Rachele Foschi - Academia.edu (original) (raw)

Papers by Rachele Foschi

Social Science Research Network, 2020

Having observed a cluster of jumps produced by an exponential Hawkes process, we study and quanti... more Having observed a cluster of jumps produced by an exponential Hawkes process, we study and quantify the residual length of the cluster. We then formalize the stochastic increasingness property of the durations between two consecutive jumps, which strengthens their positive correlation. Finally we consider the case where the process is only observed discretely and provide bounds for the probability of observing a given number of consecutive jumps. As an empirical exercise, we apply our results to a record of JPM's asset prices. First, we show that the identified jumps display dependence and clustering behavior. Second, we find that, under the exponential Hawkes model delivering the best QQ-plot, our formulas indicate a very high probability that an observed cluster of more than 1 jump did not exhaust yet.

We consider a class of transformations of Mixed Spatial Poisson processes that can be seen as the... more We consider a class of transformations of Mixed Spatial Poisson processes that can be seen as the spatial analog of the transformation induced by the construction of the departure process from an infinite server queue. We point out that some basic aspects are maintained also in the spatial case. This can have useful consequences in applications, in particular in the field of Insurance

An interesting problem in reliability is to determine the optimal burn-in time. In a previous wor... more An interesting problem in reliability is to determine the optimal burn-in time. In a previous work, the authors studied the solution of such a problem under a particular cost structure. It has been shown there that a key role in the problem is played by a function , representing the reward coming from the use of a component in the eld. A relevant case in this investigation is the one when is linear. In this paper, we explore further the linear case and use its solutions as a benchmark for determining the locally optimal times when the function is not linear or under a dierent cost structure.

We consider a pair of exchangeable lifetimes X;Y and the families of the conditional survival fun... more We consider a pair of exchangeable lifetimes X;Y and the families of the conditional survival functions F t (x;y) of (X t;Y t) given (X > t;Y > t). We analyze some properties of dependence and of ageing for F t (x;y) and some relations among them.

As a motivating problem, we aim to study some special aspects of the marginal distributions of th... more As a motivating problem, we aim to study some special aspects of the marginal distributions of the order statistics for exchangeable and (more generally) for minimally stable non-negative random variables T1, ..., Tr. In any case, we assume that T1, ..., Tr are identically distributed, with a common survival function G and their survival copula is denoted by K. The diagonal’s and subdiagonals’ sections of K, along with G, are possible tools to describe the information needed to recover the laws of order statistics. When attention is restricted to the absolutely continuous case, such a joint distribution can be described in terms of the associated multivariate conditional hazard rate (m.c.h.r.) functions. We then study the distributions of the order statistics of T1, ..., Tr also in terms of the system of the m.c.h.r. functions. We compare and, in a sense, we combine the two different approaches in order to obtain different detailed formulas and to analyze some probabilistic aspects ...

In previous papers, evolution of dependence and ageing, for vectors of non-negative random variab... more In previous papers, evolution of dependence and ageing, for vectors of non-negative random variables, have been separately considered. Some analogies between the two evolutions emerge however in those studies. In the present paper, we propose a unified approach, based on semigroup arguments, explaining the origin of such analogies and relations among properties of stochastic dependence and ageing.

Proportional Hazard Models arise from a straig htforward generalization of the simple case of con... more Proportional Hazard Models arise from a straig htforward generalization of the simple case of conditionally i.i.d., exponentially distributed random variables and, in a sense, can be considered as the idealized models in the statistical analysis of failure and survival data for lifetimes. For these rea sons, they have been extensively studied in the literature. Despite of the richness of related contributions, there are still special aspects of these models that are worthwhile focusing. In this discussion paper we aim to present some contributions, in th e frame of a Bayesian approach and by using some very basic notions of stochastic ordering.

Methodology and Computing in Applied Probability

Poisson processes are widely used to model the occurrence of similar and independent events. Howe... more Poisson processes are widely used to model the occurrence of similar and independent events. However they turn out to be an inadequate tool to describe a sequence of (possibly differently) interacting events. Many phenomena can be modelled instead by Hawkes processes. In this paper we aim at quantifying how much a Hawkes process departs from a Poisson one with respect to different aspects, namely, the behaviour of the stochastic intensity at jump times, the cumulative intensity and the interarrival times distribution. We show how the behaviour of Hawkes processes with respect to these three aspects may be very irregular. Therefore, we believe that developing a single measure describing them is not efficient, and that, instead, the departure from a Poisson process with respect to any different aspect should be separately quantified, by means of as many different measures. Key to defining these measures will be the stochastic intensity and the integrated intensity of a Hawkes process,...

