The transform likelihood ratio method for rare event simulation with heavy tails

We present a novel method, called the transform likelihood ratio (TLR) method, for estimation of rare event probabilities with heavy-tailed distributions. Via a simple transformation ( change of variables) technique the TLR method reduces the original rare event probability estimation with heavy tail distributions to an equivalent one with light tail distributions. Once this transformation has been established we estimate the rare event probability via importance sampling, using the classical exponential change of measure or the standard likelihood ratio change of measure. In the latter case the importance sampling distribution is chosen from the same parametric family as the transformed distribution. We estimate the optimal parameter vector of the importance sampling distribution using the cross-entropy method. We prove the polynomial complexity of the TLR method for certain heavy-tailed models and demonstrate numerically its high efficiency for various heavy-tailed models previously thought to be intractable. We also show that the TLR method can be viewed as a universal tool in the sense that not only it provides a unified view for heavy-tailed simulation but also can be efficiently used in simulation with light-tailed distributions. We present extensive simulation results which support the efficiency of the TLR method.

Improved algorithms for rare event simulation with heavy tails

The estimation of P(S-n > u) by simulation, where S, is the sum of independent. identically distributed random varibles Y-1,..., Y-n, is of importance in many applications. We propose two simulation estimators based upon the identity P(S-n > u) = nP(S, > u, M-n = Y-n), where M-n = max(Y-1,..., Y-n). One estimator uses importance sampling (for Y-n only), and the other uses conditional Monte Carlo conditioning upon Y1,..., Yn-1. Properties of the relative error of the estimators are derived and a numerical study given in terms of the M/G/1 queue in which n is replaced by an independent geometric random variable N. The conclusion is that the new estimators compare extremely favorably with previous ones. In particular, the conditional Monte Carlo estimator is the first heavy-tailed example of an estimator with bounded relative error. Further improvements are obtained in the random-N case, by incorporating control variates and stratification techniques into the new estimation procedures.

Evaluating the Survey of Professional Forecasters probability distributions of expected inflation based on derived event probability forecasts

Techniques are proposed for evaluating forecast probabilities of events. The tools are especially useful when, as in the case of the Survey of Professional Forecasters (SPF) expected probability distributions of inflation, recourse cannot be made to the method of construction in the evaluation of the forecasts. The tests of efficiency and conditional efficiency are applied to the forecast probabilities of events of interest derived from the SPF distributions, and supplement a whole-density evaluation of the SPF distributions based on the probability integral transform approach.

Evaluating Risks of Dam-Reservoir Systems Using Efficient Importance Sampling

The occurrence frequency of failure events serve as critical indexes representing the safety status of dam-reservoir systems. Although overtopping is the most common failure mode with significant consequences, this type of event, in most cases, has a small probability. Estimation of such rare event risks for dam-reservoir systems with crude Monte Carlo (CMC) simulation techniques requires a prohibitively large number of trials, where significant computational resources are required to reach the satisfied estimation results. Otherwise, estimation of the disturbances would not be accurate enough. In order to reduce the computation expenses and improve the risk estimation efficiency, an importance sampling (IS) based simulation approach is proposed in this dissertation to address the overtopping risks of dam-reservoir systems. Deliverables of this study mainly include the following five aspects: 1) the reservoir inflow hydrograph model; 2) the dam-reservoir system operation model; 3) the CMC simulation framework; 4) the IS-based Monte Carlo (ISMC) simulation framework; and 5) the overtopping risk estimation comparison of both CMC and ISMC simulation. In a broader sense, this study meets the following three expectations: 1) to address the natural stochastic characteristics of the dam-reservoir system, such as the reservoir inflow rate; 2) to build up the fundamental CMC and ISMC simulation frameworks of the dam-reservoir system in order to estimate the overtopping risks; and 3) to compare the simulation results and the computational performance in order to demonstrate the ISMC simulation advantages. The estimation results of overtopping probability could be used to guide the future dam safety investigations and studies, and to supplement the conventional analyses in decision making on the dam-reservoir system improvements. At the same time, the proposed methodology of ISMC simulation is reasonably robust and proved to improve the overtopping risk estimation. The more accurate estimation, the smaller variance, and the reduced CPU time, expand the application of Monte Carlo (MC) technique on evaluating rare event risks for infrastructures.

