bounds

This paper is concerned with the derivation of new estimators and performance bounds for the problem of timing estimation of (linearly) digitally modulated signals. The conditional maximum likelihood (CML) method is adopted, in contrast to the classical low-SNR unconditional ML (UML) formulationthat is systematically applied in the literature for the derivationof non-data-aided (NDA) timing-error-detectors (TEDs). A new CML TED is derived and proved to be self-noise free, in contrast to the conventional low-SNR-UML TED. In addition, the paper provides a derivation of the conditional Cramér–Rao Bound (CRB ), which is higher (less optimistic) than the modified CRB (MCRB)[which is only reached by decision-directed (DD) methods]. It is shown that the CRB is a lower bound on the asymptotic statisticalaccuracy of the set of consistent estimators that are quadratic with respect to the received signal. Although the obtained boundis not general, it applies to most NDA synchronizers proposed in the literature. A closed-form expression of the conditional CRBis obtained, and numerical results confirm that the CML TED attains the new bound for moderate to high Eg/No.

Sensory characteristics of 'Douradão' peaches submitted to modified atmosphere packaging

The sensory, physical and chemical characteristics of 'Douradão' peaches cold stored in different modified atmosphere packaging (LDPE bags of 30, 50, 60, 75µm thickness) were studied. After 14, 21 and 28 days of cold storage (1 ± 1 ºC and 90 ± 5% RH), samples were withdrawn from MAP and kept during 4 days in ambient air for ripening. Descriptive terminology and sensory profile of the peaches were developed by methodology based on the Quantitative Descriptive Analysis (QDA). The assessors consensually defined the sensory descriptors, their respective reference materials and the descriptive evaluation ballot. Fourteen individuals were selected as judges based on their discrimination capacity and reproducibility. Seven descriptors were generated showing similarities and differences among the samples. The data were analysed by ANOVA, Tukey test and Principal Component Analysis (PCA). The atmospheres that developed inside the different packaging materials during cold storage differed significantly. The PCA showed that MA50 and MA60 treatments were more characterized by the fresh peach flavour, fresh appearance, juiciness and flesh firmness, and were effective for keeping good quality of 'Douradão' peaches during 28 d of cold storage. The Control and MA30 treatments were characterized by the mealiness, the MA75 treatment showed lower intensity for all attributes evaluated and they were ineffective to maintain good quality of the fruits during cold storage. Higher correlation coefficients (positive) were found between fresh appearance and flesh firmness (0.95), fresh appearance and juiciness (0.97), ratio and intensity of fresh peach smell (0.81), as well as higher correlation coefficients (negative) between Hue angle and intensity of yellow colour (-0.91), fresh appearance and mealiness (-0.92), juiciness and mealiness (-0.95), firmness and mealiness (-0.94).

Indirect likelihood inference (revised)

Standard indirect Inference (II) estimators take a given finite-dimensional statistic, Z_{n} , and then estimate the parameters by matching the sample statistic with the model-implied population moment. We here propose a novel estimation method that utilizes all available information contained in the distribution of Z_{n} , not just its first moment. This is done by computing the likelihood of Z_{n}, and then estimating the parameters by either maximizing the likelihood or computing the posterior mean for a given prior of the parameters. These are referred to as the maximum indirect likelihood (MIL) and Bayesian Indirect Likelihood (BIL) estimators, respectively. We show that the IL estimators are first-order equivalent to the corresponding moment-based II estimator that employs the optimal weighting matrix. However, due to higher-order features of Z_{n} , the IL estimators are higher order efficient relative to the standard II estimator. The likelihood of Z_{n} will in general be unknown and so simulated versions of IL estimators are developed. Monte Carlo results for a structural auction model and a DSGE model show that the proposed estimators indeed have attractive finite sample properties.

