Maximum likelihood estimation for stochastic rth -order reactions

In this paper we derive the probability distributions of the number of molecules and of the lifetime of a molecule in a stochastic rth-order system by direct evaluation of probabilities, avoiding the use of differential-difference equations. Maximum likelihood estimation of the rate constant is based on an observation of the level of the system at time t > 0. We find the asymptotic solution of the likelihood equation for a large initial number of molecules. By comparison with the numerical solution of the likelihood equation, the asymptotic estimator is shown to be a satisfactory approximation for second order reactions which are far from completion.

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A Maximum Entropy Procedure to Solve Likelihood Equations

Entropy ◽

10.3390/e21060596 ◽

2019 ◽

Vol 21 (6) ◽

pp. 596

Author(s):

Antonio Calcagnì ◽

Livio Finos ◽

Gianmarco Altoé ◽

Massimiliano Pastore

Keyword(s):

Logistic Regression ◽

Maximum Likelihood ◽

Maximum Entropy ◽

Probability Distributions ◽

Score Function ◽

Likelihood Estimation ◽

Likelihood Equation ◽

Equation Problem ◽

Standard Procedures ◽

Data Separation

In this article, we provide initial findings regarding the problem of solving likelihood equations by means of a maximum entropy (ME) approach. Unlike standard procedures that require equating the score function of the maximum likelihood problem at zero, we propose an alternative strategy where the score is instead used as an external informative constraint to the maximization of the convex Shannon’s entropy function. The problem involves the reparameterization of the score parameters as expected values of discrete probability distributions where probabilities need to be estimated. This leads to a simpler situation where parameters are searched in smaller (hyper) simplex space. We assessed our proposal by means of empirical case studies and a simulation study, the latter involving the most critical case of logistic regression under data separation. The results suggested that the maximum entropy reformulation of the score problem solves the likelihood equation problem. Similarly, when maximum likelihood estimation is difficult, as is the case of logistic regression under separation, the maximum entropy proposal achieved results (numerically) comparable to those obtained by the Firth’s bias-corrected approach. Overall, these first findings reveal that a maximum entropy solution can be considered as an alternative technique to solve the likelihood equation.

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Maximum likelihood estimation for stochastic rth-order reactions. II

Journal of Applied Probability ◽

10.1017/s0021900200095449 ◽

1973 ◽

Vol 10 (02) ◽

pp. 441-446

Author(s):

John P. Mullooly

Keyword(s):

Maximum Likelihood ◽

Maximum Likelihood Estimation ◽

Waiting Time ◽

Likelihood Estimation ◽

Initial Number ◽

Asymptotically Normal ◽

Reaction Event ◽

Wide Range ◽

Order Case ◽

Exact Maximum Likelihood

In this paper we consider maximum likelihood estimation of the rate constant for stochastic rth-order reactions based on observation of the level of the system at time t > 0. Conditions are found for which the waiting time until the nth. reaction event is asymptotically normal, as the initial number of molecules and the number of reaction events become large. This distributional result is used to derive an approximate estimator which is shown for the second-order case to be close to the exact maximum likelihood estimate over a wide range of percentage completion.

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Maximum-likelihood estimation of the parameters of a four-parameter class of probability distributions

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500003400 ◽

1988 ◽

Vol 31 (2) ◽

pp. 271-283 ◽

Cited By ~ 2

Author(s):

Siegfried H. Lehnigk

Keyword(s):

Maximum Likelihood ◽

Probability Density Function ◽

Maximum Likelihood Estimation ◽

Probability Density ◽

Density Function ◽

Probability Distributions ◽

Likelihood Estimation ◽

Initial Shape ◽

Definition Of ◽

Image Position

We shall concern ourselves with the class of continuous, four-parameter, one-sided probability distributions which can be characterized by the probability density function (pdf) classIt depends on the four parameters: shift c ∈ R, scale b > 0, initial shape p < 1, and terminal shape β > 0. For p ≦ 0, the definition of f(x) can be completed by setting f(c) = β/bΓ(β−1)>0 if p = 0, and f(c) = 0 if p < 0. For 0 < p < 1, f(x) remains undefined at x = c; f(x)↑ + ∞ as x↓c.