Stochastic simulation of rare events

Stochastic simulation is an important and practical technique for computing probabilities of rare events, like the payoff probability of a financial option, the probability that a queue exceeds a certain level or the probability of ruin of the insurer's risk process. Rare events occur so infrequently, that they cannot be reasonably recorded during a standard simulation procedure: specifc simulation algorithms which thwart the rarity of the event to simulate are required. An important algorithm in this context is based on changing the sampling distribution and it is called importance sampling. Optimal Monte Carlo algorithms for computing rare event probabilities are either logarithmic eficient or possess bounded relative error.

Mutations in exon 3 of the beta-catenin gene are rare in melanoma cell lines

Mutations in exon 3 of the CTNNB1 gene encoding beta-catenin have been reported in colorectal cancer cell lines and tumours. Although one study reported mutations or deletions affecting beta-catenin in 20% of melanoma cell lines, subsequent reports detected a much lower frequency of aberrations in uncultured melanomas. To determine whether this difference in mutation frequency reflected an in vitro culturing artefact, exon 3 of CTNNB1 was screened in a panel of 62 melanoma cell lines. In addition, reverse transcription-polymerase chain reaction (RT-PCR) was performed to detect intragenic deletions affecting exon 3. One out of 62 (1.6%) cell lines was found to carry a mutation, indicating that aberration of the Wnt-1/wingless pathway through activation of beta-catenin is a rare event, even in melanoma cell lines.

Detecting rare events using Kullback-Leibler divergence

One main challenge in developing a system for visual surveillance event detection is the annotation of target events in the training data. By making use of the assumption that events with security interest are often rare compared to regular behaviours, this paper presents a novel approach by using Kullback-Leibler (KL) divergence for rare event detection in a weakly supervised learning setting, where only clip-level annotation is available. It will be shown that this approach outperforms state-of-the-art methods on a popular real-world dataset, while preserving real time performance.

Report on Marine Mammal Unusual Mortality Event UMESE0501Sp: Multispecies Mass Stranding of Pilot Whales (Globicephala macrorhynchus), Minke Whale (Balaenoptera acutorostrata), and Dwarf Sperm Whales (Kogia sima) in North Carolina on 15-16 January 2005

