Hawkes processes are temporal self-exciting point processes. They are well established in earthquake modelling or finance and their application is spreading to diverse areas. Most models from the literature have two m...
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Hawkes processes are temporal self-exciting point processes. They are well established in earthquake modelling or finance and their application is spreading to diverse areas. Most models from the literature have two major drawbacks regarding their potential application to insurance. First, they use an exponentially-decaying form of excitation, which does not allow a delay between the occurrence of an event and its excitation effect on the process and does not fit well on insurance data consequently. Second, theoretical results developed from these models are valid only when time of observation tends to infinity, whereas the time horizon for an insurance use case is of several months or years. In this paper, we define a complete framework of Hawkes processes with a Gamma density excitation function (i.e. estimation, simulation, goodness-of-fit) instead of an exponential-decaying function and we demonstrate some mathematical properties (i.e. expectation, variance) about the transient regime of the process. We illustrate our results with real insurance data about natural disasters in Luxembourg.
The existing expectation maximization (em) and space-alternating generalized em (SAGE) algorithms are only applied to direction of arrival (DOA) estimation in known noise. In this paper, the two algorithms are designe...
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The existing expectation maximization (em) and space-alternating generalized em (SAGE) algorithms are only applied to direction of arrival (DOA) estimation in known noise. In this paper, the two algorithms are designed for DOA estimation in unknown uniform noise. Both the deterministic and random signal models are considered. In addition, a new modified em (Mem) algorithm applicable to the noise assumption is also proposed. Next, these em-type algorithms are improved to ensure the stability when the powers of sources are not equal. After being improved, simulation results illustrate that the em algorithm has similar convergence with the Mem algorithm, the SAGE algorithm outperforms the em and Mem algorithms for the deterministic signal model, and the SAGE algorithm cannot always outperform the em and Mem algorithms for the random signal model. Furthermore, simulation results show that processing the same snapshots from the random signal model, the SAGE algorithm for the deterministic signal model can require the fewest computations.
The tick structure of the financial markets entails discreteness of stock price changes. Based on this empirical evidence, we develop a multivariate model for discrete price changes featuring a mechanism to account fo...
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The tick structure of the financial markets entails discreteness of stock price changes. Based on this empirical evidence, we develop a multivariate model for discrete price changes featuring a mechanism to account for the large share of zero returns at high frequency. We assume that the observed price changes are independent conditional on the realization of two hidden Markov chains determining the dynamics and the distribution of the multivariate time series at hand. We study the properties of the model, which is a dynamic mixture of zero-inflated Skellam distributions. We develop an expectation-maximization algorithm with closed-form M-step that allows us to estimate the model by maximum likelihood. In the empirical application, we study the joint distribution of the price changes of a number of assets traded on NYSE. Particular focus is dedicated to the assessment of the quality of univariate and multivariate density forecasts, and of the precision of the predictions of moments like volatility and correlations. Finally, we look at the predictability of price staleness and its determinants in relation to the trading activity on the financial markets.
This paper considers the latent Gaussian graphical model, which extends the Gaussian graphical model to handle discrete data as well as mixed data with both continuous and discrete variables by assuming that discrete ...
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This paper considers the latent Gaussian graphical model, which extends the Gaussian graphical model to handle discrete data as well as mixed data with both continuous and discrete variables by assuming that discrete variables are generated by discretizing latent Gaussian variables. We propose a modified expectation-maximization (em) algorithm to estimate parameters in the latent Gaussian model for binary data. We also extend the proposed modified em algorithm to the latent Gaussian model for mixed data. The conditional dependence structure can be consequently constructed by exploring the sparsity pattern of the precision matrix of the latent variables. We illustrate the performance of our proposed estimator through comprehensive numerical studies and an application to voting data of the United Nations General Assembly.
Graphical models have received an increasing amount of attention in network psychometrics as a promising probabilistic approach to study the conditional relations among variables using graph theory. Despite recent adv...
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Graphical models have received an increasing amount of attention in network psychometrics as a promising probabilistic approach to study the conditional relations among variables using graph theory. Despite recent advances, existing methods on graphical models usually assume a homogeneous population and focus on binary or continuous variables. However, ordinal variables are very popular in many areas of psychological science, and the population often consists of several different groups based on the heterogeneity in ordinal data. Driven by these needs, we introduce the finite mixture of ordinal graphical models to effectively study the heterogeneous conditional dependence relationships of ordinal data. We develop a penalized likelihood approach for model estimation, and design a generalized expectation-maximization (em) algorithm to solve the significant computational challenges. We examine the performance of the proposed method and algorithm in simulation studies. Moreover, we demonstrate the potential usefulness of the proposed method in psychological science through a real application concerning the interests and attitudes related to fan avidity for students in a large public university in the United States.
