SUMMARYWe propose truncated and power-transformed (TPT) models for daily rainfall and we derive the generalized extreme value (GEV) limit distributions for these models. We find that these limit distributions belong to the domain of attraction of the Fréchet family when the parent distribution of the daily values is a TPT t-Student model. In this case, the shape parameter of the limiting GEV model depends on the degrees of freedom and the power transformation parameter. When the parent distribution of the daily values is a TPT Normal model, the limiting GEV model is independent of the parameters of the parent model. We perform a detailed inference and predictive analysis to verify these theoretical results using a Bayesian approach. Markov Chain Monte Carlo methods (MCMC) were used to estimate the posterior distribution of the parameters of the t-Student model for daily rainfall on one hand, and to estimate the posterior distribution of the parameters of the GEV model for the annual maxima on the other hand. Numerical results are presented for two locations: Maiquetía (Vargas State), and La Mariposa (Miranda State), Venezuela. Simulations from the predictive distribution of the daily values suggest a good approximation between the extreme distribution of the TPT t-Student model and the Fréchet model found by standard extreme value limit theory.
In this paper, we describe and implement two recursive filtering algorithms, the optimized particle filter, and the Viterbi algorithm, which allow the joint estimation of states and parameters of continuous-time stochastic volatility models, such as the Cox Ingersoll Ross and Heston model. In practice, good parameter estimates are required so that the models are able to generate accurate forecasts. To achieve the objectives the proposed algorithms were implemented using daily empirical data from the time series of the $S\&P500$ returns of the stock exchange index. The proposed methodology facilitates computational calculations of the marginal likelihood of states and allows the reconstruction of unknown states in a suitable way, and reliable estimation of the parameters. To measure the quality of estimation of the algorithms, we used the square root of the mean square error and relative deviation standard as measures of goodness of fit. The estimated errors are insignificant for the analyzed data and the two models considered. We also calculated the execution times of the algorithms, demonstrating that the Viterbi algorithm has less execution time than the optimized particle filter.
The growth dynamics that a population follows is mainly due to births, deaths or migrations, each of these phenomena is affected by other factors such as public health, birth control, work sources, economy, safety and conditions of quality of life in neighboring countries, among many others. In this paper is proposed two statistical models based on a system of stochastic differential equations (SDE) that model the dynamics of population growth, and three computational algorithms that allow the generation of probability distribution samples in high dimensions, in models that have non-linear structures and that are useful for making inferences. The algorithms allow to estimate simultaneously states solutions and parameters in SDE models. The interpretation of the parameters is important because they are related to the variables of growth, mortality, migration, physical-chemical conditions of the environment, among other factors. The algorithms are illustrated using real data from a sector of the population of the Republic of Ecuador, and are compared with the results obtained with the models used by the World Bank for the same data, which shows that stochastic models Proposals based on an SDE more adequately and reliably adjust the dynamics of demographic randomness, sampling errors and environmental randomness in comparison with the deterministic models used by the World Bank. It is observed that the population grows year by year and seems to have a definite tendency; that is, a clearly growing behavior is seen. To measure the relative success of the algorithms, the relative error was estimated, obtaining small percentage errors.
Stochastic differential equations arise in a variety of contexts. There are many techniques for approximation of the solutions of these equations that include statistical methods, numerical methods, and other approximations. This article implements a parametric and a nonparametric method to approximate the probability density of the solutions of stochastic differential equation from observations of a discrete dynamic system. To achieve these objectives, mixtures of Dirichlet process and Gaussian mixtures are considered. The methodology uses computational techniques based on Gaussian mixtures filters, nonparametric particle filters and Gaussian particle filters to establish the relationship between the theoretical processes of unobserved and observed states. The approximations obtained by this proposal are attractive because the hierarchical structures used for modeling are flexible, easy to interpret and computationally feasible. The methodology is illustrated by means of two problems: the synthetic Lorenz model and a real model of rainfall. Experiments show that the proposed filters provide satisfactory results, demonstrating that the algorithms work well in the context of processes with nonlinear dynamics which require the joint estimation of states and parameters. The estimated measure of goodness of fit validates the estimation quality of the algorithms. 290APPROXIMATIONS OF THE SOLUTIONS OF A STOCHASTIC DIFFERENTIAL EQUATION nonlinear stochastic differential equations, for example [60] and [5] have used an approach based on a Gaussian process to model the time evolution of the solution of a general stochastic differential equation, the methodology uses data assimilation techniques ([31]). Recently [57] introduced a nonparametric method to estimate the function of drift in a stochastic differential equation where a measure of random probability was considered to model the drift and a Expectation-Maximization (EM) algorithm was developed to try to establish a link between unobserved and observed states. This paper aims to approximate the solutions of a stochastic differential equation by using the Gaussian mixture distribution model, Dirichlet process mixture models, together with a non-parametric density estimation algorithm and three sequential filters. The most representative references about the Gaussian mixture distribution model are [61]; [3]; [68] and [2]. Dirichlet processes were treated in [16], and they have been identified in the literature as a useful tool for estimating densities in the field of non-parametric models. The idea of using Dirichlet process is not new, and has been considered in previous works such as [15]; [46]; [42]; [45]; [70] among others.The rest of article is summarized as follows: Section 2 presents the formulation of the problem; Section 3 contains the description of the filtering algorithms defined to approximate a probability density of the solutions; In Section 4, the results obtained for two different examples are shown, and lastly, Section 5 contains a final discussion an...
In this article, an estimation methodology based on the sequential Monte Carlo algorithm is proposed, thatjointly estimate the states and parameters, the relationship between the prices of futures contracts and the spot prices of primary products is determined, the evolution of prices and the volatility of the historical data of the primary market (Gold and Soybean) are analyzed. Two stochastic models for an estimate the states and parameters are considered, the parameters and states describe physical measure (associated with the price) and risk-neutral measure (associated with the markets to futures), the price dynamics in the short-term through the reversion to the mean and volatility are determined, while that in the long term through markets to futures. Other characteristics such as seasonal patterns, price spikes, market dependent volatilities, and non-seasonality can also be observed. In the methodology, a parameter learning algorithm is used, specifically, three algorithms are proposed, that is the sequential Monte Carlo estimation (SMC) for state space modelswith unknown parameters: the first method is considered a particle filter that is based on the sampling algorithm of sequential importance with resampling (SISR). The second implemented method is the Storvik algorithm [19], the states and parameters of the posterior distribution are estimated that have supported in low-dimensional spaces, a sufficient statistics from the sample of the filtered distribution is considered. The third method is (PLS) Carvalho’s Particle Learning and Smoothing algorithm [31]. The cash prices of the contracts with future delivery dates are analyzed. The results indicate postponement of payment, the future prices on different maturity dates with the spot price are highly correlated. Likewise, the contracts with a delivery date for the last periods of the year 2017, the spot price lower than the prices of the contracts with expiration date for 12 and 24 months is found, opposite occurs in the contracts with expiration date for 1 and 6 months.
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