On 15-16 January 2005, three offshore species of cetaceans (33 short-finned pilot whales, Globicephala macrorhynchus, one minke whale, Balaenoptera acutorostrata, and two dwarf sperm whales, Kogia sima) stranded alive on the beaches of North Carolina. The pilot whales stranded near Oregon Inlet, the minke whale in northern North Carolina, and the dwarf sperm whales near Cape Hatteras. Live strandings of three species in one weekend was unique in North Carolina and qualified as an Unusual Mortality Event. Gross necropsies were conducted on 16-17 January 2005 on 27 pilot whales, two dwarf sperm whales, and the minke whale. Samples were collected for clinical pathology, parasitology, gross pathology, histopathology, microbiology and serology. There was variation in the number of animals sampled for each collection type, however, due to carcasses washing off the beach or degradation in carcass condition during the course of the response. Comprehensive histologic examination was conducted on 16 pilot whales, both dwarf sperm whales, and the minke whale. Limited organ or only head tissue suites were obtained from nine pilot whales. Histologic examination of tissues began in February 2005 and concluded in December 2005 when final sampling was concluded. Neither the pilot whales nor dwarf sperm whales were emaciated although none had recently ingested prey in their stomachs. The minke whale was emaciated; it was likely a dependent calf that became separated from the female. Most serum biochemistry abnormalities appear to have resulted from the stranding and indicated deteriorating condition from being on land for an extended period. Three pilot whales had clinical evidence of pre-existing systemic inflammation, which was supported by histopathologic findings. Although gross and histologic lesions involving all organ systems were noted, consistent lesions were not observed across species. Verminous pterygoid sinusitis and healed fishery interactions were seen in pilot whales but neither of these changes were causes of debilitation or death. In three pilot whales and one dwarf sperm whale there was evidence of clinically significant disease in postcranial tissues which led to chronic debilitation. Cardiovascular disease was present in one pilot whale and one dwarf sperm whale; musculoskeletal disease and intra-abdominal granulomas were present in two pilot whales. These lesions were possible, but not definitive, causal factors in the stranding. Remaining lesions were incidental or post-stranding. The minke whale and three of five tested pilot whales had positive morbillivirus titers (≥1:8 with one at >1:256), but there was no histologic evidence of active viral infection. Parasites (nematodes, cestodes, and trematodes) were collected from 26 pilot whales and two dwarf sperm whales. Sites of collection included stomach, nasal/pterygoid, peribullar sinuses, blubber, and abdominal cavity. Parasite species, locations and loads were within normal limits for free-ranging cetaceans and were not considered causative for the stranding event. Gas emboli lesions which were considered consistent with or diagnostic of sonarassociated strandings of beaked whales or small cetaceans were not found in the whales stranded as part of UMESE0501Sp. Twenty-five heads were examined with nine specific anatomic locations of interest: extramandibular fat, intramandibular fat, auditory meatus, peribullar acoustic fat, peribullar soft tissue, peribullar sinus, pterygoid sinus, melon, and brain. The common finding in all examined heads was verminous pterygoid sinusitis. Intramandibular adipose tissue reddening, typically adjacent to the vascular plexus, was observed in some individuals and could represent localized hemorrhage resulting from vascular rete rupture, hypostatic congestion, or erythrocyte rupture during the freeze/thaw cycle. One cetacean had peracute to acute subdural hemorrhage that likely occurred from thrashing on the beach post-stranding, although its occurrence prior to stranding cannot be excluded. Information provided to NMFS by the U.S. Navy indicated routine tactical mid-frequency sonar operations from individual surface vessels over relatively short durations and small spatial scales within the area and time period investigated. No marine mammals were detected by marine mammal observers on operational vessels; standard operating procedure for surface naval vessels operating mid-frequency sonar is the use of trained visual lookouts using high-powered binoculars. Sound propagation modeling using information provided to NMFS indicated that acoustic conditions in the vicinity likely depended heavily on position of the receivers (e.g., range, bearing, depth) relative to that of the sources. Absent explicit information on the location of animals meant that it was not possible to estimate received acoustic exposures from active sonar transmissions. Nonetheless, the event was associated in time and space with naval activity using mid-frequency active sonar. It also had a number of features in common (e.g., the “atypical” distribution of strandings involving multiple offshore species, all stranding alive, and without evidence of common infectious or other disease process) with other sonar-related cetacean mass stranding events. Given that this event was the only stranding of offshore species to occur within a 2-3 day period in the region on record (i.e., a very rare event), and given the occurrence of the event simultaneously in time and space with a naval exercise using active sonar, the association between the naval sonar activity and the location and timing of the event could be a causal rather than a coincidental relationship. However, evidence supporting a definitive association is lacking, and, in particular, there are differences in operational/environmental characteristics between this event and previous events where sonar has apparently played a role in marine mammal strandings. This does not preclude behavorial avoidance of noise exposure. No harmful algal blooms were present along the Atlantic coast south of the Chesapeake Bay during the months prior to the event. Environmental conditions, including strong winds, changes in upwelling- to downwelling-favorable conditions, and gently sloping bathymetry, were consistent with conditions which have been correlated with other mass strandings. In summary, we did not find commonality in gross and histologic lesions that would indicate a single cause for this stranding event. Three pilot whales and one dwarf sperm whale had debilitating conditions identified that could have contributed to stranding, one pilot whale had a debilitating condition (subdural hemorrhage) that could have been present prior to or resulting from stranding. While the pilot and dwarf sperm whale strandings may have had a common cause, the minke whale stranding was probably just coincidental. On the basis of examination of physical evidence in the affected whales, however, we cannot definitively conclude that there was or was not a causal link between anthropogenic sonar activity or environmental conditions (or a combination of these factors) and the strandings. Overall, the cause of UMESE0501Sp in North Carolina is not and likely will not be definitively known. (PDF contains 240 pages)

Flowering in British Lemna: A rare, cyclic or simply overlooked phenomenon?

