scholarly journals Powers of the Stochastic Gompertz and Lognormal Diffusion Processes, Statistical Inference and Simulation

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 588 ◽  
Author(s):  
Eva María Ramos-Ábalos ◽  
Ramón Gutiérrez-Sánchez ◽  
Ahmed Nafidi
Keyword(s):  
Diffusion Process ◽  
Statistical Inference ◽  
Diffusion Processes ◽  
Transition Probability ◽  
Estimation Method ◽  
Simulated Data ◽  
Likelihood Estimation ◽  
Transition Probability Density ◽  
Ergodic Properties ◽  
New Family

In this paper, we study a new family of Gompertz processes, defined by the power of the homogeneous Gompertz diffusion process, which we term the powers of the stochastic Gompertz diffusion process. First, we show that this homogenous Gompertz diffusion process is stable, by power transformation, and determine the probabilistic characteristics of the process, i.e., its analytic expression, the transition probability density function and the trend functions. We then study the statistical inference in this process. The parameters present in the model are studied by using the maximum likelihood estimation method, based on discrete sampling, thus obtaining the expression of the likelihood estimators and their ergodic properties. We then obtain the power process of the stochastic lognormal diffusion as the limit of the Gompertz process being studied and go on to obtain all the probabilistic characteristics and the statistical inference. Finally, the proposed model is applied to simulated data.

scholarly journals Restricted Gompertz-Type Diffusion Processes with Periodic Regulation Functions

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 555 ◽  
Author(s):  
Virginia Giorno ◽  
Amelia G. Nobile
Keyword(s):  
Diffusion Process ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Transition Probability ◽  
Passage Time ◽  
Time Dependent ◽  
Transition Probability Density ◽  
Special Boundaries ◽  
Special Cases ◽  
Regulation Functions

We consider two different time-inhomogeneous diffusion processes useful to model the evolution of a population in a random environment. The first is a Gompertz-type diffusion process with time-dependent growth intensity, carrying capacity and noise intensity, whose conditional median coincides with the deterministic solution. The second is a shifted-restricted Gompertz-type diffusion process with a reflecting condition in zero state and with time-dependent regulation functions. For both processes, we analyze the transient and the asymptotic behavior of the transition probability density functions and their conditional moments. Particular attention is dedicated to the first-passage time, by deriving some closed form for its density through special boundaries. Finally, special cases of periodic regulation functions are discussed.

scholarly journals The modified beta transmuted family of distributions with applications using the exponential distribution

PLoS ONE ◽  
2021 ◽  
Vol 16 (11) ◽  
pp. e0258512
Author(s):  
Phillip Oluwatobi Awodutire ◽  
Oluwafemi Samson Balogun ◽  
Akintayo Kehinde Olapade ◽  
Ethelbert Chinaka Nduka
Keyword(s):  
Exponential Distribution ◽  
Real Life ◽  
Estimation Method ◽  
Likelihood Estimation ◽  
Life Time ◽  
Time Data ◽  
New Family ◽  
Special Cases ◽  
The Family ◽  
Family Of Distributions

In this work, a new family of distributions, which extends the Beta transmuted family, was obtained, called the Modified Beta Transmuted Family of distribution. This derived family has the Beta Family of Distribution and the Transmuted family of distribution as subfamilies. The Modified beta transmuted frechet, modified beta transmuted exponential, modified beta transmuted gompertz and modified beta transmuted lindley were obtained as special cases. The analytical expressions were studied for some statistical properties of the derived family of distribution which includes the moments, moments generating function and order statistics. The estimates of the parameters of the family were obtained using the maximum likelihood estimation method. Using the exponential distribution as a baseline for the family distribution, the resulting distribution (modified beta transmuted exponential distribution) was studied and its properties. The modified beta transmuted exponential distribution was applied to a real life time data to assess its flexibility in which the results shows a better fit when compared to some competitive models.

Transition probability density of a certain diffusion process concentrated on a finite spatial interval

1992 ◽  
Vol 29 (2) ◽  
pp. 334-342
Author(s):  
A. Milian
Keyword(s):  
Diffusion Process ◽  
Probability Density ◽  
Transition Probability ◽  
Transition Probability Density ◽  
One Dimensional ◽  
Spatial Interval

We show that under some assumptions a diffusion process satisfying a one-dimensional Itô's equation has a transition probability density concentrated on a finite spatial interval. We give a formula for this density.

scholarly journals Time-Inhomogeneous Feller-Type Diffusion Process in Population Dynamics

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1879
Author(s):  
Virginia Giorno ◽  
Amelia G. Nobile
Keyword(s):  
Diffusion Process ◽  
Transition Probability ◽  
Death Process ◽  
Continuous Approximation ◽  
Periodic Functions ◽  
Transition Probability Density ◽  
Asymptotic Behaviors ◽  
Conditional Moments ◽  
Intensity Functions ◽  
Random Fluctuations

The time-inhomogeneous Feller-type diffusion process, having infinitesimal drift α(t)x+β(t) and infinitesimal variance 2r(t)x, with a zero-flux condition in the zero-state, is considered. This process is obtained as a continuous approximation of a birth-death process with immigration. The transition probability density function and the related conditional moments, with their asymptotic behaviors, are determined. Special attention is paid to the cases in which the intensity functions α(t), β(t), r(t) exhibit some kind of periodicity due to seasonal immigration, regular environmental cycles or random fluctuations. Various numerical computations are performed to illustrate the role played by the periodic functions.