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A Study on Properties and Applications of a Lomax Gompertz-Makeham Distribution

Asian Research Journal of Mathematics ◽

10.9734/arjom/2019/v15i430155 ◽

2019 ◽

pp. 1-27

Author(s):

Innocent Boyle Eraikhuemen ◽

Terna Godfrey Ieren ◽

Tajan Mashingil Mabur ◽

Mohammed Sa’ad ◽

Samson Kuje ◽

...

Keyword(s):

Maximum Likelihood ◽

Maximum Likelihood Estimation ◽

Probability Distributions ◽

Real Life ◽

Likelihood Estimation ◽

The Other ◽

Unknown Parameters ◽

Method Of Maximum Likelihood ◽

Compound Model ◽

New Distribution

The article presents an extension of the Gompertz-Makeham distribution using the Lomax generator of probability distributions. This generalization of the Gompertz-Makeham distribution provides a more skewed and flexible compound model called Lomax Gompertz-Makeham distribution. The paper derives and discusses some Mathematical and Statistical properties of the new distribution. The unknown parameters of the new model are estimated via the method of maximum likelihood estimation. In conclusion, the new distribution is applied to two real life datasets together with two other related models to check its flexibility or performance and the results indicate that the proposed extension is more flexible compared to the other two distributions considered in the paper based on the two datasets used.

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Maximum likelihood estimation for stochastic rth-order reactions. II

Journal of Applied Probability ◽

10.2307/3212361 ◽

1973 ◽

Vol 10 (2) ◽

pp. 441-446 ◽

Cited By ~ 2

Author(s):

John P. Mullooly

Keyword(s):

Maximum Likelihood ◽

Maximum Likelihood Estimation ◽

Waiting Time ◽

Likelihood Estimation ◽

Initial Number ◽

Asymptotically Normal ◽

Reaction Event ◽

Wide Range ◽

Order Case ◽

Exact Maximum Likelihood

In this paper we consider maximum likelihood estimation of the rate constant for stochastic rth-order reactions based on observation of the level of the system at time t > 0. Conditions are found for which the waiting time until the nth. reaction event is asymptotically normal, as the initial number of molecules and the number of reaction events become large. This distributional result is used to derive an approximate estimator which is shown for the second-order case to be close to the exact maximum likelihood estimate over a wide range of percentage completion.

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Probabilistic Non-Suppression Model of Electrical Fire in Nuclear Power Plant Using Maximum Likelihood Estimation Method

Korean Society of Hazard Mitigation ◽

10.9798/kosham.2020.20.5.123 ◽

2020 ◽

Vol 20 (5) ◽

pp. 123-134

Author(s):

Sunghyun Kim ◽

Sungsu Lee

Keyword(s):

Maximum Likelihood ◽

Maximum Likelihood Estimation ◽

Nuclear Power ◽

Power Plants ◽

Probability Distributions ◽

Probability Model ◽

Estimation Method ◽

Likelihood Estimation ◽

Fire Event ◽

Adequate Model

Electrical fire is the most significant fire-induced risk in nuclear power plants. In this study, we developed a probabilistic non-suppression model of electric fire by estimating the relevant parameters using the Maximum Likelihood Estimation (MLE) method to secure and improve the realism in Fire PRA in application to the Risk Informed Fire Protection Program. Based on the results of fitting 32 continuous probability distributions into fire incident data, 3-Parameter Weibull probability model was selected as the best fitted and adequate model and was compared with the existing exponential probability model. The comparison shows that the level of model adequacy is improved with decreases in BIC by 17.7%, in RSS by 24.4%, and in MSE by 58.2%. The 3-Parameter Weibull probability model drawn in this study is expected to contribute to the enhancement of the Fire PRA realism for supporting risk-based decision by reflecting real fire event experiences.

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