It is largely presumed that reproduction in British Lemna, as in other British Lemnaceae, is almost entirely asexual, with new daughter fronds being produced from the side pouches of older mother fronds. Sexual reproduction is considered to be a rather rare event or even absent and because of this rarity the sexual features of Lemna, such as anthers and fruit, are often considered to be of little taxonomic value. It was with some surprise, therefore, that widespread flowering was observed in all British Lemna during the summer of 1995. Initial observations in Shropshire during June recorded flowers in minor and trisulca, with fruit production in trisulca. L.gibba, minor and minuta were noted as being in flower on several occasions in Kent, during July and August, probably fruit production occurring in both species. To what extent these events are truly representative of the sexual reproduction rate of British Lemna on a year-to-year basis, or simply reflect the unusually high summer temperatures of 1995, is unclear.

Tail risk in the hedge fund industry

The dissertation goal is to quantify the tail risk premium embedded into hedge funds' returns. Tail risk is the probability of extreme large losses. Although it is a rare event, asset pricing theory suggests that investors demand compensation for holding assets sensitive to extreme market downturns. By de nition, such events have a small likelihood to be represented in the sample, what poses a challenge to estimate the e ects of tail risk by means of traditional approaches such as VaR. The results show that it is not su cient to account for the tail risk stemming from equities markets. Active portfolio management employed by hedge funds demand a speci c measure to estimate and control tail risk. Our proposed factor lls that void inasmuch it presents explanatory power both over the time series as well as the cross-section of funds' returns.

A rare non-Robertsonian translocation involving chromosomes 15 and 21

CONTEXTO:Translocações robertsonianas (TR) estão entre os rearranjos estruturais balanceados mais comuns em humanos e compreendem a fusão da cromatina completa do braço longo de dois cromossomos acrocêntricos. No entanto, são raras as translocações não Robertsonianas envolvendo esses cromossomos.RELATO DE CASO:Nós descrevemos uma translocação não balanceada de novo envolvendo os cromossomos 15 e 21. A recém-nascida era filha de uma mãe de 29 anos e de um pai de 42 anos, casal não consanguíneo. Os achados clínicos levaram ao diagnóstico de síndrome de Down (SD) com defeitos cardíacos congênitos graves (persistência do canal arterial e defeito do septo atrioventricular completo), além de baixos comprimento e peso ao nascimento (< 5o e < 10o percentil em curvas de medidas específicas para SD, respectivamente). A análise citogenética convencional revelou o cariótipo 46,XX,der(15)(15pter→15q26.2

Somatic CEBPA mutations are a frequent second event in families with germline CEBPA mutations and familial acute myeloid leukemia

PURPOSE: The transcription factor CCAAT/enhancer binding protein-alpha (CEBPA) is crucial for normal myeloid differentiation. Mutations in the CEBPA gene are found in subsets of patients with acute myeloid leukemia (AML). Recently, three families were reported in whom several family members had germline CEBPA mutations and subsequently developed AML. Whereas familial AML is considered a rare event, the frequency of CEBPA germline mutations in AML is not known. PATIENTS AND METHODS: In this study, we screened 187 consecutive AML patients for CEBPA mutations at diagnosis. We detected 18 patients (9.6%) with CEBPA mutations. We then analyzed remission samples and constitutive DNA from these patients. RESULTS: We found that two (11.1%) of 18 AML patients with CEBPA mutations carried a germline N-terminal frameshift CEBPA mutation. Interestingly, additional members in the families of both of these patients have been affected by AML, and the germline CEBPA mutations were also observed in these patients. Additional somatic mutations in AML patients with germline CEBPA mutations in the two families comprised in-frame C-terminal CEBPA mutations in two patients, two nonsilent CEBPA point mutations in one patient, and monosomy 7 in one patient. CONCLUSION: This study shows, for the first time to our knowledge, that germline CEBPA mutations are frequently observed among AML patients with CEBPA mutations. Including the families with germline CEBPA mutations reported previously, additional somatic CEBPA mutations represent a frequent second event in AML with germline CEBPA mutations. Our data strongly indicate that germline CEBPA mutations predispose to AML and that additional somatic CEBPA mutations contribute to the development of the disease.

