Uniqueness of the Embedding Continuous Convolution Semigroup of a Gaussian Probability Measure on the Affine Group and an Application in Mathematical Finance

Let {μ(i)t}t≥0 ( i=1,2 ) be continuous convolution semigroups (c.c.s.) of probability measures on Aff(1) (the affine group on the real line). Suppose that μ(1)1=μ(2)1 . Assume furthermore that {μ(1)t}t≥0 is a Gaussian c.c.s. (in the sense that its generating distribution is a sum of a primitive distribution and a second-order differential operator). Then μ(1)t=μ(2)t for all t≥0 . We end up with a possible application in mathematical finance.

Bounding Standard Gaussian Tail Probabilities

We review various inequalities for Mills' ratio (1 - Φ)= Ø, where Ø and Φ denote the standard Gaussian density and distribution function, respectively. Elementary considerations involving finite continued fractions lead to a general approximation scheme which implies and refines several known bounds.

Left, right and interval bivariate censored data: Evaluating screening mammography in the presence of lead -time bias, length bias and over-detection

Of the large clinical trials evaluating screening mammography efficacy, none included women ages 75 and older. Recommendations on an upper age limit at which to discontinue screening are based on indirect evidence and are not consistent. Screening mammography is evaluated using observational data from the SEER-Medicare linked database. Measuring the benefit of screening mammography is difficult due to the impact of lead-time bias, length bias and over-detection. The underlying conceptual model divides the disease into two stages: pre-clinical (T0) and symptomatic (T1) breast cancer. Treating the time in these phases as a pair of dependent bivariate observations, (t0,t1), estimates are derived to describe the distribution of this random vector. To quantify the effect of screening mammography, statistical inference is made about the mammography parameters that correspond to the marginal distribution of the symptomatic phase duration (T1). This shows the hazard ratio of death from breast cancer comparing women with screen-detected tumors to those detected at their symptom onset is 0.36 (0.30, 0.42), indicating a benefit among the screen-detected cases. ^

Univariate and bivariate truncated von Mises distributions

In this article we study the univariate and bivariate truncated von Mises distribution, as a generalization of the von Mises distribution (\cite{jupp1989}), (\cite{mardia2000directional}). This implies the addition of two or four new truncation parameters in the univariate and, bivariate cases, respectively. The results include the definition, properties of the distribution and maximum likelihood estimators for the univariate and bivariate cases. Additionally, the analysis of the bivariate case shows how the conditional distribution is a truncated von Mises distribution, whereas the marginal distribution that generalizes the distribution introduced in \cite{repe}. From the viewpoint of applications, we test the distribution with simulated data, as well as with data regarding leaf inclination angles (\cite{safari}) and dihedral angles in protein chains (\cite{prote}). This research aims to assert this probability distribution as a potential option for modelling or simulating any kind of phenomena where circular distributions are applicable.\par