Application of a modified sequential organ failure assessment score to critically ill patients

The purpose of the present study was to explore the usefulness of the Mexican sequential organ failure assessment (MEXSOFA) score for assessing the risk of mortality for critically ill patients in the ICU. A total of 232 consecutive patients admitted to an ICU were included in the study. The MEXSOFA was calculated using the original SOFA scoring system with two modifications: the PaO2/FiO2 ratio was replaced with the SpO2/FiO2 ratio, and the evaluation of neurologic dysfunction was excluded. The ICU mortality rate was 20.2%. Patients with an initial MEXSOFA score of 9 points or less calculated during the first 24 h after admission to the ICU had a mortality rate of 14.8%, while those with an initial MEXSOFA score of 10 points or more had a mortality rate of 40%. The MEXSOFA score at 48 h was also associated with mortality: patients with a score of 9 points or less had a mortality rate of 14.1%, while those with a score of 10 points or more had a mortality rate of 50%. In a multivariate analysis, only the MEXSOFA score at 48 h was an independent predictor for in-ICU death with an OR = 1.35 (95%CI = 1.14-1.59, P < 0.001). The SOFA and MEXSOFA scores calculated 24 h after admission to the ICU demonstrated a good level of discrimination for predicting the in-ICU mortality risk in critically ill patients. The MEXSOFA score at 48 h was an independent predictor of death; with each 1-point increase, the odds of death increased by 35%.

Particle size and cholesterol content of a mayonnaise formulated by OSA-modified potato starch

Egg yolk was partially replaced (0, 25, 50, 75, and 100%) with octenyl succinic anhydride (OSA)-modified potato starch in a reduced-fat mayonnaise formulation to curtail the problems associated with high cholesterol and induced allergic reactions. The physicochemical properties included parameters such as: pH, fat content, and emulsion stability of the formulations analyzed. The samples with 75% and 100% egg yolk substitute showed the maximum emulsion stability (>95% after two of months storage), and they were selected according to cholesterol content, particle size distributions, dynamic rheological properties, microstructure, and sensory characteristic. A significant reduction (84-97%) in the cholesterol content was observed in the selected samples. Particle size analysis showed that by increasing the amount of OSA starch, the oil droplets with the peak size of 70 µm engulfed by this compound became larger. The rheological tests elucidated that in the absence of egg yolk, OSA starch may not result in a final product with consistent texture and that the best ratio of the two emulsifiers (OSA starch/egg yolk) to produce stable reduced-fat, low cholesterol mayonnaise is 75/25. The microscopic images confirmed the formation of a stable cohesive layer of starch surrounding the oil droplets emulsified in the samples selected.

A modified nominal stress method for fatigue assessment of steel plates with thermally cut edges

Thermal cutting methods, are commonly used in the manufacture of metal parts. Thermal cutting processes separate materials by using heat. The process can be done with or without a stream of cutting oxygen. Common processes are Oxygen, plasma and laser cutting. It depends on the application and material which cutting method is used. Numerically-controlled thermal cutting is a cost-effective way of prefabricating components. One design aim is to minimize the number of work steps in order to increase competitiveness. This has resulted in the holes and openings in plate parts manufactured today being made using thermal cutting methods. This is a problem from the fatigue life perspective because there is local detail in the as-welded state that causes a rise in stress in a local area of the plate. In a case where the static utilization of a net section is full used, the calculated linear local stresses and stress ranges are often over 2 times the material yield strength. The shakedown criteria are exceeded. Fatigue life assessment of flame-cut details is commonly based on the nominal stress method. For welded details, design standards and instructions provide more accurate and flexible methods, e.g. a hot-spot method, but these methods are not universally applied to flame cut edges. Some of the fatigue tests of flame cut edges in the laboratory indicated that fatigue life estimations based on the standard nominal stress method can give quite a conservative fatigue life estimate in cases where a high notch factor was present. This is an undesirable phenomenon and it limits the potential for minimizing structure size and total costs. A new calculation method is introduced to improve the accuracy of the theoretical fatigue life prediction method of a flame cut edge with a high stress concentration factor. Simple equations were derived by using laboratory fatigue test results, which are published in this work. The proposed method is called the modified FAT method (FATmod). The method takes into account the residual stress state, surface quality, material strength class and true stress ratio in the critical place.