Metodología de análisis estadístico de roturas en redes de distribución de agua

La presente Tesis plantea una metodología de análisis estadístico de roturas de tubería en redes de distribución de agua, que analiza la relación entre las roturas y la presión de agua y que propone la implantación de una gestión de presiones que reduzca el número de roturas que se producen en dichas redes. Las redes de distribución de agua se deterioran y una de sus graves consecuencias es la aparición de roturas frecuentes en sus tuberías. Las roturas llevan asociados elevados costes sociales, económicos y medioambientales y es por ello por lo que las compañías gestoras del agua tratan de reducirlas en la medida de lo posible. Las redes de distribución de agua se pueden dividir en zonas o sectores que facilitan su control y que pueden ser independientes o aislarse mediante válvulas, como ocurre en las redes de países más desarrollados, o pueden estar intercomunicados hidráulicamente. La implantación de una gestión de presiones suele llevarse a cabo a través de las válvulas reductoras de presión (VPR), que se instalan en las cabeceras de estos sectores y que controlan la presión aguas abajo de la misma, aunque varíe su caudal de entrada. Los métodos más conocidos de la gestión de presiones son la reducción de presiones, que es el control más habitual, el mantenimiento de la presión, la prevención y/o alivio de los aumentos repentinos de presión y el establecimiento de un control por alturas. A partir del año 2005 se empezó a reconocer el efecto de la gestión de presiones sobre la disminución de las roturas. En esta Tesis, se sugiere una gestión de presiones que controle los rangos de los indicadores de la presión de cabecera que más influyan en la probabilidad de roturas de tubería. Así, la presión del agua se caracteriza a través de indicadores obtenidos de la presión registrada en la cabecera de los sectores, debido a que se asume que esta presión es representativa de la presión de operación de todas las tuberías porque las pérdidas de carga son relativamente bajas y las diferencias topográficas se tienen en cuenta en el diseño de los sectores. Y los indicadores de presión, que se pueden definir como el estadístico calculado a partir de las series de la presión de cabecera sobre una ventana de tiempo, pueden proveer la información necesaria para ayudar a la toma de decisiones a los gestores del agua con el fin de reducir las roturas de tubería en las redes de distribución de agua. La primera parte de la metodología que se propone en esta Tesis trata de encontrar los indicadores de presión que influyen más en la probabilidad de roturas de tuberías. Para conocer si un indicador es influyente en la probabilidad de las roturas se comparan las estimaciones de las funciones de distribución acumulada (FDAs) de los indicadores de presiones, considerando dos situaciones: cuando se condicionan a la ocurrencia de una rotura (suceso raro) y cuando se calculan en la situación normal de operación (normal operación). Por lo general, las compañías gestoras cuentan con registros de roturas de los años más recientes y al encontrarse las tuberías enterradas se complica el acceso a la información. Por ello, se propone el uso de funciones de probabilidad que permiten reducir la incertidumbre asociada a los datos registrados. De esta forma, se determinan las funciones de distribución acumuladas (FDAs) de los valores del indicador de la serie de presión (situación normal de operación) y las FDAs de los valores del indicador en el momento de ocurrencia de las roturas (condicionado a las roturas). Si las funciones de distribución provienen de la misma población, no se puede deducir que el indicador claramente influya en la probabilidad de roturas. Sin embargo, si se prueba estadísticamente que las funciones proceden de la misma población, se puede concluir que existe una relación entre el indicador analizado y la ocurrencia de las roturas. Debido a que el número de valores del indicador de la FDA condicionada a las roturas es mucho menor que el número de valores del indicador de la FDA incondicional a las roturas, se generan series aleatorias a partir de los valores de los indicadores con el mismo número de valores que roturas registradas hay. De esta forma, se comparan las FDAs de series aleatorias del indicador con la FDA condicionada a las roturas del mismo indicador y se deduce si el indicador es influyente en la probabilidad de las roturas. Los indicadores de presión pueden depender de unos parámetros. A través de un análisis de sensibilidad y aplicando un test estadístico robusto se determina la situación en la que estos parámetros dan lugar a que el indicador sea más influyente en la probabilidad de las roturas. Al mismo tiempo, los indicadores se pueden calcular en función de dos parámetros de cálculo que se denominan el tiempo de anticipación y el ancho de ventana. El tiempo de anticipación es el tiempo (en horas) entre el final del periodo de computación del indicador de presión y la rotura, y el ancho de ventana es el número de valores de presión que se requieren para calcular el indicador de presión y que es múltiplo de 24 horas debido al comportamiento