scholarly journals On Local Linear Approximations to Diffusion Processes

2011 ◽  
Vol 2011 ◽  
pp. 1-26
Author(s):  
X. L. Duan ◽  
Z. M. Qian ◽  
W. A. Zheng
Keyword(s):  
Maximum Likelihood ◽  
Diffusion Processes ◽  
Gaussian Approximation ◽  
Diffusion Models ◽  
Likelihood Method ◽  
Unknown Parameters ◽  
Likelihood Functions ◽  
The Past ◽  
Local Linear ◽  
Exact Maximum Likelihood

Diffusion models have been used extensively in many applications. These models, such as those used in the financial engineering, usually contain unknown parameters which we wish to determine. One way is to use the maximum likelihood method with discrete samplings to devise statistics for unknown parameters. In general, the maximum likelihood functions for diffusion models are not available, hence it is difficult to derive the exact maximum likelihood estimator (MLE). There are many different approaches proposed by various authors over the past years, see, for example, the excellent books and Kutoyants (2004), Liptser and Shiryayev (1977), Kushner and Dupuis (2002), and Prakasa Rao (1999), and also the recent works by Aït-Sahalia (1999), (2004), (2002), and so forth. Shoji and Ozaki (1998; see also Shoji and Ozaki (1995) and Shoji and Ozaki (1997)) proposed a simple local linear approximation. In this paper, among other things, we show that Shoji's local linear Gaussian approximation indeed yields a good MLE.

scholarly journals Two Stochastic Differential Equations for Modeling Oscillabolastic-Type Behavior

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 155
Author(s):  
Antonio Barrera ◽  
Patricia Román-Román ◽  
Francisco Torres-Ruiz
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Maximum Likelihood ◽  
Numerical Methods ◽  
Stochastic Models ◽  
Maximum Likelihood Method ◽  
Diffusion Processes ◽  
Oscillatory Behavior ◽  
Likelihood Method ◽  
Systems Of Equations

Stochastic models based on deterministic ones play an important role in the description of growth phenomena. In particular, models showing oscillatory behavior are suitable for modeling phenomena in several application areas, among which the field of biomedicine stands out. The oscillabolastic growth curve is an example of such oscillatory models. In this work, two stochastic models based on diffusion processes related to the oscillabolastic curve are proposed. Each of them is the solution of a stochastic differential equation obtained by modifying, in a different way, the original ordinary differential equation giving rise to the curve. After obtaining the distributions of the processes, the problem of estimating the parameters is analyzed by means of the maximum likelihood method. Due to the parametric structure of the processes, the resulting systems of equations are quite complex and require numerical methods for their resolution. The problem of obtaining initial solutions is addressed and a strategy is established for this purpose. Finally, a simulation study is carried out.

scholarly journals Parameter estimation in a Black Scholes

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 117-122
Author(s):  
Mustafa Bayram ◽  
Buyukoz Orucova ◽  
Tugcem Partal
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Parametric Method ◽  
Estimation Method ◽  
Parametric Estimation ◽  
Likelihood Method ◽  
Observation Data ◽  
Unknown Parameters ◽  
Black Scholes ◽  
Non Parametric

In this paper we discuss parameter estimation in black scholes model. A non-parametric estimation method and well known maximum likelihood estimator are considered. Our aim is to estimate the unknown parameters for stochastic differential equation with discrete time observation data. In simulation study we compare the non-parametric method with maximum likelihood method using stochastic numerical scheme named with Euler Maruyama.

scholarly journals Multivariate Skew-Power-Normal Distributions: Properties and Associated Inference

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1509
Author(s):  
Guillermo Martínez-Flórez ◽  
Artur J. Lemonte ◽  
Hugo S. Salinas
Keyword(s):  
Maximum Likelihood ◽  
Normal Distribution ◽  
Real Data ◽  
Likelihood Method ◽  
Model Parameters ◽  
Unknown Parameters ◽  
Multivariate Distributions ◽  
Likelihood Approach ◽  
Information Matrices ◽  
Univariate Distribution

