Lafuente Blasco, Miguel
Sanz Sáiz, Gerardo (dir.) ; López Lorente, Francisco Javier (dir.)
Universidad de Zaragoza,
2022
Resumen: Records, defined as observations that exceed all previous observations, are ubiquitous in modern everyday life. They have also attracted much research and attention, due to their intrinsic interest and the mathematical challenges they pose. In 1952, a seminal paper by Chandler launched what has now grown to become a rich body of literature on the mathematical properties of record observations. The study of the probabilistic properties of records has attained an important degree of maturity and it is therefore natural that significant effort has been devoted to statistical inference with records over recent decades.
The classical probabilistic setting of records in independent and identically distributed (i.i.d.) continuous random variables (r.v.), reflects the scarcity of this kind of observations. Indeed, for sequences of i.i.d. continuous r.v., it is known that the probability that the n-th observation is a record is 1/n, and the expected number of records is of the order of the logarithm of n, where n is the number of observations. Note however that this universal property is lost when the underlying r.v. are discrete.
The connection of records with many interesting problems led to a considerable interest in the study of record observations, especially from the perspective of physics. Records have proved their worth in many areas such as athletics, risk theory, financial modeling and evolutionary biology. One of the main fields of application is climatology, where the i.i.d. model fails to predict the number of high temperature records, with these observations being significantly higher than expected. In this thesis, we are going to consider two distinct generalizations related to the study of usual records with the aim of enabling a greater number of problems to be addressed.
The first concerns the mathematical definition of a record. In this monograph we focus on two of the record-related concepts that have been most studied ¿ near-records introduced in 2005 by Balakrishnan et al. and delta-records proposed by Gouet et al. in 2007. Given a sequence of observations (X_n), we say that the n-th observation is a delta-record if X_n > M_{n-1} +delta, where M_{n-1} is the maximum among the first n-1 observations, and delta is a real parameter. If delta = 0 records and delta-records are equivalent, while in the case delta < 0 ( delta > 0), delta-records are more (less) frequent than records.
An observation is considered to be a near-record if it is not a record but is at a distance of less than a units from the last record. Consequently, the study of delta-records and near-records is closely related, and obtaining properties of one of these notions generally results in obtaining properties of the other. The other kind of generalization that we consider concerns the model of the underlying variables. Adding a deterministic linear trend to the observations we obtain what is known as a Linear Drift Model (LDM), first introduced by Ballerini and Resnick, and studied later by other authors. The LDM has proven particularly useful in the study of global warming to explain the actual number of upper records observed.
In this monograph we address some open problems for near-records and delta-records.
Chapter 1 presents the known properties of records and delta-records, as well as establishing the notation that will be used later.
In Chapter 2 we study the point process of near-record values when the observations are discrete, taking values in the integers. This problem was already studied in for continuous distributions. While in the discrete setting the resulting process is also a cluster process, it is no longer a Poisson process, which makes the study of the point process and its characterization more difficult. Laws of large numbers and central limit theorems for the number of near-records with a value in a set are also obtained. Finally, we characterize which discrete distributions fulfill a martingale condition relating the partial maxima and the number of delta-records at time n. We relate this characterization to the open problem of the positivity of the terms in a recurrence relation.
The LDM is studied extensively in Chapter 3 for delta-records. From the basic properties and derivations of the probability of delta-record, we study the asymptotic delta-record probability and its analytic properties. We derive exact expressions for some distributions, some of them also unknown in the case of usual records, and we use these results to assess the effect of the delta parameter when the underlying variables are heavy-tailed. For distributions where an analytic expression is not available, we propose first order approximations to study delta-record probabilities. We also compute the correlation of delta-record observations as a function of the number of observations and the delta parameter. We study the asymptotics of the counting process of delta-records in the LDM. The finiteness of the number of delta-records in the LDM is completely characterized. In particular, this result solves a conjecture posed by Franke et al. for records, proving the result not only for usual records but also in the general setting with delta != 0. Finally, we obtain laws of large numbers and a central limit theorem under mild conditions, extending previous known results to the case delta != 0, and we prove a law of large numbers for a random trend model.
In Chapter 4 we develop statistical inference methods for delta-records in the LDM. We propose two estimators for the variance of the number of delta-records and discuss their properties, proving consistency. We also study Maximum Likelihood Estimation based on delta-records in the LDM. We develop a general framework for Maximum Likelihood Estimation and we find analytic solutions for particular cases. We use Montecarlo simulation to compare the performance of the Maximum Likelihood Estimators using delta-records with those using records only. Finally, the results in Chapter 3 and in this chapter are applied to a real dataset of temperatures, where the LDM is consistent with the findings of other authors and the phenomenon of global warming. In particular, we find good agreement between the theoretical results and the data observed in the example.
