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20 pages, 511 KiB

Open AccessArticle

T-Growth Stochastic Model: Simulation and Inference via Metaheuristic Algorithms

by Antonio Barrera, Patricia Román-Román and Francisco Torres-Ruiz

Mathematics 2021, 9(9), 959; https://doi.org/10.3390/math9090959 - 25 Apr 2021

Cited by 2 | Viewed by 1844

The main objective of this work is to introduce a stochastic model associated with the one described by the T-growth curve, which is in turn a modification of the logistic curve. By conveniently reformulating the T curve, it may be obtained as a [...] Read more.

The main objective of this work is to introduce a stochastic model associated with the one described by the T-growth curve, which is in turn a modification of the logistic curve. By conveniently reformulating the T curve, it may be obtained as a solution to a linear differential equation. This greatly simplifies the mathematical treatment of the model and allows a diffusion process to be defined, which is derived from the non-homogeneous lognormal diffusion process, whose mean function is a T curve. This allows the phenomenon under study to be viewed in a dynamic way. In these pages, the distribution of the process is obtained, as are its main characteristics. The maximum likelihood estimation procedure is carried out by optimization via metaheuristic algorithms. Thanks to an exhaustive study of the curve, a strategy is obtained to bound the parametric space, which is a requirement for the application of various swarm-based metaheuristic algorithms. A simulation study is presented to show the validity of the bounding procedure and an example based on real data is provided. Full article

(This article belongs to the Special Issue Stochastic Statistics and Modeling)

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25 pages, 4557 KiB

Open AccessArticle

On the Simulation of a Special Class of Time-Inhomogeneous Diffusion Processes

by Virginia Giorno and Amelia G. Nobile

Mathematics 2021, 9(8), 818; https://doi.org/10.3390/math9080818 - 9 Apr 2021

Cited by 2 | Viewed by 1923

Abstract

General methods to simulate probability density functions and first passage time densities are provided for time-inhomogeneous stochastic diffusion processes obtained via a composition of two Gauss–Markov processes conditioned on the same initial state. Many diffusion processes with time-dependent infinitesimal drift and infinitesimal variance [...] Read more.

General methods to simulate probability density functions and first passage time densities are provided for time-inhomogeneous stochastic diffusion processes obtained via a composition of two Gauss–Markov processes conditioned on the same initial state. Many diffusion processes with time-dependent infinitesimal drift and infinitesimal variance are included in the considered class. For these processes, the transition probability density function is explicitly determined. Moreover, simulation procedures are applied to the diffusion processes obtained starting from Wiener and Ornstein–Uhlenbeck processes. Specific examples in which the infinitesimal moments include periodic functions are discussed. Full article

(This article belongs to the Special Issue Stochastic Statistics and Modeling)

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14 pages, 674 KiB

Open AccessArticle

Using First-Passage Times to Analyze Tumor Growth Delay

by Patricia Román-Román, Sergio Román-Román, Juan José Serrano-Pérez and Francisco Torres-Ruiz

Mathematics 2021, 9(6), 642; https://doi.org/10.3390/math9060642 - 17 Mar 2021

Cited by 3 | Viewed by 2288

Abstract

A central aspect of in vivo experiments with anticancer therapies is the comparison of the effect of different therapies, or doses of the same therapeutic agent, on tumor growth. One of the most popular clinical endpoints is tumor growth delay, which measures the [...] Read more.

A central aspect of in vivo experiments with anticancer therapies is the comparison of the effect of different therapies, or doses of the same therapeutic agent, on tumor growth. One of the most popular clinical endpoints is tumor growth delay, which measures the effect of treatment on the time required for tumor volume to reach a specific value. This effect has been analyzed through a variety of statistical methods: conventional descriptive analysis, linear regression, Cox regression, etc. We propose a new approach based on stochastic modeling of tumor growth and the study of first-passage time variables. This approach allows us to prove that the time required for tumor volume to reach a specific value must be determined empirically as the average of the times required for the volume of individual tumors to reach said value instead of the time required for the average volume of the tumors to reach the value of interest. In addition, we define several measures in random environments to compare the time required for the tumor volume to multiply k times its initial volume in control, as well as treated groups, and the usefulness of these measures is illustrated by means of an application to real data. Full article