Computational Management Science

An interesting problem in reliability is to determine the optimal burn-in time. In \cite{FS11DecA... more An interesting problem in reliability is to determine the optimal burn-in time. In \cite{FS11DecA}, the solution of such a problem under a particular cost structure has been studied.

Eusflat, 2007

We consider a pair of exchangeable lifetimes X, Y and the families of the conditional sur- vival ... more We consider a pair of exchangeable lifetimes X, Y and the families of the conditional sur- vival functions Ft (x,y) of (X t,Y t) given (X > t,Y > t). We analyze some properties of dependence and of ageing for Ft (x,y) and some relations among them.

Advances in Intelligent Systems and Computing, 2013

We investigate the dependence properties of a vector of residual lifetimes by means of the copula... more We investigate the dependence properties of a vector of residual lifetimes by means of the copula associated with the conditional distribution function. In particular, the evolution of positive dependence properties (like quadrant dependence and total positivity) are analyzed and expressions for the evolution of measures of association are given.

ABSTRACT We consider a class of transformations of Mixed Spatial Poisson pro-cesses that can be s... more ABSTRACT We consider a class of transformations of Mixed Spatial Poisson pro-cesses that can be seen as the spatial analog of the transformation induced by the construction of the departure process from an infinite server queue. We point out that some basic aspects are maintained also in the spatial case. This can have useful consequences in applications, in particular in the field of Insurance Keywords. Spatial M/G/∞ queues, an invariance result, conditional mixing dis-tribution.

Mathematical and Statistical Models and Methods in Reliability, 2010

ABSTRACT We consider non-negative conditionally independent and identically distributed random va... more ABSTRACT We consider non-negative conditionally independent and identically distributed random variables and analyze conditions for monotonicity of survival copulas of residual lifetimes. Concentrating attention on the bivariate copula, we compare its behavior at the instant of default with its evolution between two defaults. The assumptions for our results will be expressed in terms of conditional hazard rates. Keywords and phrasesSurvival Copulas of residual lifetimes-Default contagion-Multivariate stochastic orders-Longitudinal observations-Posterior distributions

Social Science Research Network, 2020

Having observed a cluster of jumps produced by an exponential Hawkes process, we study and quanti... more Having observed a cluster of jumps produced by an exponential Hawkes process, we study and quantify the residual length of the cluster. We then formalize the stochastic increasingness property of the durations between two consecutive jumps, which strengthens their positive correlation. Finally we consider the case where the process is only observed discretely and provide bounds for the probability of observing a given number of consecutive jumps. As an empirical exercise, we apply our results to a record of JPM's asset prices. First, we show that the identified jumps display dependence and clustering behavior. Second, we find that, under the exponential Hawkes model delivering the best QQ-plot, our formulas indicate a very high probability that an observed cluster of more than 1 jump did not exhaust yet.

We consider a class of transformations of Mixed Spatial Poisson processes that can be seen as the... more We consider a class of transformations of Mixed Spatial Poisson processes that can be seen as the spatial analog of the transformation induced by the construction of the departure process from an infinite server queue. We point out that some basic aspects are maintained also in the spatial case. This can have useful consequences in applications, in particular in the field of Insurance

An interesting problem in reliability is to determine the optimal burn-in time. In a previous wor... more An interesting problem in reliability is to determine the optimal burn-in time. In a previous work, the authors studied the solution of such a problem under a particular cost structure. It has been shown there that a key role in the problem is played by a function , representing the reward coming from the use of a component in the eld. A relevant case in this investigation is the one when is linear. In this paper, we explore further the linear case and use its solutions as a benchmark for determining the locally optimal times when the function is not linear or under a dierent cost structure.

We consider a pair of exchangeable lifetimes X;Y and the families of the conditional survival fun... more We consider a pair of exchangeable lifetimes X;Y and the families of the conditional survival functions F t (x;y) of (X t;Y t) given (X > t;Y > t). We analyze some properties of dependence and of ageing for F t (x;y) and some relations among them.

