Parameterschätzung in zeitdiskreten ergodischen Markov-Prozessen am Beispiel des Cox-Ingersoll-Ross-Modells

In dieser Arbeit geht es um die Schätzung von Parametern in zeitdiskreten ergodischen Markov-Prozessen im allgemeinen und im CIR-Modell im besonderen. Beim CIR-Modell handelt es sich um eine stochastische Differentialgleichung, die von Cox, Ingersoll und Ross (1985) zur Beschreibung der Dynamik von Zinsraten vorgeschlagen wurde. Problemstellung ist die Schätzung der Parameter des Drift- und des Diffusionskoeffizienten aufgrund von äquidistanten diskreten Beobachtungen des CIR-Prozesses. Nach einer kurzen Einführung in das CIR-Modell verwenden wir die insbesondere von Bibby und Sørensen untersuchte Methode der Martingal-Schätzfunktionen und -Schätzgleichungen, um das Problem der Parameterschätzung in ergodischen Markov-Prozessen zunächst ganz allgemein zu untersuchen. Im Anschluss an Untersuchungen von Sørensen (1999) werden hinreichende Bedingungen (im Sinne von Regularitätsvoraussetzungen an die Schätzfunktion) für die Existenz, starke Konsistenz und asymptotische Normalität von Lösungen einer Martingal-Schätzgleichung angegeben. Angewandt auf den Spezialfall der Likelihood-Schätzung stellen diese Bedingungen zugleich lokal-asymptotische Normalität des Modells sicher. Ferner wird ein einfaches Kriterium für Godambe-Heyde-Optimalität von Schätzfunktionen angegeben und skizziert, wie dies in wichtigen Spezialfällen zur expliziten Konstruktion optimaler Schätzfunktionen verwendet werden kann. Die allgemeinen Resultate werden anschließend auf das diskretisierte CIR-Modell angewendet. Wir analysieren einige von Overbeck und Rydén (1997) vorgeschlagene Schätzer für den Drift- und den Diffusionskoeffizienten, welche als Lösungen quadratischer Martingal-Schätzfunktionen definiert sind, und berechnen das optimale Element in dieser Klasse. Abschließend verallgemeinern wir Ergebnisse von Overbeck und Rydén (1997), indem wir die Existenz einer stark konsistenten und asymptotisch normalen Lösung der Likelihood-Gleichung zeigen und lokal-asymptotische Normalität für das CIR-Modell ohne Einschränkungen an den Parameterraum beweisen.

Assessing the effectiveness of ISO 3382 for evaluating the acoustical quality of italian historical theatres: the case study of the Bonci Theatre

The present study concerns the acoustical characterisation of Italian historical theatres. It moved from the ISO 3382 which provides the guidelines for the measurement of a well established set of room acoustic parameters inside performance spaces. Nevertheless, the peculiarity of Italian historical theatres needs a more specific approach. The Charter of Ferrara goes in this direction, aiming at qualifying the sound field in this kind of halls and the present work pursues the way forward. Trying to understand how the acoustical qualification should be done, the Bonci Theatre in Cesena has been taken as a case study. In September 2012 acoustical measurements were carried out in the theatre, recording monaural e binaural impulse responses at each seat in the hall. The values of the time criteria, energy criteria and psycho-acoustical and spatial criteria have been extracted according to ISO 3382. Statistics were performed and a 3D model of the theatre was realised and tuned. Statistical investigations were carried out on the whole set of measurement positions and on carefully chosen reduced subsets; it turned out that these subsets are representative only of the “average” acoustics of the hall. Normality tests were carried out to verify whether EDT, T30 and C80 could be described with some degree of reliability with a theoretical distribution. Different results, according to the varying assumptions underlying each test, were found. Finally, an attempt was made to correlate the numerical results emerged from the statistical analysis to the perceptual sphere. Looking for “acoustical equivalent areas”, relative difference limens were considered as threshold values. No rule of thumb emerged. Finally, the significance of the usual representation through mean values and standard deviation, which may be meaningful for normal distributed data, was investigated.