The univariate power-normal distribution is quite useful for modeling many types of real data. On the other hand, multivariate extensions of this univariate distribution are not common in the statistic literature, mainly skewed multivariate extensions that can be bimodal, for example. In this paper, based on the univariate power-normal distribution, we extend the univariate power-normal distribution to the multivariate setup. Structural properties of the new multivariate distributions are established. We consider the maximum likelihood method to estimate the unknown parameters, and the observed and expected Fisher information matrices are also derived. Monte Carlo simulation results indicate that the maximum likelihood approach is quite effective to estimate the model parameters. An empirical application of the proposed multivariate distribution to real data is provided for illustrative purposes.

scholarly journals An algorithm for the assessment of several small rates

2012 ◽  
Vol 53 ◽  
Author(s):  
Leonidas Sakalauskas ◽  
Ingrida Vaičiulytė
Keyword(s):  
Maximum Likelihood ◽  
Bayesian Approach ◽  
Maximum Likelihood Method ◽  
Rare Events ◽  
Maximum Likelihood Estimates ◽  
Likelihood Method ◽  
Model Parameters ◽  
Unknown Parameters ◽  
Empirical Bayesian ◽  
Empirical Bayesian Approach

The present paper describes the empirical Bayesian approach applied in the estimation of several small rates. Modeling by empirical Bayesian approach the probabilities of several rare events, it is assumed that the frequencies of events follow to Poisson’s law with different parameters, which are correlated Gaussian random values. The unknown parameters are estimated by the maximum likelihood method computing the integrals appeared here by Hermite–Gauss quadratures. The equations derived that are satisfied by maximum likelihood estimates of model parameters.

Semiparametric estimation for nonparametric frailty models using nonparametric maximum likelihood approach

2021 ◽  
pp. 096228022110370
Author(s):  
Chew-Seng Chee ◽  
Il Do Ha ◽  
Byungtae Seo ◽  
Youngjo Lee
Keyword(s):  
Maximum Likelihood ◽  
Regression Coefficient ◽  
Model Fitting ◽  
Estimation Procedure ◽  
Bias Reduction ◽  
Frailty Models ◽  
Likelihood Method ◽  
Nonparametric Maximum Likelihood ◽  
Unknown Parameters ◽  
Baseline Hazard

A consequence of using a parametric frailty model with nonparametric baseline hazard for analyzing clustered time-to-event data is that its regression coefficient estimates could be sensitive to the underlying frailty distribution. Recently, there has been a proposal for specifying both the baseline hazard and the frailty distribution nonparametrically, and estimating the unknown parameters by the maximum penalized likelihood method. Instead, in this paper, we propose the nonparametric maximum likelihood method for a general class of nonparametric frailty models, i.e. models where the frailty distribution is completely unspecified but the baseline hazard can be either parametric or nonparametric. The implementation of the estimation procedure can be based on a combination of either the Broyden–Fletcher–Goldfarb–Shanno or expectation-maximization algorithm and the constrained Newton algorithm with multiple support point inclusion. Simulation studies to investigate the performance of estimation of a regression coefficient by several different model-fitting methods were conducted. The simulation results show that our proposed regression coefficient estimator generally gives a reasonable bias reduction when the number of clusters is increased under various frailty distributions. Our proposed method is also illustrated with two data examples.

Adaptive control of a production-inventory system

1981 ◽  
Vol 18 (1) ◽  
pp. 204-215 ◽  
Author(s):  
Bharat T. Doshi
Keyword(s):  
Maximum Likelihood ◽  
Asymptotic Properties ◽  
Inventory System ◽  
Maximum Likelihood Estimates ◽  
Likelihood Method ◽  
Unknown Parameters ◽  
True Value ◽  
Adaptive Policy ◽  
Production Inventory ◽  
Production Inventory System

We study a production-inventory system in which input is deterministic, and its rate is the controlled parameter. The output is a compound Poisson process with exponential jump-size distribution. The parameters of the output process are unknown. The following adaptive policy is used: At each time t the unknown parameters are estimated by maximum likelihood method and the input rate which would be optimal if these were the true values of the parameter is used. The issues investigated are (1) the convergence of maximum likelihood estimates to the true value, and (2) the asymptotic properties of the cost under the adaptive policy.