As a motivating problem, we aim to study some special aspects of the marginal distributions of th... more As a motivating problem, we aim to study some special aspects of the marginal distributions of the order statistics for exchangeable and (more generally) for minimally stable non-negative random variables T1, ..., Tr. In any case, we assume that T1, ..., Tr are identically distributed, with a common survival function G and their survival copula is denoted by K. The diagonal’s and subdiagonals’ sections of K, along with G, are possible tools to describe the information needed to recover the laws of order statistics. When attention is restricted to the absolutely continuous case, such a joint distribution can be described in terms of the associated multivariate conditional hazard rate (m.c.h.r.) functions. We then study the distributions of the order statistics of T1, ..., Tr also in terms of the system of the m.c.h.r. functions. We compare and, in a sense, we combine the two different approaches in order to obtain different detailed formulas and to analyze some probabilistic aspects ...

In previous papers, evolution of dependence and ageing, for vectors of non-negative random variab... more In previous papers, evolution of dependence and ageing, for vectors of non-negative random variables, have been separately considered. Some analogies between the two evolutions emerge however in those studies. In the present paper, we propose a unified approach, based on semigroup arguments, explaining the origin of such analogies and relations among properties of stochastic dependence and ageing.

Proportional Hazard Models arise from a straig htforward generalization of the simple case of con... more Proportional Hazard Models arise from a straig htforward generalization of the simple case of conditionally i.i.d., exponentially distributed random variables and, in a sense, can be considered as the idealized models in the statistical analysis of failure and survival data for lifetimes. For these rea sons, they have been extensively studied in the literature. Despite of the richness of related contributions, there are still special aspects of these models that are worthwhile focusing. In this discussion paper we aim to present some contributions, in th e frame of a Bayesian approach and by using some very basic notions of stochastic ordering.

Methodology and Computing in Applied Probability

Poisson processes are widely used to model the occurrence of similar and independent events. Howe... more Poisson processes are widely used to model the occurrence of similar and independent events. However they turn out to be an inadequate tool to describe a sequence of (possibly differently) interacting events. Many phenomena can be modelled instead by Hawkes processes. In this paper we aim at quantifying how much a Hawkes process departs from a Poisson one with respect to different aspects, namely, the behaviour of the stochastic intensity at jump times, the cumulative intensity and the interarrival times distribution. We show how the behaviour of Hawkes processes with respect to these three aspects may be very irregular. Therefore, we believe that developing a single measure describing them is not efficient, and that, instead, the departure from a Poisson process with respect to any different aspect should be separately quantified, by means of as many different measures. Key to defining these measures will be the stochastic intensity and the integrated intensity of a Hawkes process,...

Computational Management Science

An interesting problem in reliability is to determine the optimal burn-in time. In \cite{FS11DecA... more An interesting problem in reliability is to determine the optimal burn-in time. In \cite{FS11DecA}, the solution of such a problem under a particular cost structure has been studied.

Eusflat, 2007

We consider a pair of exchangeable lifetimes X, Y and the families of the conditional sur- vival ... more We consider a pair of exchangeable lifetimes X, Y and the families of the conditional sur- vival functions Ft (x,y) of (X t,Y t) given (X > t,Y > t). We analyze some properties of dependence and of ageing for Ft (x,y) and some relations among them.

Advances in Intelligent Systems and Computing, 2013

We investigate the dependence properties of a vector of residual lifetimes by means of the copula... more We investigate the dependence properties of a vector of residual lifetimes by means of the copula associated with the conditional distribution function. In particular, the evolution of positive dependence properties (like quadrant dependence and total positivity) are analyzed and expressions for the evolution of measures of association are given.

ABSTRACT We consider a class of transformations of Mixed Spatial Poisson pro-cesses that can be s... more ABSTRACT We consider a class of transformations of Mixed Spatial Poisson pro-cesses that can be seen as the spatial analog of the transformation induced by the construction of the departure process from an infinite server queue. We point out that some basic aspects are maintained also in the spatial case. This can have useful consequences in applications, in particular in the field of Insurance Keywords. Spatial M/G/∞ queues, an invariance result, conditional mixing dis-tribution.

Mathematical and Statistical Models and Methods in Reliability, 2010

ABSTRACT We consider non-negative conditionally independent and identically distributed random va... more ABSTRACT We consider non-negative conditionally independent and identically distributed random variables and analyze conditions for monotonicity of survival copulas of residual lifetimes. Concentrating attention on the bivariate copula, we compare its behavior at the instant of default with its evolution between two defaults. The assumptions for our results will be expressed in terms of conditional hazard rates. Keywords and phrasesSurvival Copulas of residual lifetimes-Default contagion-Multivariate stochastic orders-Longitudinal observations-Posterior distributions