Utilzzo dei parametri ecocardiografici tradizionali ed a Doppler tissutale nella valutazione della disfunzione atrio-ventricolare in cani affetti da malattia mitralica cronica

In questo lavoro ci siamo posti come obiettivo lo studio della disfunzione atrio-ventricolare mediante tecniche ecocardiografiche avanzate (come il Tissue Doppler Imaging - TDI) in cani affetti da malattia mitralica cronica (MVD). Una prima parte è volta alla valutazione della funzionalità diastolica del ventricolo destro. Ci siamo proposti di analizzare la funzione del ventricolo destro in cani affetti da malattia del cuore sinistro per comprendere se quest’ultima possa condizionare direttamente la performance del settore cardiaco controlaterale. I risultati più importanti che abbiamo riscontrato sono: l’assenza di differenze significative nella disfunzione sisto-diastolica del ventricolo destro in cani con MVD a diverso stadio; la diretta correlazione tra le variabili TDI di funzionalità del ventricolo destro con il grado di disfunzione del ventricolo sinistro, come indicatori di interdipendenza ventricolare; ed infine il riscontro di una maggior tendenza ad alterazioni diastoliche del ventricolo sinistro in cani con ipertensione polmonare. A quest’ultimo proposito, per quanto riguarda le variabili TDI, il rapporto E/e’ dell’anulus mitralico laterale e settale è risultato avere una differenza significativa tra i cani con ipertensione polmonare e quelli privi di ipertensione polmonare (P<0,01). Nel secondo studio abbiamo applicato il TDI per l’analisi della funzione sisto-diastolica dell’atrio sinistro. Il lavoro è stato articolato in una parte di validazione della metodica su cani normali ed una su animali affetti da MDV. I risultati ottenuti mostrano che la valutazione ecocardiografica delle proprietà di deformazione dell’atrio sinistro basata sul TDI è attuabile e riproducibile nel cane. Abbiamo fornito dei valori di normalità per questa specie e confrontato questi dati con quelli ricavati in cani portatori di MVD. Le differenza tra le varie classi di malattia, nei diversi gradi di dilatazione atriale, sono risultate limitate, ma abbiamo individuato delle correlazioni tra i parametri TDI ed alcune variabili di funzionalità atriale.

Chapter I: Definitions, epidemiology, clinical presentation and prognosis

The concept of chronic critical limb ischaemia (CLI) emerged late in the history of peripheral arterial occlusive disease (PAOD). The historical background and changing definitions of CLI over the last decades are important to know in order to understand why epidemiologic data are so difficult to compare between articles and over time. The prevalence of CLI is probably very high and largely underestimated, and significant differences exist between population studies and clinical series. The extremely high costs associated with management of these patients make CLI a real public health issue for the future. In the era of emerging vascular surgery in the 1950s, the initial classification of PAOD by Fontaine, with stages III and IV corresponding to CLI, was based only on clinical symptoms. Later, with increasing access to non-invasive haemodynamic measurements (ankle pressure, toe pressure), the need to prove a causal relationship between PAOD and clinical findings suggestive of CLI became a real concern, and the Rutherford classification published in 1986 included objective haemodynamic criteria. The first consensus document on CLI was published in 1991 and included clinical criteria associated with ankle and toe pressure and transcutaneous oxygen pressure (TcPO(2)) cut-off levels <50 mmHg, <30 mmHg and <10 mmHg respectively). This rigorous definition reflects an arterial insufficiency that is so severe as to cause microcirculatory changes and compromise tissue integrity, with a high rate of major amputation and mortality. The TASC I consensus document published in 2000 used less severe pressure cut-offs (≤ 50-70 mmHg, ≤ 30-50 mmHg and ≤ 30-50 mmHg respectively). The thresholds for toe pressure and especially TcPO(2) (which will be also included in TASC II consensus document) are however just below the lower limit of normality. It is therefore easy to infer that patients qualifying as CLI based on TASC criteria can suffer from far less severe disease than those qualifying as CLI in the initial 1991 consensus document. Furthermore, inclusion criteria of many recent interventional studies have even shifted further from the efforts of definition standardisation with objective criteria, by including patients as CLI based merely on Fontaine classification (stage III and IV) without haemodynamic criteria. The differences in the natural history of patients with CLI, including prognosis of the limb and the patient, are thus difficult to compare between studies in this context. Overall, CLI as defined by clinical and haemodynamic criteria remains a severe condition with poor prognosis, high medical costs and a major impact in terms of public health and patients' loss of functional capacity. The major progresses in best medical therapy of arterial disease and revascularisation procedures will certainly improve the outcome of CLI patients. In the future, an effort to apply a standardised definition with clinical and objective haemodynamic criteria will be needed to better demonstrate and compare the advances in management of these patients.