cíclico diario de la presión. Un análisis de sensibilidad de los parámetros de cálculo explica cuándo los indicadores de presión influyen más en la probabilidad de roturas. En la segunda parte de la metodología se presenta un modelo de diagnóstico bayesiano. Este tipo de modelo forma parte de los modelos estadísticos de prevención de roturas, parten de los datos registrados para establecer patrones de fallo y utilizan el teorema de Bayes para determinar la probabilidad de fallo cuando se condiciona la red a unas determinadas características. Así, a través del teorema de Bayes se comparan la FDA genérica del indicador con la FDA condicionada a las roturas y se determina cuándo la probabilidad de roturas aumenta para ciertos rangos del indicador que se ha inferido como influyente en las roturas. Se determina un ratio de probabilidad (RP) que cuando es superior a la unidad permite distinguir cuándo la probabilidad de roturas incrementa para determinados intervalos del indicador. La primera parte de la metodología se aplica a la red de distribución de la Comunidad de Madrid (España) y a la red de distribución de Ciudad de Panamá (Panamá). Tras el filtrado de datos se deduce que se puede aplicar la metodología en 15 sectores en la Comunidad de Madrid y en dos sectores, llamados corregimientos, en Ciudad de Panamá. Los resultados demuestran que en las dos redes los indicadores más influyentes en la probabilidad de las roturas son el rango de la presión, que supone la diferencia entre la presión máxima y la presión mínima, y la variabilidad de la presión, que considera la propiedad estadística de la desviación típica. Se trata, por tanto, de indicadores que hacen referencia a la dispersión de los datos, a la persistencia de la variación de la presión y que se puede asimilar en resistencia de materiales a la fatiga. La segunda parte de la metodología se ha aplicado a los indicadores influyentes en la probabilidad de las roturas de la Comunidad de Madrid y se ha deducido que la probabilidad de roturas aumenta para valores extremos del indicador del rango de la presión y del indicador de la variabilidad de la presión. Finalmente, se recomienda una gestión de presiones que limite los intervalos de los indicadores influyentes en la probabilidad de roturas que incrementen dicha probabilidad. La metodología propuesta puede aplicarse a otras redes de distribución y puede ayudar a las compañías gestoras a reducir el número de fallos en el sistema a través de la gestión de presiones. This Thesis presents a methodology for the statistical analysis of pipe breaks in water distribution networks. The methodology studies the relationship between pipe breaks and water pressure, and proposes a pressure management procedure to reduce the number of breaks that occur in such networks. One of the manifestations of the deterioration of water supply systems is frequent pipe breaks. System failures are one of the major challenges faced by water utilities, due to their associated social, economic and environmental costs. For all these reasons, water utilities aim at reducing the problem of break occurrence to as great an extent as possible. Water distribution networks can be divided into areas or sectors, which facilitates the control of the network. These areas may be independent or isolated by valves, as it usually happens in developing countries. Alternatively, they can be hydraulically interconnected. The implementation of pressure management strategies is usually carried out through pressure-reducing valves (PRV). These valves are installed at the head of the sectors and, although the inflow may vary significantly, they control the downstream pressure. The most popular methods of pressure management consist of pressure reduction, which is the common form of control, pressure sustaining, prevention and/or alleviation of pressure surges or large variations in pressure, and level/altitude control. From 2005 onwards, the effects of pressure management on burst frequencies have become more widely recognized in the technical literature. This thesis suggests a pressure management that controls the pressure indicator ranges most influential on the probability of pipe breaks. Operating pressure in a sector is characterized by means of a pressure indicator at the head of the DMA, as head losses are relatively small and topographical differences were accounted for at the design stage. The pressure indicator, which may be defined as the calculated statistic from the time series of pressure head over a specific time window, may provide necessary information to help water utilities to make decisions to reduce pipe breaks in water distribution networks. The first part of the methodology presented in this Thesis provides the pressure indicators which have the greatest impact on the probability of pipe breaks to be determined. In order to know whether a pressure indicator influences the probability of pipe breaks, the proposed methodology compares estimates of cumulative distribution functions (CDFs) of a pressure indicator through consideration of two situations: when they are conditioned to the occurrence of a pipe break (a rare event), and when they are not (a