Testing the existence of clustering in the extreme values

This paper introduces an estimator for the extremal index as the ratio of the number of elements of two point processes defined by threshold sequences un, vn and a partition of the sequence in different blocks of the same size. The first point process is defined by the sequence of the block maxima that exceed un. This paper introduces a thinning of this point process, defined by a threshold vn with vn > un, and with the appealing property that under some mild conditions the ratio of the number of elements of both point processes is a consistent estimator of the extremal index. The method supports a hypothesis test for the extremal index, and hence for testing the existence of clustering in the extreme values. Other advantages are that it allows some freedom to choose un, and it is not very sensitive to the choice of the partition. Finally, the stylized facts found in financial returns (clustering, skewness, heavy tails) are tested via the extremal index, in this case for the DaX return

CROSS-LAYER ASPECTS OF COGNITIVE WIRELESS NETWORKS

We study cognitive wireless networks from a cross-layer perspective, where we investigate the effects of the PHY layer parameters and enhancements on the MAC layer performance. We quantify the benefit of using sophisticated techniques such as cooperative communications and network coding in cognitive networks. The first part deals with unicast scenarios. We first study the problem of random access over time varying channels with cognitive nodes adjusting their access probabilities according to the decentralized channel state information they acquire at the PHY layer. We derive the conditions for our random access scheme to outperform orthogonal access. We then study the case where a set of secondary users (SUs) opportunistically accesses the primary user's (PU) spectrum whenever it is idle. Since sensing errors are unavoidable, we study the effect of the interference from the SUs on the stable throughput of the PU. We then compute the range of the SUs' transmission parameters that guarantees the stability of the PU queue. In order to balance the negative effects of the interference from the SUs, we propose a PHY layer relaying protocol between the PU and SU networks that is based on distributed orthogonal space-time block codes. Under this protocol, it is shown that the PU's throughput gain from relaying increases with the number of SUs. Moreover, the SUs might benefit from relaying the PU's packets as well. Next, we propose and analyze access schemes at the SUs aiming at exploiting the SU's knowledge of the statistics of various channels and of the average arrival rate to the PU. The motivation is that although the traditional opportunistic spectrum access (OSA) guarantees full protection to the PUs, it is sometimes too conservative if the interference caused by the SUs at the PU receiver is negligible. We derive the conditions under which schemes without sensing outperform schemes with sensing since they offer to the SU more data transmission duration. The second part of the dissertation deals with cognitive multicasting networks. First, we study relay assisted multicasting. The relay delivers the unsuccessful packets of the source during the idle slots of the source which are determined by sensing. This avoids allocating any explicit resources to the relay. We then substantiate the benefit of using network coding (NC) at the relay. Finally, we study the problem of reliable spectrum sensing and opportunistic access on channels with stochastic traffic in batch processing systems such as NC. We show how an SU can leverage the structure induced by block-based NC on PUs' channels to mitigate the effects of channel sensing errors and improve the throughput. We consider two different objectives at the SU: quickest detection of an idle slot and throughput maximization. We validate our results with real radio measurements taken in software-defined radio based wireless network tests