Synthesis, Characterization and Applications of Modified TiO2 Systems

This thesis is divided in to 9 chapters and deals with the modification of TiO2 for various applications include photocatalysis, thermal reaction, photovoltaics and non-linear optics. Chapter 1 involves a brief introduction of the topic of study. An introduction to the applications of modified titania systems in various fields are discussed concisely. Scope and objectives of the present work are also discussed in this chapter. Chapter 2 explains the strategy adopted for the synthesis of metal, nonmetal co-doped TiO2 systems. Hydrothermal technique was employed for the preparation of the co-doped TiO2 system, where Ti[OCH(CH3)2]4, urea and metal nitrates were used as the sources for TiO2, N and metals respectively. In all the co-doped systems, urea to Ti[OCH(CH3)2]4 was taken in a 1:1 molar ratio and varied the concentration of metals. Five different co-doped catalytic systems and for each catalysts, three versions were prepared by varying the concentration of metals. A brief explanation of physico-chemical techniques used for the characterization of the material was also presented in this chapter. This includes X-ray Diffraction (XRD), Raman Spectroscopy, FTIR analysis, Thermo Gravimetric Analysis, Energy Dispersive X-ray Analysis (EDX), Scanning Electron Microscopy(SEM), UV-Visible Diffuse Reflectance Spectroscopy (UV-Vis DRS), Transmission Electron Microscopy (TEM), BET Surface Area Measurements and X-ray Photoelectron Spectroscopy (XPS). Chapter 3 contains the results and discussion of characterization techniques used for analyzing the prepared systems. Characterization is an inevitable part of materials research. Determination of physico-chemical properties of the prepared materials using suitable characterization techniques is very crucial to find its exact field of application. It is clear from the XRD pattern that photocatalytically active anatase phase dominates in the calcined samples with peaks at 2θ values around 25.4°, 38°, 48.1°, 55.2° and 62.7° corresponding to (101), (004), (200), (211) and (204) crystal planes (JCPDS 21-1272) respectively. But in the case of Pr-N-Ti sample, a new peak was observed at 2θ = 30.8° corresponding to the (121) plane of the polymorph brookite. There are no visible peaks corresponding to dopants, which may be due to their low concentration or it is an indication of the better dispersion of impurities in the TiO2. Crystallite size of the sample was calculated from Scherrer equation byusing full width at half maximum (FWHM) of the (101) peak of the anatase phase. Crystallite size of all the co-doped TiO2 was found to be lower than that of bare TiO2 which indicates that the doping of metal ions having higher ionic radius into the lattice of TiO2 causes some lattice distortion which suppress the growth of TiO2 nanoparticles. The structural identity of the prepared system obtained from XRD pattern is further confirmed by Raman spectra measurements. Anatase has six Raman active modes. Band gap of the co-doped system was calculated using Kubelka-Munk equation and that was found to be lower than pure TiO2. Stability of the prepared systems was understood from thermo gravimetric analysis. FT-IR was performed to understand the functional groups as well as to study the surface changes occurred during modification. EDX was used to determine the impurities present in the system. The EDX spectra of all the co-doped samples show signals directly related to the dopants. Spectra of all the co-doped systems contain O and Ti as the main components with low concentrations of doped elements. Morphologies of the prepared systems were obtained from SEM and TEM analysis. Average particle size of the systems was drawn from histogram data. Electronic structures of the samples were identified perfectly from XPS measurements. Chapter 4 describes the photocatalytic degradation of herbicides Atrazine and Metolachlor using metal, non-metal co-doped titania systems. The percentage of degradation was analyzed by HPLC technique. Parameters such as effect of different catalysts, effect of time, effect of catalysts amount and reusability studies were discussed. Chapter 5 deals with the photo-oxidation of some anthracene derivatives by co-doped catalytic systems. These anthracene derivatives come underthe category of polycyclic aromatic hydrocarbons (PAH). Due to the presence of stable benzene rings, most of the PAH show strong inhibition towards biological degradation and the common methods employed for their removal. According to environmental protection agency, most of the PAH are highly toxic in nature. TiO2 photochemistry has been extensively investigated as a method for the catalytic conversion of such organic compounds, highlighting the potential of thereof in the green