Finally, in Chapter 5, we set out some conclusions of the results reached in previous chapters and offer some ideas for future work.
Resumen (otro idioma):
The classical probabilistic setting of records in independent and identically distributed (i.i.d.) continuous random variables (r.v.), reflects the scarcity of this kind of observations. Indeed, for sequences of i.i.d. continuous r.v., it is known that the probability that the n-th observation is a record is 1/n, and the expected number of records is of the order of the logarithm of n, where n is the number of observations. Note however that this universal property is lost when the underlying r.v. are discrete.
The connection of records with many interesting problems led to a considerable interest in the study of record observations, especially from the perspective of physics. Records have proved their worth in many areas such as athletics, risk theory, financial modeling and evolutionary biology. One of the main fields of application is climatology, where the i.i.d. model fails to predict the number of high temperature records, with these observations being significantly higher than expected. In this thesis, we are going to consider two distinct generalizations related to the study of usual records with the aim of enabling a greater number of problems to be addressed.
The first concerns the mathematical definition of a record. In this monograph we focus on two of the record-related concepts that have been most studied ¿ near-records introduced in 2005 by Balakrishnan et al. and delta-records proposed by Gouet et al. in 2007. Given a sequence of observations (X_n), we say that the n-th observation is a delta-record if X_n > M_{n-1} +delta, where M_{n-1} is the maximum among the first n-1 observations, and delta is a real parameter. If delta = 0 records and delta-records are equivalent, while in the case delta < 0 ( delta > 0), delta-records are more (less) frequent than records.
An observation is considered to be a near-record if it is not a record but is at a distance of less than a units from the last record. Consequently, the study of delta-records and near-records is closely related, and obtaining properties of one of these notions generally results in obtaining properties of the other. The other kind of generalization that we consider concerns the model of the underlying variables. Adding a deterministic linear trend to the observations we obtain what is known as a Linear Drift Model (LDM), first introduced by Ballerini and Resnick, and studied later by other authors. The LDM has proven particularly useful in the study of global warming to explain the actual number of upper records observed.
In this monograph we address some open problems for near-records and delta-records.
Chapter 1 presents the known properties of records and delta-records, as well as establishing the notation that will be used later.
In Chapter 2 we study the point process of near-record values when the observations are discrete, taking values in the integers. This problem was already studied in for continuous distributions. While in the discrete setting the resulting process is also a cluster process, it is no longer a Poisson process, which makes the study of the point process and its characterization more difficult. Laws of large numbers and central limit theorems for the number of near-records with a value in a set are also obtained. Finally, we characterize which discrete distributions fulfill a martingale condition relating the partial maxima and the number of delta-records at time n. We relate this characterization to the open problem of the positivity of the terms in a recurrence relation.
The LDM is studied extensively in Chapter 3 for delta-records. From the basic properties and derivations of the probability of delta-record, we study the asymptotic delta-record probability and its analytic properties. We derive exact expressions for some distributions, some of them also unknown in the case of usual records, and we use these results to assess the effect of the delta parameter when the underlying variables are heavy-tailed. For distributions where an analytic expression is not available, we propose first order approximations to study delta-record probabilities. We also compute the correlation of delta-record observations as a function of the number of observations and the delta parameter. We study the asymptotics of the counting process of delta-records in the LDM. The finiteness of the number of delta-records in the LDM is completely characterized. In particular, this result solves a conjecture posed by Franke et al. for records, proving the result not only for usual records but also in the general setting with delta != 0. Finally, we obtain laws of large numbers and a central limit theorem under mild conditions, extending previous known results to the case delta != 0, and we prove a law of large numbers for a random trend model.
In Chapter 4 we develop statistical inference methods for delta-records in the LDM. We propose two estimators for the variance of the number of delta-records and discuss their properties, proving consistency. We also study Maximum Likelihood Estimation based on delta-records in the LDM. We develop a general framework for Maximum Likelihood Estimation and we find analytic solutions for particular cases. We use Montecarlo simulation to compare the performance of the Maximum Likelihood Estimators using delta-records with those using records only. Finally, the results in Chapter 3 and in this chapter are applied to a real dataset of temperatures, where the LDM is consistent with the findings of other authors and the phenomenon of global warming. In particular, we find good agreement between the theoretical results and the data observed in the example.
Finally, in Chapter 5, we set out some conclusions of the results reached in previous chapters and offer some ideas for future work.
Resumen (otro idioma):
Pal. clave: probabilidad ; aplicación de la probabilidad ; teoremas del limite ; estadística
Titulación: Programa de Doctorado en Matemáticas y Estadística
Plan(es): Plan 490
Área de conocimiento: Ciencias
Nota: Presentado: 18 03 2022
Nota: Tesis-Univ. Zaragoza, , 2022
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