Heritability, determinants and reference values of renal length: a family-based population study

OBJECTIVES: In this population-based study, reference values were generated for renal length, and the heritability and factors associated with kidney length were assessed. METHODS: Anthropometric parameters and renal ultrasound measurements were assessed in randomly selected nuclear families of European ancestry (Switzerland). The adjusted narrow sense heritability of kidney size parameters was estimated by maximum likelihood assuming multivariate normality after power transformation. Gender-specific reference centiles were generated for renal length according to body height in the subset of non-diabetic non-obese participants with normal renal function. RESULTS: We included 374 men and 419 women (mean ± SD, age 47 ± 18 and 48 ± 17 years, BMI 26.2 ± 4 and 24.5 ± 5 kg/m(2), respectively) from 205 families. Renal length was 11.4 ± 0.8 cm in men and 10.7 ± 0.8 cm in women; there was no difference between right and left renal length. Body height, weight and estimated glomerular filtration rate (eGFR) were positively associated with renal length, kidney function negatively, age quadratically, whereas gender and hypertension were not. The adjusted heritability estimates of renal length and volume were 47.3 ± 8.5 % and 45.5 ± 8.8 %, respectively (P < 0.001). CONCLUSION: The significant heritability of renal length and volume highlights the familial aggregation of this trait, independently of age and body size. Population-based references for renal length provide a useful guide for clinicians. KEY POINTS: • Renal length and volume are heritable traits, independent of age and size. • Based on a European population, gender-specific reference values/percentiles are provided for renal length. • Renal length correlates positively with body length and weight. • There was no difference between right and left renal lengths in this study. • This negates general teaching that the left kidney is larger and longer.

Estimating the number of motor units using random sums with independently thinned terms

The problem of estimating the numbers of motor units N in a muscle is embedded in a general stochastic model using the notion of thinning from point process theory. In the paper a new moment type estimator for the numbers of motor units in a muscle is denned, which is derived using random sums with independently thinned terms. Asymptotic normality of the estimator is shown and its practical value is demonstrated with bootstrap and approximative confidence intervals for a data set from a 31-year-old healthy right-handed, female volunteer. Moreover simulation results are presented and Monte-Carlo based quantiles, means, and variances are calculated for N in{300,600,1000}.

Power and Robustness of Linkage Tests for Quantitative Traits in General Pedigrees

There are numerous statistical methods for quantitative trait linkage analysis in human studies. An ideal such method would have high power to detect genetic loci contributing to the trait, would be robust to non-normality in the phenotype distribution, would be appropriate for general pedigrees, would allow the incorporation of environmental covariates, and would be appropriate in the presence of selective sampling. We recently described a general framework for quantitative trait linkage analysis, based on generalized estimating equations, for which many current methods are special cases. This procedure is appropriate for general pedigrees and easily accommodates environmental covariates. In this paper, we use computer simulations to investigate the power robustness of a variety of linkage test statistics built upon our general framework. We also propose two novel test statistics that take account of higher moments of the phenotype distribution, in order to accommodate non-normality. These new linkage tests are shown to have high power and to be robust to non-normality. While we have not yet examined the performance of our procedures in the context of selective sampling via computer simulations, the proposed tests satisfy all of the other qualities of an ideal quantitative trait linkage analysis method.

Inference on Survival Data with Covariate Measurement Error - An Imputation-based Approach

We propose a new method for fitting proportional hazards models with error-prone covariates. Regression coefficients are estimated by solving an estimating equation that is the average of the partial likelihood scores based on imputed true covariates. For the purpose of imputation, a linear spline model is assumed on the baseline hazard. We discuss consistency and asymptotic normality of the resulting estimators, and propose a stochastic approximation scheme to obtain the estimates. The algorithm is easy to implement, and reduces to the ordinary Cox partial likelihood approach when the measurement error has a degenerative distribution. Simulations indicate high efficiency and robustness. We consider the special case where error-prone replicates are available on the unobserved true covariates. As expected, increasing the number of replicate for the unobserved covariates increases efficiency and reduces bias. We illustrate the practical utility of the proposed method with an Eastern Cooperative Oncology Group clinical trial where a genetic marker, c-myc expression level, is subject to measurement error.