chemistry. There are actually two methods for the removal of pollutants from the ecosystem. Complete mineralization is the one way to remove pollutants. Conversion of toxic compounds to another compound having toxicity less than the initial starting compound is the second way. Here in this chapter, we are concentrating on the second aspect. The catalysts used were Gd(1wt%)-N-Ti, Pd(1wt%)-N-Ti and Ag(1wt%)-N-Ti. Here we were very successfully converted all the PAH to anthraquinone, a compound having diverse applications in industrial as well as medical fields. Substitution of 10th position of desired PAH by phenyl ring reduces the feasibility of photo reaction and produced 9-hydroxy 9-phenyl anthrone (9H9PA) as an intermediate species. The products were separated and purified by column chromatography using 70:30 hexane/DCM mixtures as the mobile phase and the resultant products were characterized thoroughly by 1H NMR, IR spectroscopy and GCMS analysis. Chapter 6 elucidates the heterogeneous Suzuki coupling reaction by Cu/Pd bimetallic supported on TiO2. Sol-Gel followed by impregnation method was adopted for the synthesis of Cu/Pd-TiO2. The prepared system was characterized by XRD, TG-DTG, SEM, EDX, BET Surface area and XPS. The product was separated and purified by column chromatography using hexane as the mobile phase. Maximum isolated yield of biphenyl of around72% was obtained in DMF using Cu(2wt%)-Pd(4wt%)-Ti as the catalyst. In this reaction, effective solvent, base and catalyst were found to be DMF, K2CO3 and Cu(2wt%)-Pd(4wt%)-Ti respectively. Chapter 7 gives an idea about the photovoltaic (PV) applications of TiO2 based thin films. Due to energy crisis, the whole world is looking for a new sustainable energy source. Harnessing solar energy is one of the most promising ways to tackle this issue. The present dominant photovoltaic (PV) technologies are based on inorganic materials. But the high material, low power conversion efficiency and manufacturing cost limits its popularization. A lot of research has been conducted towards the development of low-cost PV technologies, of which organic photovoltaic (OPV) devices are one of the promising. Here two TiO2 thin films having different thickness were prepared by spin coating technique. The prepared films were characterized by XRD, AFM and conductivity measurements. The thickness of the films was measured by Stylus Profiler. This chapter mainly concentrated on the fabrication of an inverted hetero junction solar cell using conducting polymer MEH-PPV as photo active layer. Here TiO2 was used as the electron transport layer. Thin films of MEH-PPV were also prepared using spin coating technique. Two fullerene derivatives such as PCBM and ICBA were introduced into the device in order to improve the power conversion efficiency. Effective charge transfer between the conducting polymer and ICBA were understood from fluorescence quenching studies. The fabricated Inverted hetero junction exhibited maximum power conversion efficiency of 0.22% with ICBA as the acceptor molecule. Chapter 8 narrates the third order order nonlinear optical properties of bare and noble metal modified TiO2 thin films. Thin films were fabricatedby spray pyrolysis technique. Sol-Gel derived Ti[OCH(CH3)2]4 in CH3CH2OH/CH3COOH was used as the precursor for TiO2. The precursors used for Au, Ag and Pd were the aqueous solutions of HAuCl4, AgNO3 and Pd(NO3)2 respectively. The prepared films were characterized by XRD, SEM and EDX. The nonlinear optical properties of the prepared materials were investigated by Z-Scan technique comprising of Nd-YAG laser (532 nm,7 ns and10 Hz). The non-linear coefficients were obtained by fitting the experimental Z-Scan plot with the theoretical plots. Nonlinear absorption is a phenomenon defined as a nonlinear change (increase or decrease) in absorption with increasing of intensity. This can be mainly divided into two types: saturable absorption (SA) and reverse saturable absorption (RSA). Depending on the pump intensity and on the absorption cross- section at the excitation wavelength, most molecules show non- linear absorption. With increasing intensity, if the excited states show saturation owing to their long lifetimes, the transmission will show SA characteristics. Here absorption decreases with increase of intensity. If, however, the excited state has strong absorption compared with that of the ground state, the transmission will show RSA characteristics. Here in our work most of the materials show SA behavior and some materials exhibited RSA behavior. Both these properties purely depend on the nature of the materials and alignment of energy states within them. Both these SA and RSA have got immense applications in electronic devices. The important results obtained from various studies are presented in chapter 9.