Flood frequency analysis by a bivariate model based on copulas

La adecuada estimación de avenidas de diseño asociadas a altos periodos de retorno es necesaria para el diseño y gestión de estructuras hidráulicas como presas. En la práctica, la estimación de estos cuantiles se realiza normalmente a través de análisis de frecuencia univariados, basados en su mayoría en el estudio de caudales punta. Sin embargo, la naturaleza de las avenidas es multivariada, siendo esencial tener en cuenta características representativas de las avenidas, tales como caudal punta, volumen y duración del hidrograma, con el fin de llevar a cabo un análisis apropiado; especialmente cuando el caudal de entrada se transforma en un caudal de salida diferente durante el proceso de laminación en un embalse o llanura de inundación. Los análisis de frecuencia de avenidas multivariados han sido tradicionalmente llevados a cabo mediante el uso de distribuciones bivariadas estándar con el fin de modelar variables correlacionadas. Sin embargo, su uso conlleva limitaciones como la necesidad de usar el mismo tipo de distribuciones marginales para todas las variables y la existencia de una relación de dependencia lineal entre ellas. Recientemente, el uso de cópulas se ha extendido en hidrología debido a sus beneficios en relación al contexto multivariado, permitiendo superar los inconvenientes de las técnicas tradicionales. Una copula es una función que representa la estructura de dependencia de las variables de estudio, y permite obtener la distribución de frecuencia multivariada de dichas variables mediante sus distribuciones marginales, sin importar el tipo de distribución marginal utilizada. La estimación de periodos de retorno multivariados, y por lo tanto, de cuantiles multivariados, también se facilita debido a la manera en la que las cópulas están formuladas. La presente tesis doctoral busca proporcionar metodologías que mejoren las técnicas tradicionales usadas por profesionales para estimar cuantiles de avenida más adecuados para el diseño y la gestión de presas, así como para la evaluación del riesgo de avenida, mediante análisis de frecuencia de avenidas bivariados basados en cópulas. Las variables consideradas para ello son el caudal punta y el volumen del hidrograma. Con el objetivo de llevar a cabo un estudio completo, la presente investigación abarca: (i) el análisis de frecuencia de avenidas local bivariado centrado en examinar y comparar los periodos de retorno teóricos basados en la probabilidad natural de ocurrencia de una avenida, con el periodo de retorno asociado al riesgo de sobrevertido de la presa bajo análisis, con el fin de proporcionar cuantiles en una estación de aforo determinada; (ii) la extensión del enfoque local al regional, proporcionando un procedimiento completo para llevar a cabo un análisis de frecuencia de avenidas regional bivariado para proporcionar cuantiles en estaciones sin aforar o para mejorar la estimación de dichos cuantiles en estaciones aforadas; (iii) el uso de cópulas para investigar tendencias bivariadas en avenidas debido al aumento de los niveles de urbanización en una cuenca; y (iv) la extensión de series de avenida observadas mediante la combinación de los beneficios de un modelo basado en cópulas y de un modelo hidrometeorológico. Accurate design flood estimates associated with high return periods are necessary to design and manage hydraulic structures such as dams. In practice, the estimate of such quantiles is usually done via univariate flood frequency analyses, mostly based on the study of peak flows. Nevertheless, the nature of floods is multivariate, being essential to consider representative flood characteristics, such as flood peak, hydrograph volume and hydrograph duration to carry out an appropriate analysis; especially when the inflow peak is transformed into a different outflow peak during the routing process in a reservoir or floodplain. Multivariate flood frequency analyses have been traditionally performed by using standard bivariate distributions to model correlated variables, yet they entail some shortcomings such as the need of using the same kind of marginal distribution for all variables and the assumption of a linear dependence relation between them. Recently, the use of copulas has been extended in hydrology because of their benefits regarding dealing with the multivariate context, as they overcome the drawbacks of the traditional approach. A copula is a function that represents the dependence structure of the studied variables, and allows obtaining the multivariate frequency distribution of them by using their marginal distributions, regardless of the kind of marginal distributions considered. The estimate of multivariate return periods, and therefore multivariate quantiles, is also facilitated by the way in which copulas are formulated. The present doctoral thesis seeks to provide methodologies that improve traditional techniques used by practitioners, in order to estimate more appropriate flood quantiles for dam design, dam management and flood risk assessment, through bivariate flood frequency analyses based on the copula approach. The flood variables considered for that goal are peak flow and hydrograph volume. In order to accomplish a complete study, the present research addresses: (i) a bivariate local flood frequency analysis focused on examining and comparing theoretical return periods based on the natural probability of occurrence of a flood, with the return period associated with the risk of dam overtopping, to estimate quantiles at a given gauged site; (ii) the extension of the local to the regional approach, supplying a complete procedure for performing a bivariate regional flood frequency analysis to either estimate quantiles at ungauged sites or improve at-site estimates at gauged sites; (iii) the use of copulas to investigate bivariate flood trends due to increasing urbanisation levels in a catchment; and (iv) the extension of observed flood series by combining the benefits of a copula-based model and a hydro-meteorological model.

The asymptotic distribution of canonical correlations and vectors in higher-order cointegrated models

The study of the large-sample distribution of the canonical correlations and variates in cointegrated models is extended from the first-order autoregression model to autoregression of any (finite) order. The cointegrated process considered here is nonstationary in some dimensions and stationary in some other directions, but the first difference (the “error-correction form”) is stationary. The asymptotic distribution of the canonical correlations between the first differences and the predictor variables as well as the corresponding canonical variables is obtained under the assumption that the process is Gaussian. The method of analysis is similar to that used for the first-order process.