Fixed-Width Output Analysis for Markov Chain Monte Carlo

Markov chain Monte Carlo is a method of producing a correlated sample in order to estimate features of a complicated target distribution via simple ergodic averages. A fundamental question in MCMC applications is when should the sampling stop? That is, when are the ergodic averages good estimates of the desired quantities? We consider a method that stops the MCMC sampling the first time the width of a confidence interval based on the ergodic averages is less than a user-specified value. Hence calculating Monte Carlo standard errors is a critical step in assessing the output of the simulation. In particular, we consider the regenerative simulation and batch means methods of estimating the variance of the asymptotic normal distribution. We describe sufficient conditions for the strong consistency and asymptotic normality of both methods and investigate their finite sample properties in a variety of examples.

Novel pressure-to-cornea index in glaucoma

BACKGROUND: Several conversion tables and formulas have been suggested to correct applanation intraocular pressure (IOP) for central corneal thickness (CCT). CCT is also thought to represent an independent glaucoma risk factor. In an attempt to integrate IOP and CCT into a unified risk factor and avoid uncertain correction for tonometric inaccuracy, a new pressure-to-cornea index (PCI) is proposed. METHODS: PCI (IOP/CCT(3)) was defined as the ratio between untreated IOP and CCT(3) in mm (ultrasound pachymetry). PCI distribution in 220 normal controls, 53 patients with normal-tension glaucoma (NTG), 76 with ocular hypertension (OHT), and 89 with primary open-angle glaucoma (POAG) was investigated. PCI's ability to discriminate between glaucoma (NTG+POAG) and non-glaucoma (controls+OHT) was compared with that of three published formulae for correcting IOP for CCT. Receiver operating characteristic (ROC) curves were built. RESULTS: Mean PCI values were: Controls 92.0 (SD 24.8), NTG 129.1 (SD 25.8), OHT 134.0 (SD 26.5), POAG 173.6 (SD 40.9). To minimise IOP bias, eyes within the same 2 mm Hg range between 16 and 29 mm Hg (16-17, 18-19, etc) were separately compared: control and NTG eyes as well as OHT and POAG eyes differed significantly. PCI demonstrated a larger area under the ROC curve (AUC) and significantly higher sensitivity at fixed 80% and 90% specificities compared with each of the correction formulas; optimum PCI cut-off value 133.8. CONCLUSIONS: A PCI range of 120-140 is proposed as the upper limit of "normality", 120 being the cut-off value for eyes with untreated pressures or=22 mm Hg. PCI may reflect individual susceptibility to a given IOP level, and thus represent a glaucoma risk factor. Longitudinal studies are needed to prove its prognostic value.

Berry-Esseen bounds for nonlinear statistics, and asymptotic relative efficiency between correlation statistics

Four papers, written in collaboration with the author’s graduate school advisor, are presented. In the first paper, uniform and non-uniform Berry-Esseen (BE) bounds on the convergence to normality of a general class of nonlinear statistics are provided; novel applications to specific statistics, including the non-central Student’s, Pearson’s, and the non-central Hotelling’s, are also stated. In the second paper, a BE bound on the rate of convergence of the F-statistic used in testing hypotheses from a general linear model is given. The third paper considers the asymptotic relative efficiency (ARE) between the Pearson, Spearman, and Kendall correlation statistics; conditions sufficient to ensure that the Spearman and Kendall statistics are equally (asymptotically) efficient are provided, and several models are considered which illustrate the use of such conditions. Lastly, the fourth paper proves that, in the bivariate normal model, the ARE between any of these correlation statistics possesses certain monotonicity properties; quadratic lower and upper bounds on the ARE are stated as direct applications of such monotonicity patterns.

Effects of mild vitamin a deficiency on lung maturation in newborn rats: a morphometric and morphologic study