(This article belongs to the Special Issue Stochastic Statistics and Modeling)

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12 pages, 576 KiB

Open AccessArticle

Demographic Dynamics in Multitype Populations with Migrations

by Manuel Molina-Fernández and Manuel Mota-Medina

Mathematics 2021, 9(3), 246; https://doi.org/10.3390/math9030246 - 27 Jan 2021

Viewed by 1860

Abstract

This research work deals with mathematical modeling in complex biological systems in which several types of individuals coexist in various populations. Migratory phenomena among the populations are allowed. We propose a class of mathematical models to describe the demographic dynamics of these type [...] Read more.

This research work deals with mathematical modeling in complex biological systems in which several types of individuals coexist in various populations. Migratory phenomena among the populations are allowed. We propose a class of mathematical models to describe the demographic dynamics of these type of complex systems. The probability model is defined through a sequence of random matrices in which rows and columns represent the various populations and the several types of individuals, respectively. We prove that this stochastic sequence can be studied under the general setting provided by the multitype branching process theory. Probabilistic properties and limiting results are then established. As application, we present an illustrative example about the population dynamics of biological systems formed by long-lived raptor colonies. Full article

(This article belongs to the Special Issue Stochastic Statistics and Modeling)

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10 pages, 325 KiB

Open AccessArticle

Option Pricing under Double Heston Jump-Diffusion Model with Approximative Fractional Stochastic Volatility

by Ying Chang, Yiming Wang and Sumei Zhang

Mathematics 2021, 9(2), 126; https://doi.org/10.3390/math9020126 - 8 Jan 2021

Cited by 7 | Viewed by 4063

Abstract

Based on the present studies about the application of approximative fractional Brownian motion in the European option pricing models, our goal in the article is that we adopt the creative model by adding approximative fractional stochastic volatility to double Heston model with jumps [...] Read more.

Based on the present studies about the application of approximative fractional Brownian motion in the European option pricing models, our goal in the article is that we adopt the creative model by adding approximative fractional stochastic volatility to double Heston model with jumps since approximative fractional Brownian motion is more proper for application than Brownian motion in building option pricing models based on financial market data. We are the first to adopt the creative model. We derive the pricing formula for the options and the formula for the characteristic function. We also estimate the parameters with the loss function for the model and two nested models and compare the performance among those models based on the market data. The outcome illustrates that the model offers the best performance among the three models. It demonstrates that approximative fractional Brownian motion is more proper for application than Brownian motion. Full article

(This article belongs to the Special Issue Stochastic Statistics and Modeling)

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14 pages, 296 KiB

Open AccessArticle

Estimating General Parameters from Non-Probability Surveys Using Propensity Score Adjustment

by Luis Castro-Martín, María del Mar Rueda and Ramón Ferri-García

Mathematics 2020, 8(11), 2096; https://doi.org/10.3390/math8112096 - 23 Nov 2020

Cited by 8 | Viewed by 2679

Abstract

This study introduces a general framework on inference for a general parameter using nonprobability survey data when a probability sample with auxiliary variables, common to both samples, is available. The proposed framework covers parameters from inequality measures and distribution function estimates but the [...] Read more.

This study introduces a general framework on inference for a general parameter using nonprobability survey data when a probability sample with auxiliary variables, common to both samples, is available. The proposed framework covers parameters from inequality measures and distribution function estimates but the scope of the paper is broader. We develop a rigorous framework for general parameter estimation by solving survey weighted estimating equations which involve propensity score estimation for units in the non-probability sample. This development includes the expression of the variance estimator, as well as some alternatives which are discussed under the proposed framework. We carried a simulation study using data from a real-world survey, on which the application of the estimation methods showed the effectiveness of the proposed design-based inference on several general parameters. Full article

(This article belongs to the Special Issue Stochastic Statistics and Modeling)

15 pages, 530 KiB

Open AccessArticle

New Modeling Approaches Based on Varimax Rotation of Functional Principal Components

by Christian Acal, Ana M. Aguilera and Manuel Escabias

Mathematics 2020, 8(11), 2085; https://doi.org/10.3390/math8112085 - 22 Nov 2020

Cited by 52 | Viewed by 6227

Abstract

Functional Principal Component Analysis (FPCA) is an important dimension reduction technique to interpret the main modes of functional data variation in terms of a small set of uncorrelated variables. The principal components can not always be simply interpreted and rotation is one of [...] Read more.