The asymmetric exponential power (AEP) distribution has received much attention in economics and finance. Simulation study shows that iterative methods developed for finding the maximum likelihood (ML) estimates of th...
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The asymmetric exponential power (AEP) distribution has received much attention in economics and finance. Simulation study shows that iterative methods developed for finding the maximum likelihood (ML) estimates of the AEP distribution sometimes fail to converge. In this paper, the expectation-maximization (em) algorithm is proposed to find the ML estimates of the AEP distribution which always converges. Performance of the em algorithm is demonstrated by simulations and a real data illustration. As an application, the proposed em algorithm is applied to find the ML estimates for the regression coefficients when the error term in a linear regression model follows the AEP distribution. Performance of the AEP distribution in robust simple regression modelling is established through a real data illustration.
Finite Gamma mixture models are often used to describe randomness in income data, insurance data, and data in applications where the response values are intrinsically positive. The popular likelihood approach for mode...
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Finite Gamma mixture models are often used to describe randomness in income data, insurance data, and data in applications where the response values are intrinsically positive. The popular likelihood approach for model fitting, however, does not work for this model because its likelihood function is unbounded. Because of this, the maximum likelihood estimator is not well-defined. Other approaches have been developed to achieve consistent estimation of the mixing distribution, such as placing an upper bound on the shape parameter or adding a penalty to the log-likelihood function. In this paper, we show that if the shape parameter in the finite Gamma mixture model is structural, then the direct maximum likelihood estimator of the mixing distribution is well-defined and strongly consistent. We also present simulation results demonstrating the consistency of the estimator. We illustrate the application of the model with a structural shape parameter to household income data. The fitted mixture distribution leads to several possible subpopulation structures with regard to the level of disposable income.
Random effect models have been popularly used as a mainstream statistical technique over several decades;and the same can be said for response transformation models such as the Box-Cox transformation. The latter aims ...
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Random effect models have been popularly used as a mainstream statistical technique over several decades;and the same can be said for response transformation models such as the Box-Cox transformation. The latter aims at ensuring that the assumptions of normality and of homoscedasticity of the response distribution are fulfilled, which are essential conditions for inference based on a linear model or a linear mixed model. However, methodology for response transformation and simultaneous inclusion of random effects has been developed and implemented only scarcely, and is so far restricted to Gaussian random effects. We develop such methodology, thereby not requiring parametric assumptions on the distribution of the random effects. This is achieved by extending the 'Nonparametric Maximum Likelihood' towards a 'Nonparametric profile maximum likelihood' technique, allowing to deal with overdispersion as well as two-level data scenarios.
The stochastic blockmodel (SBM) models the connectivity within and between disjoint subsets of nodes in networks. Prior work demonstrated that the rows of an SBM's adjacency spectral embedding (ASE) and Laplacian ...
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The stochastic blockmodel (SBM) models the connectivity within and between disjoint subsets of nodes in networks. Prior work demonstrated that the rows of an SBM's adjacency spectral embedding (ASE) and Laplacian spectral embedding (LSE) both converge in law to Gaussian mixtures where the components are curved exponential families. Maximum likelihood estimation via the Expectation-Maximization (em) algorithm for a full Gaussian mixture model (GMM) can then perform the task of clustering graph nodes, albeit without appealing to the components' curvature. Noting that em is a special case of the Expectation-Solution (ES) algorithm, we propose two ES algorithms that allow us to take full advantage of these curved structures. After presenting the ES algorithm for the general curved-Gaussian mixture, we develop those corresponding to the ASE and LSE limiting distributions. Simulating from artificial SBMs and a brain connectome SBM reveals that clustering graph nodes via our ES algorithms can improve upon that of em for a full GMM for a wide range of settings.
Change point estimation in standard process observed over time is an important problem in literature with applications in various fields. We study this problem in a heterogeneous population. A model-based clustering p...
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Change point estimation in standard process observed over time is an important problem in literature with applications in various fields. We study this problem in a heterogeneous population. A model-based clustering procedure relying on skewed matrix-variate mixture is proposed. It is capable of capturing the heterogeneity pattern and estimating change points from all data groups simultaneously. The appeal of such approach also lies in its flexibility to model the skewness and dependence in data with good interpretability. Two novel algorithms called matrix power mixture with abrupt change model and matrix power mixture with gradual change model are developed. The approaches are illustrated by simulation studies across a variety of settings. The models are then tested on the US crime data with promising results.
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