Extreme value theory in risk management

La intención de esta tesis es conocer más sobre la gestión del riesgo por medio de una metodología muy diferente de las técnicas estadísticas normalmente utilizadas: varianza y correlación. La alternativa utilizada es la teoría de valores extremos, que se presenta como el medio natural para cuantificar el riesgo en enocometría financiera. La tesis se concentra en el riesgo. Hay diferentes interpretaciones de este concepto que dan lugar a diversas metodologías para cuantificar su magnitud e impacto en diferentes características de la econometría financiera. En la introducción de la tesis se discute la distinción entre incertidumbre y riesgo desde diferentes puntos de vista, teoría de la decisión y gestión del riesgo. Se sigue con una definición formal del riesgo motivada por teoría de la decisión pero consistente con la metodología usada en la gestión del riesgo. El riesgo se puede cuantificar por medio de técnicas estadísticas. Se caracteriza por las colas de la distribución de los datos, en particular por la verosimilitud de cualquier suceso que conlleve una característica negativa. En econometría financiera esta definición de riesgo se denomina normalmente “downside risk” y se asocia con la cola izquierda de la distribución de los rendimientos. El objetivo del segundo capítulo es dar medidas adecuadas para cuantificar el riesgo en series financieras. Para conseguir esto, se aplican herramientas derivadas de la teoría de valores extremos. Todas estas medidas del riesgo recientemente consideradas en la literatura basadas en valores extremos se caracterizan en la práctica por métodos de selección ad-hoc de los valores extremos (5 %, 1 %, etc.) La principal contribución en el segundo capítulo es proponer una definición formal para estos valores. Los valores extremos de una muestra aleatoria simple de tamaño n de una distribución F se definen como las observaciones que exceden cierto umbral y siguen una distribución Generalizada de Pareto (GPD) donde el “tail index” de F juega un papel principal. El umbral es el estadístico de orden que minimiza un estadístico del tipo de Kolmogorov-Smirnov entre la distribución empírica de las correspondientes observaciones mayores y la correspondiente GPD. Para formalizar la definición usamos un bootstrap semiparamétrico para contrastar la correspondiente aproximación por la distribución Generalizada de Pareto. Finalmente, usamos nuestra metodología para cuantificar el riesgo estimando el tail index (es decir, el ratio de decaimiento de la cola negativa), y el Valor en Riesgo (VaR) de algunos índices financieros de los principales mercados de acciones. Una vez que el riesgo se define y es formalmente cuantificado el siguiente objetivo de la tesis es analizar los mecanismos de transmisión del riesgo en diferentes marcos. El capítulo III se dedica a la transmisión del riesgo en series temporales. El riesgo se mide por la ocurrencia de observaciones de gran magnitud y el canal de transmisión es la dependencia temporal que se encuentra en los valores extremos y que pueden originar el agrupamiento de estas observaciones. En este contexto existe un parámetro, el “extremal index” que gobierna la dependencia temporal en las observaciones más altas, y tal que su recíproco mide el nivel de agrupamiento (clustering) en los extremos. La contribución de la tesis en este capítulo comienza por redefinir este parámetro. La definición provee un sencillo e inmediato método de estimación para el extremal index con interesantes propiedades estadísticas como son la consistencia y la distribución asintótica gasussiana. La existencia de clustering en las observaciones más grandes es una consecuencia de la transmisión del riesgo derivado de la ocurrencia de sucesos extremos. Una contribución muy importante en esta parte es la posibilidad de contrastar la transmisión del riesgo en series financieras mediante el contraste del clustering en los valores extremos. Esta teoría contrasta con teorías fundadas en modelos para la volatilidad que modelizan la dependencia condicional en los segundos momentos. El siguiente capítulo trata sobre la transmisión del riesgo entre mercados financieros. El interés en esta sección radica en distinguir interdependencia entre mercados, que surge de los lazos normales entre diferentes economías, de los efectos de contagio, originados por unas conexiones que se hacen más fuertes en periodos de crisis. Para hacer esto, las nociones de interdependencia y contagio se revisan. La contribución en este punto se basa en nuevas definiciones para estos conceptos basados en propiedades de las funciones cópula y en monotonicidad en las colas, que se usarán para analizar el contagio direccional (causalidad entre extremos). Esto es posible gracias a una innovadora función cópula que se deriva de la teoría de valores extremos multivariante. Esta cópula nos permite modelizar diferentes patrones de dependencia entre las variables de acuerdo al estado de los mercados, por ejemplo en mercados a la baja o en mercados al alza. Este modelo es suficientemente flexible para describir asimetrías entre las variables de tal manera que el contagio direccional se puede contrastar. El modelo se aplica para medir el fenómeno de vuelo hacia la calidad (flight to quality), es decir, flujos de capital que salen de los mercados de acciones hacia los mercados de bonos cuando los primeros afrontan periodos de crisis. Finalmente el capítulo V esboza las lineas de investigación futuras que implican diferentes aspectos del análisis del riesgo.