Estimators and Tests based on Likelihood-Depth with Application to Weibull Distribution, Gaussian and Gumbel Copula

In dieser Arbeit werden mithilfe der Likelihood-Tiefen, eingeführt von Mizera und Müller (2004), (ausreißer-)robuste Schätzfunktionen und Tests für den unbekannten Parameter einer stetigen Dichtefunktion entwickelt. Die entwickelten Verfahren werden dann auf drei verschiedene Verteilungen angewandt. Für eindimensionale Parameter wird die Likelihood-Tiefe eines Parameters im Datensatz als das Minimum aus dem Anteil der Daten, für die die Ableitung der Loglikelihood-Funktion nach dem Parameter nicht negativ ist, und dem Anteil der Daten, für die diese Ableitung nicht positiv ist, berechnet. Damit hat der Parameter die größte Tiefe, für den beide Anzahlen gleich groß sind. Dieser wird zunächst als Schätzer gewählt, da die Likelihood-Tiefe ein Maß dafür sein soll, wie gut ein Parameter zum Datensatz passt. Asymptotisch hat der Parameter die größte Tiefe, für den die Wahrscheinlichkeit, dass für eine Beobachtung die Ableitung der Loglikelihood-Funktion nach dem Parameter nicht negativ ist, gleich einhalb ist. Wenn dies für den zu Grunde liegenden Parameter nicht der Fall ist, ist der Schätzer basierend auf der Likelihood-Tiefe verfälscht. In dieser Arbeit wird gezeigt, wie diese Verfälschung korrigiert werden kann sodass die korrigierten Schätzer konsistente Schätzungen bilden. Zur Entwicklung von Tests für den Parameter, wird die von Müller (2005) entwickelte Simplex Likelihood-Tiefe, die eine U-Statistik ist, benutzt. Es zeigt sich, dass für dieselben Verteilungen, für die die Likelihood-Tiefe verfälschte Schätzer liefert, die Simplex Likelihood-Tiefe eine unverfälschte U-Statistik ist. Damit ist insbesondere die asymptotische Verteilung bekannt und es lassen sich Tests für verschiedene Hypothesen formulieren. Die Verschiebung in der Tiefe führt aber für einige Hypothesen zu einer schlechten Güte des zugehörigen Tests. Es werden daher korrigierte Tests eingeführt und Voraussetzungen angegeben, unter denen diese dann konsistent sind. Die Arbeit besteht aus zwei Teilen. Im ersten Teil der Arbeit wird die allgemeine Theorie über die Schätzfunktionen und Tests dargestellt und zudem deren jeweiligen Konsistenz gezeigt. Im zweiten Teil wird die Theorie auf drei verschiedene Verteilungen angewandt: Die Weibull-Verteilung, die Gauß- und die Gumbel-Copula. Damit wird gezeigt, wie die Verfahren des ersten Teils genutzt werden können, um (robuste) konsistente Schätzfunktionen und Tests für den unbekannten Parameter der Verteilung herzuleiten. Insgesamt zeigt sich, dass für die drei Verteilungen mithilfe der Likelihood-Tiefen robuste Schätzfunktionen und Tests gefunden werden können. In unverfälschten Daten sind vorhandene Standardmethoden zum Teil überlegen, jedoch zeigt sich der Vorteil der neuen Methoden in kontaminierten Daten und Daten mit Ausreißern.

Aitchison Geometry for Probability and Likelihood as a new approach to mathematical statistics

The Aitchison vector space structure for the simplex is generalized to a Hilbert space structure A2(P) for distributions and likelihoods on arbitrary spaces. Central notations of statistics, such as Information or Likelihood, can be identified in the algebraical structure of A2(P) and their corresponding notions in compositional data analysis, such as Aitchison distance or centered log ratio transform. In this way very elaborated aspects of mathematical statistics can be understood easily in the light of a simple vector space structure and of compositional data analysis. E.g. combination of statistical information such as Bayesian updating, combination of likelihood and robust M-estimation functions are simple additions/ perturbations in A2(Pprior). Weighting observations corresponds to a weighted addition of the corresponding evidence. Likelihood based statistics for general exponential families turns out to have a particularly easy interpretation in terms of A2(P). Regular exponential families form finite dimensional linear subspaces of A2(P) and they correspond to finite dimensional subspaces formed by their posterior in the dual information space A2(Pprior). The Aitchison norm can identified with mean Fisher information. The closing constant itself is identified with a generalization of the cummulant function and shown to be Kullback Leiblers directed information. Fisher information is the local geometry of the manifold induced by the A2(P) derivative of the Kullback Leibler information and the space A2(P) can therefore be seen as the tangential geometry of statistical inference at the distribution P. The discussion of A2(P) valued random variables, such as estimation functions or likelihoods, give a further interpretation of Fisher information as the expected squared norm of evidence and a scale free understanding of unbiased reasoning