The distribution of shortrun commodity price movements /

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No-switching quantum key distribution using broadband modulated coherent light

We realize an end-to-end no-switching quantum key distribution protocol using continuous-wave coherent light. We encode weak broadband Gaussian modulations onto the amplitude and phase quadratures of light beams. Our no-switching protocol achieves high secret key rate via a post-selection protocol that utilizes both quadrature information simultaneously. We establish a secret key rate of 25 Mbits/s for a lossless channel and 1 kbit/s for 90% channel loss, per 17 MHz of detected bandwidth, assuming individual Gaussian eavesdropping attacks. Since our scheme is truly broadband, it can potentially deliver orders of magnitude higher key rates by extending the encoding bandwidth with higher-end telecommunication technology.

Coherent-state quantum key distribution without random basis switching

The random switching of measurement bases is commonly assumed to be a necessary step of quantum key distribution protocols. In this paper we present a no-switching protocol and show that switching is not required for coherent-state continuous-variable quantum key distribution. Further, this protocol achieves higher information rates and a simpler experimental setup compared to previous protocols that rely on switching. We propose an optimal eavesdropping attack against this protocol, assuming individual Gaussian attacks. Finally, we investigate and compare the no-switching protocol applied to the original Bennett-Brassard 1984 scheme.

Gaussian operator bases for correlated fermions

We formulate a general multi-mode Gaussian operator basis for fermions, to enable a positive phase-space representation of correlated Fermi states. The Gaussian basis extends existing bosonic phase-space methods to Fermi systems and thus allows first-principles dynamical or equilibrium calculations in quantum many-body Fermi systems. We prove the completeness of the basis and derive differential forms for products with one- and two-body operators. Because the basis satisfies fermionic superselection rules, the resulting phase space involves only c-numbers, without requiring anticommuting Grassmann variables. Furthermore, because of the overcompleteness of the basis, the phase-space distribution can always be chosen positive. This has important consequences for the sign problem in fermion physics.

Gaussian processes for regression

The Bayesian analysis of neural networks is difficult because a simple prior over weights implies a complex prior distribution over functions. In this paper we investigate the use of Gaussian process priors over functions, which permit the predictive Bayesian analysis for fixed values of hyperparameters to be carried out exactly using matrix operations. Two methods, using optimization and averaging (via Hybrid Monte Carlo) over hyperparameters have been tested on a number of challenging problems and have produced excellent results.

General gaussian priors for improved generalisation

We explore the dependence of performance measures, such as the generalization error and generalization consistency, on the structure and the parameterization of the prior on `rules', instanced here by the noisy linear perceptron. Using a statistical mechanics framework, we show how one may assign values to the parameters of a model for a `rule' on the basis of data instancing the rule. Information about the data, such as input distribution, noise distribution and other `rule' characteristics may be embedded in the form of general gaussian priors for improving net performance. We examine explicitly two types of general gaussian priors which are useful in some simple cases. We calculate the optimal values for the parameters of these priors and show their effect in modifying the most probable, MAP, values for the rules.

Regression with input-dependent noise: A Gaussian process treatment

Gaussian processes provide natural non-parametric prior distributions over regression functions. In this paper we consider regression problems where there is noise on the output, and the variance of the noise depends on the inputs. If we assume that the noise is a smooth function of the inputs, then it is natural to model the noise variance using a second Gaussian process, in addition to the Gaussian process governing the noise-free output value. We show that prior uncertainty about the parameters controlling both processes can be handled and that the posterior distribution of the noise rate can be sampled from using Markov chain Monte Carlo methods. Our results on a synthetic data set give a posterior noise variance that well-approximates the true variance.

General bounds on Bayes errors for regression with Gaussian processes

Based on a simple convexity lemma, we develop bounds for different types of Bayesian prediction errors for regression with Gaussian processes. The basic bounds are formulated for a fixed training set. Simpler expressions are obtained for sampling from an input distribution which equals the weight function of the covariance kernel, yielding asymptotically tight results. The results are compared with numerical experiments.