OBJECTIVE To evaluate the effects of a 60% vitamin A deficiency (VAD) on the two postnatal stages of lung development: alveolarization and microvascular maturation. Lungs from deficient rats were compared to age-matched controls. STUDY DESIGN Starting at 3 weeks before mating, female rats were maintained under a diet lacking vitamin A. Due to the slow depletion of the vitamin A liver stores the pregnant rats carried to term and delivered pups under mild VAD conditions. Mothers and offspring were then kept under the same diet what resulted in a mean reduction of vitamin A plasma concentration of about 60% vs. controls during the whole experimental period. Pups were sacrificed on days 4, 10 and 21 and their lungs fixed and analyzed by means of a combined morphologic and morphometric investigation at light and electron microscopic levels. RESULTS During the whole experiment, body weights of VAD animals were lower than controls with a significant decrease on day 10. On days 4, 10 and 21 the pulmonary structure was in a comparable gross morphologic state in both groups. Despite this morphologic normality, quantitative alterations in some functional parameters could be detected. On day 4, lung volume and the volume and surface area of air spaces were decreased, while the arithmetic mean barrier thickness and type 2 pneumocyte volume were increased in the VAD group. On day 21, some changes were again manifest mainly consisting in an augmentation of the vascularization and a decrease in interstitial volume in deficient animals. CONCLUSIONS Mild VAD causes no gross disturbances in the postnatal phases of lung development in rats. However, a body weight-related transient retardation of lung maturation was detectable in the first postnatal week. At 3 weeks, the VAD lungs showed a more mature vascular system substantiated by an increase in volume of both capillary volume and the large non-parenchymal vessels. In view of these quantitative alterations, we suspect that mild VAD deregulates the normal phases of body and lung growth, but does not induce serious functional impairments.

Detection of nonrandom association of alleles from the distribution of the number of heterozygous loci in a sample.

The distribution of the number of heterozygous loci in two randomly chosen gametes or in a random diploid zygote provides information regarding the nonrandom association of alleles among different genetic loci. Two alternative statistics may be employed for detection of nonrandom association of genes of different loci when observations are made on these distributions: observed variance of the number of heterozygous loci (s2k) and a goodness-of-fit criterion (X2) to contrast the observed distribution with that expected under the hypothesis of random association of genes. It is shown, by simulation, that s2k is statistically more efficient than X2 to detect a given extent of nonrandom association. Asymptotic normality of s2k is justified, and X2 is shown to follow a chi-square (chi 2) distribution with partial loss of degrees of freedom arising because of estimation of parameters from the marginal gene frequency data. Whenever direct evaluations of linkage disequilibrium values are possible, tests based on maximum likelihood estimators of linkage disequilibria require a smaller sample size (number of zygotes or gametes) to detect a given level of nonrandom association in comparison with that required if such tests are conducted on the basis of s2k. Summarization of multilocus genotype (or haplotype) data, into the different number of heterozygous loci classes, thus, amounts to appreciable loss of information.

Statistical characterization and development of summary measures of non-normal distributions of nuclear morphometry data from prostate cancer cells for use as prognostic indicators

Nuclear morphometry (NM) uses image analysis to measure features of the cell nucleus which are classified as: bulk properties, shape or form, and DNA distribution. Studies have used these measurements as diagnostic and prognostic indicators of disease with inconclusive results. The distributional properties of these variables have not been systematically investigated although much of the medical data exhibit nonnormal distributions. Measurements are done on several hundred cells per patient so summary measurements reflecting the underlying distribution are needed.^ Distributional characteristics of 34 NM variables from prostate cancer cells were investigated using graphical and analytical techniques. Cells per sample ranged from 52 to 458. A small sample of patients with benign prostatic hyperplasia (BPH), representing non-cancer cells, was used for general comparison with the cancer cells.^ Data transformations such as log, square root and 1/x did not yield normality as measured by the Shapiro-Wilks test for normality. A modulus transformation, used for distributions having abnormal kurtosis values, also did not produce normality.^ Kernel density histograms of the 34 variables exhibited non-normality and 18 variables also exhibited bimodality. A bimodality coefficient was calculated and 3 variables: DNA concentration, shape and elongation, showed the strongest evidence of bimodality and were studied further.^ Two analytical approaches were used to obtain a summary measure for each variable for each patient: cluster analysis to determine significant clusters and a mixture model analysis using a two component model having a Gaussian distribution with equal variances. The mixture component parameters were used to bootstrap the log likelihood ratio to determine the significant number of components, 1 or 2. These summary measures were used as predictors of disease severity in several proportional odds logistic regression models. The disease severity scale had 5 levels and was constructed of 3 components: extracapsulary penetration (ECP), lymph node involvement (LN+) and seminal vesicle involvement (SV+) which represent surrogate measures of prognosis. The summary measures were not strong predictors of disease severity. There was some indication from the mixture model results that there were changes in mean levels and proportions of the components in the lower severity levels. ^