----------------The intention of this dissertation is to provide some insight about risk management by using a methodology far from the standard statistical techniques: variance and correlation. The alternative is Extreme Value Theory, that is presented as the natural setup to quantify risk in financial econometrics. The thesis concentrates on risk. There are different interpretations of this concept that result in diverse methodologies to quantify its magnitude and impact on different characteristics of financial econometrics. In the introduction of the thesis the distinction between uncertainty and risk is discussed, regarding the point of view: decision theory or risk management. It follows with a formal definition of risk motivated by decision theory but consistent with the methodology used in risk management. Risk can be quantified by means of statistical techniques. Risk is characterized by the tails of the distribution of the data, in particular by the likelihood of any event entailing a negative feature. In financial econometrics this definition of risk is usually denominated downside risk and is associated with the left tail of the distribution of returns. The aim of the second chapter is to provide reliable measures to quantify the risk found in financial sequences. In order to achieve this, standard tools of extreme value theory are applied. All the risk measures recently considered in the literature based on extreme values are characterized in practice by ad-hoc selection methods for the extreme values (5%, 1%, etc.) The main contribution in the second chapter is to propose a formal definition for these values. The extreme values of any random sample of size n from a distribution function F are defined as the observations exceeding a threshold and following a type of generalized Pareto distribution (GPD) involving the tail index of F. The threshold is the order statistic that minimizes a Kolmogorov-Smirnov statistic between the empirical distribution of the corresponding largest observations and the corresponding GPD. To formalize the definition we use a semiparametric bootstrap to test the corresponding GPD approximation. Finally, we use our methodology to quantify risk by estimating the tail index (ratio of decay of the negative tail), and the value at risk (VaR) of some financial indexes of major stock markets. Once risk is defined and formally quantified the following aim in the thesis is analyzing its transmission mechanisms in different settings. Chapter 3 is devoted to the transmission of risk in time series. The risk is measured by the occurrence of significant large observations and the transmission channel is the serial dependence found in the extreme values that can originate clustering of data. In this context there exists a parameter, the extremal index, that governs the serial dependence in the largest observations, and such that its reciprocal measures the level of clustering in the extremes. The contribution of the thesis in this chapter starts by redefining this parameter. This definition provides a straightforward estimation method for the extremal index with appealing statistical properties; consistency, and asymptotic gaussian distribution. The existence of clustering in the largest observations is a byproduct of the transmission of risk derived from the occurrence of the largest observations. An outstanding contribution in this part is the possibility of testing the transmission of risk in financial sequences by testing the clustering in the extreme values. This theory contrasts with theories founded on volatility models that claim that serial dependence found in financial series is due to the conditional dependence on second moments. The next chapter involves the transmission of risk between financial markets. The interest lies in this section on distinguishing interdependence between markets, that surges from regular links between economies, from contagion effects, originated by increasing links between the markets in crises periods. In order to do this, the notions of interdependence and contagion are revisited. The contribution of the authors lies on new definitions for these concepts based on copula properties and tail monotonicity, that will be used to analyze directional contagion (causality between extremes). This is possible due to an innovative copula function that is derived from the multivariate extreme value theory. This copula allows us to model different patterns of dependence according to the state of the markets, e.g. bear or bull markets. This model is sufficiently flexible to describe asymmetries between variables in such a way that directional contagion can be tested. The model is applied to the flight to quality phenomenon, outflows of capital from the stocks markets to the bonds markets when the first ones are facing crisis periods. Finally Chapter 5 sketches future lines of research involving different aspects of the analysis of risk