scholarly journals A Symmetry-Based Approach for First-Passage-Times of Gauss-Markov Processes through Daniels-Type Boundaries

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 279
Author(s):  
Enrica Pirozzi
Keyword(s):  
Probability Density ◽  
Markov Processes ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Transition Probability ◽  
Passage Time ◽  
First Passage ◽  
Transition Probability Density ◽  
Symmetry Properties ◽  
Time Density

Symmetry properties of the Brownian motion and of some diffusion processes are useful to specify the probability density functions and the first passage time density through specific boundaries. Here, we consider the class of Gauss-Markov processes and their symmetry properties. In particular, we study probability densities of such processes in presence of a couple of Daniels-type boundaries, for which closed form results exit. The main results of this paper are the alternative proofs to characterize the transition probability density between the two boundaries and the first passage time density exploiting exclusively symmetry properties. Explicit expressions are provided for Wiener and Ornstein-Uhlenbeck processes.

scholarly journals A Stochastic Lomax Diffusion Process: Statistical Inference and Application

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 100
Author(s):  
Ahmed Nafidi ◽  
Ilyasse Makroz ◽  
Ramón Gutiérrez Sánchez
Keyword(s):  
Diffusion Process ◽  
Density Function ◽  
Transition Probability ◽  
Transition Probability Density ◽  
Trend Function ◽  
Non Linear Equation ◽  
Proposed Model ◽  
Adolescent Fertility ◽  
Trend Functions ◽  
Annealing Method

In this paper, we discuss a new stochastic diffusion process in which the trend function is proportional to the Lomax density function. This distribution arises naturally in the studies of the frequency of extremely rare events. We first consider the probabilistic characteristics of the proposed model, including its analytic expression as the unique solution to a stochastic differential equation, the transition probability density function together with the conditional and unconditional trend functions. Then, we present a method to address the problem of parameter estimation using maximum likelihood with discrete sampling. This estimation requires the solution of a non-linear equation, which is achieved via the simulated annealing method. Finally, we apply the proposed model to a real-world example concerning adolescent fertility rate in Morocco.

scholarly journals A Generalized Modification of the Kumaraswamy Distribution for Modeling and Analyzing Real-Life Data

2020 ◽  
Vol 8 (2) ◽  
pp. 521-548
Author(s):  
Rafid Alshkaki
Keyword(s):  
Real Life ◽  
Estimation Method ◽  
Simulated Data ◽  
Likelihood Estimation ◽  
Data Sets ◽  
Kumaraswamy Distribution ◽  
Life Data ◽  
Real Life Data ◽  
Beta Power ◽  
Special Cases

In this paper, a generalized modification of the Kumaraswamy distribution is proposed, and its distributional and characterizing properties are studied. This distribution is closed under scaling and exponentiation, and has some well-known distributions as special cases, such as the generalized uniform, triangular, beta, power function, Minimax, and some other Kumaraswamy related distributions. Moment generating function, Lorenz and Bonferroni curves, with its moments consisting of the mean, variance, moments about the origin, harmonic, incomplete, probability weighted, L, and trimmed L moments, are derived. The maximum likelihood estimation method is used for estimating its parameters and applied to six different simulated data sets of this distribution, in order to check the performance of the estimation method through the estimated parameters mean squares errors computed from the different simulated sample sizes. Finally, four real-life data sets are used to illustrate the usefulness and the flexibility of this distribution in application to real-life data.  

Parameter Estimation of Limited Failure Population Model With a Weibull Underlying Distribution

2020 ◽  
Vol 6 (2) ◽  
Author(s):  
Themistoklis Koutsellis ◽  
Zissimos P. Mourelatos
Keyword(s):  
Statistical Inference ◽  
Population Model ◽  
Likelihood Function ◽  
Simulated Data ◽  
Likelihood Estimation ◽  
Data Driven ◽  
Estimation Methods ◽  
Underlying Distribution ◽  
Function Fitting ◽  
Failure Statistics

Abstract For many data-driven reliability problems, the population is not homogeneous; i.e., its statistics are not described by a unimodal distribution. Also, the interval of observation may not be long enough to capture the failure statistics. A limited failure population (LFP) consists of two subpopulations, a defective and a nondefective one, with well-separated modes of the two underlying distributions. In reliability and warranty forecasting applications, the estimation of the number of defective units and the estimation of the parameters of the underlying distribution are very important. Among various estimation methods, the maximum likelihood estimation (MLE) approach is the most widely used. Its likelihood function, however, is often incomplete, resulting in an erroneous statistical inference. In this paper, we estimate the parameters of a LFP analytically using a rational function fitting (RFF) method based on the Weibull probability plot (WPP) of observed data. We also introduce a censoring factor (CF) to assess how sufficient the number of collected data is for statistical inference. The proposed RFF method is compared with existing MLE approaches using simulated data and data related to automotive warranty forecasting.

Transition probability density of a certain diffusion process concentrated on a finite spatial interval

1992 ◽  
Vol 29 (02) ◽  
pp. 334-342
Author(s):  
A. Milian
Keyword(s):  
Diffusion Process ◽  
Probability Density ◽  
Transition Probability ◽  
Transition Probability Density ◽  
One Dimensional ◽  
Spatial Interval

We show that under some assumptions a diffusion process satisfying a one-dimensional Itô's equation has a transition probability density concentrated on a finite spatial interval. We give a formula for this density.

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