normal operation). Water utilities usually have a history of failures limited to recent periods of time, and it is difficult to have access to precise information in an underground network. Therefore, the use of distribution functions to address such imprecision of recorded data is proposed. Cumulative distribution functions (CDFs) derived from the time series of pressure indicators (normal operation) and CDFs of indicator values at times coincident with a reported pipe break (conditioned to breaks) are compared. If all estimated CDFs are drawn from the same population, there is no reason to infer that the studied indicator clearly influences the probability of the rare event. However, when it is statistically proven that the estimated CDFs do not come from the same population, the analysed indicator may have an influence on the occurrence of pipe breaks. Due to the fact that the number of indicator values used to estimate the CDF conditioned to breaks is much lower in comparison with the number of indicator values to estimate the CDF of the unconditional pressure series, and that the obtained results depend on the size of the compared samples, CDFs from random sets of the same size sampled from the unconditional indicator values are estimated. Therefore, the comparison between the estimated CDFs of random sets of the indicator and the estimated CDF conditioned to breaks allows knowledge of if the indicator is influential on the probability of pipe breaks. Pressure indicators depend on various parameters. Sensitivity analysis and a robust statistical test allow determining the indicator for which these parameters result most influential on the probability of pipe breaks. At the same time, indicators can be calculated according to two model parameters, named as the anticipation time and the window width. The anticipation time refers to the time (hours) between the end of the period for the computation of the pressure indicator and the break. The window width is the number of instantaneous pressure values required to calculate the pressure indicator and is multiple of 24 hours, as water pressure has a cyclical behaviour which lasts one day. A sensitivity analysis of the model parameters explains when the pressure indicator is more influential on the probability of pipe breaks. The second part of the methodology presents a Bayesian diagnostic model. This kind of model belongs to the class of statistical predictive models, which are based on historical data, represent break behavior and patterns in water mains, and use the Bayes’ theorem to condition the probability of failure to specific system characteristics. The Bayes’ theorem allows comparing the break-conditioned FDA and the unconditional FDA of the indicators and determining when the probability of pipe breaks increases for certain pressure indicator ranges. A defined probability ratio provides a measure to establish whether the probability of breaks increases for certain ranges of the pressure indicator. The first part of the methodology is applied to the water distribution network of Madrid (Spain) and to the water distribution network of Panama City (Panama). The data filtering method suggests that the methodology can be applied to 15 sectors in Madrid and to two areas in Panama City. The results show that, in both systems, the most influential indicators on the probability of pipe breaks are the pressure range, which is the difference between the maximum pressure and the minimum pressure, and pressure variability, referred to the statistical property of the standard deviation. Therefore, they represent the dispersion of the data, the persistence of the variation in pressure and may be related to the fatigue in material resistance. The second part of the methodology has been applied to the influential indicators on the probability of pipe breaks in the water distribution network of Madrid. The main conclusion is that the probability of pipe breaks increases for the extreme values of the pressure range indicator and of the pressure variability indicator. Finally, a pressure management which limits the ranges of the pressure indicators influential on the probability of pipe breaks that increase such probability is recommended. The methodology presented here is general, may be applied to other water distribution networks, and could help water utilities reduce the number of system failures through pressure management.

Heavy tails, importance sampling and cross-entropy

We consider the problem of estimating P(Yi + (...) + Y-n > x) by importance sampling when the Yi are i.i.d. and heavy-tailed. The idea is to exploit the cross-entropy method as a toot for choosing good parameters in the importance sampling distribution; in doing so, we use the asymptotic description that given P(Y-1 + (...) + Y-n > x), n - 1 of the Yi have distribution F and one the conditional distribution of Y given Y > x. We show in some specific parametric examples (Pareto and Weibull) how this leads to precise answers which, as demonstrated numerically, are close to being variance minimal within the parametric class under consideration. Related problems for M/G/l and GI/G/l queues are also discussed.