Adaptive control of a production-inventory system

1981 ◽  
Vol 18 (01) ◽  
pp. 204-215
Author(s):  
Bharat T. Doshi
Keyword(s):  
Maximum Likelihood ◽  
Asymptotic Properties ◽  
Inventory System ◽  
Maximum Likelihood Estimates ◽  
Likelihood Method ◽  
Unknown Parameters ◽  
True Value ◽  
Adaptive Policy ◽  
Production Inventory ◽  
Production Inventory System

We study a production-inventory system in which input is deterministic, and its rate is the controlled parameter. The output is a compound Poisson process with exponential jump-size distribution. The parameters of the output process are unknown. The following adaptive policy is used: At each time t the unknown parameters are estimated by maximum likelihood method and the input rate which would be optimal if these were the true values of the parameter is used. The issues investigated are (1) the convergence of maximum likelihood estimates to the true value, and (2) the asymptotic properties of the cost under the adaptive policy.

scholarly journals Quartile Method Estimation of Two-Parameter Exponential Distribution Data with Outliers

2016 ◽  
Vol 5 (5) ◽  
pp. 12
Author(s):  
Entisar A. Elgmati ◽  
Nadia B. Gregni
Keyword(s):  
Maximum Likelihood ◽  
Exponential Distribution ◽  
Simulation Study ◽  
Maximum Likelihood Method ◽  
Likelihood Method ◽  
Distribution Data ◽  
Unknown Parameters ◽  
First Order ◽  
Minimum Statistics ◽  
Two Parameter

Several methods have been used to estimate the unknown parameters in the two-parameter exponential distribution. Here we have considered two of these methods, maximum likelihood method and median-first order statistics method. However, in the presence of outliers these methods are not valid. In this paper we propose two approaches that deal with this situation. The idea is based on using first and third quartile instead of the minimum statistics. We investigated the parameters estimate using these methods through simulation study. The new method gives similar results under the normal situation and much better results when the data has outliers.

scholarly journals Multiple Tests Based on a Gaussian Approximation of the Unitary Events Method with Delayed Coincidence Count

2014 ◽  
Vol 26 (7) ◽  
pp. 1408-1454 ◽  
Author(s):  
Christine Tuleau-Malot ◽  
Amel Rouis ◽  
Franck Grammont ◽  
Patricia Reynaud-Bouret
Keyword(s):  
Point Processes ◽  
Gaussian Approximation ◽  
Real Data ◽  
Spike Trains ◽  
Previous Method ◽  
Unknown Parameters ◽  
Multiple Tests ◽  
The Past ◽  
False Discovery ◽  
Unitary Events

The unitary events (UE) method is one of the most popular and efficient methods used over the past decade to detect patterns of coincident joint spike activity among simultaneously recorded neurons. The detection of coincidences is usually based on binned coincidence count (Grün, 1996 ), which is known to be subject to loss in synchrony detection (Grün, Diesmann, Grammont, Riehle, & Aertsen, 1999 ). This defect has been corrected by the multiple shift coincidence count (Grün et al., 1999 ). The statistical properties of this count have not been further investigated until this work, the formula being more difficult to deal with than the original binned count. First, we propose a new notion of coincidence count, the delayed coincidence count, which is equal to the multiple shift coincidence count when discretized point processes are involved as models for the spike trains. Moreover, it generalizes this notion to nondiscretized point processes, allowing us to propose a new gaussian approximation of the count. Since unknown parameters are involved in the approximation, we perform a plug-in step, where unknown parameters are replaced by estimated ones, leading to a modification of the approximating distribution. Finally the method takes the multiplicity of the tests into account via a Benjamini and Hochberg approach (Benjamini & Hochberg, 1995 ), to guarantee a prescribed control of the false discovery rate. We compare our new method, MTGAUE (multiple tests based on a gaussian approximation of the unitary events) and the UE method proposed in Grün et al. ( 1999 ) over various simulations, showing that MTGAUE extends the validity of the previous method. In particular, MTGAUE is able to detect both profusion and lack of coincidences with respect to the independence case and is robust to changes in the underlying model. Furthermore MTGAUE is applied on real data.

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