Functional Principal Component Analysis (FPCA) is an important dimension reduction technique to interpret the main modes of functional data variation in terms of a small set of uncorrelated variables. The principal components can not always be simply interpreted and rotation is one of the main solutions to improve the interpretation. In this paper, two new functional Varimax rotation approaches are introduced. They are based on the equivalence between FPCA of basis expansion of the sample curves and Principal Component Analysis (PCA) of a transformation of the matrix of basis coefficients. The first approach consists of a rotation of the eigenvectors that preserves the orthogonality between the eigenfunctions but the rotated principal component scores are not uncorrelated. The second approach is based on rotation of the loadings of the standardized principal component scores that provides uncorrelated rotated scores but non-orthogonal eigenfunctions. A simulation study and an application with data from the curves of infections by COVID-19 pandemic in Spain are developed to study the performance of these methods by comparing the results with other existing approaches. Full article

(This article belongs to the Special Issue Stochastic Statistics and Modeling)

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19 pages, 335 KiB

Open AccessArticle

Distributed Fusion Estimation with Sensor Gain Degradation and Markovian Delays

by María Jesús García-Ligero, Aurora Hermoso-Carazo and Josefa Linares-Pérez

Mathematics 2020, 8(11), 1948; https://doi.org/10.3390/math8111948 - 4 Nov 2020

Cited by 4 | Viewed by 1760

Abstract

This paper investigates the distributed fusion estimation of a signal for a class of multi-sensor systems with random uncertainties both in the sensor outputs and during the transmission connections. The measured outputs are assumed to be affected by multiplicative noises, which degrade the [...] Read more.

This paper investigates the distributed fusion estimation of a signal for a class of multi-sensor systems with random uncertainties both in the sensor outputs and during the transmission connections. The measured outputs are assumed to be affected by multiplicative noises, which degrade the signal, and delays may occur during transmission. These uncertainties are commonly described by means of independent Bernoulli random variables. In the present paper, the model is generalised in two directions:

(i)

at each sensor, the degradation in the measurements is modelled by sequences of random variables with arbitrary distribution over the interval [0, 1];

(i i)

transmission delays are described using three-state homogeneous Markov chains (Markovian delays), thus modelling dependence at different sampling times. Assuming that the measurement noises are correlated and cross-correlated at both simultaneous and consecutive sampling times, and that the evolution of the signal process is unknown, we address the problem of signal estimation in terms of covariances, using the following distributed fusion method. First, the local filtering and fixed-point smoothing algorithms are obtained by an innovation approach. Then, the corresponding distributed fusion estimators are obtained as a matrix-weighted linear combination of the local ones, using the mean squared error as the criterion of optimality. Finally, the efficiency of the algorithms obtained, measured by estimation error covariance matrices, is shown by a numerical simulation example. Full article

(This article belongs to the Special Issue Stochastic Statistics and Modeling)

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11 pages, 403 KiB

Open AccessArticle

Asymptotically Normal Estimators of the Gerber-Shiu Function in Classical Insurance Risk Model

by Wen Su and Wenguang Yu

Mathematics 2020, 8(10), 1638; https://doi.org/10.3390/math8101638 - 23 Sep 2020

Cited by 6 | Viewed by 1999

Abstract

Nonparametric estimation of the Gerber-Shiu function is a popular topic in insurance risk theory. Zhang and Su (2018) proposed a novel method for estimating the Gerber-Shiu function in classical insurance risk model by Laguerre series expansion based on the claim number and claim [...] Read more.

Nonparametric estimation of the Gerber-Shiu function is a popular topic in insurance risk theory. Zhang and Su (2018) proposed a novel method for estimating the Gerber-Shiu function in classical insurance risk model by Laguerre series expansion based on the claim number and claim sizes of sample. However, whether the estimators are asymptotically normal or not is unknown. In this paper, we give the details to verify the asymptotic normality of these estimators and present some simulation examples to support our result. Full article

(This article belongs to the Special Issue Stochastic Statistics and Modeling)

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