Estimation of the mean and the initial probabilities of a branching process

1974 ◽  
Vol 11 (4) ◽  
pp. 687-694 ◽  
Author(s):  
Jean-Pierre Dion
Keyword(s):  
Maximum Likelihood ◽  
Confidence Intervals ◽  
Branching Process ◽  
Maximum Likelihood Estimators ◽  
Asymptotic Distributions ◽  
The Mean ◽  
Watson Process

In this article, we consider maximum likelihood estimators of the initial probabilities and the mean of a supercritical Galton-Watson process; we find for these estimators, as well as for the Lotka-Nagaev estimator of the mean, the asymptotic distributions and deduce confidence intervals. As these results hold even if the independence between individuals of the same generation is not satisfied, application to living populations may be considered.

Estimation of the mean and the initial probabilities of a branching process

1974 ◽  
Vol 11 (04) ◽  
pp. 687-694 ◽  
Author(s):  
Jean-Pierre Dion
Keyword(s):  
Maximum Likelihood ◽  
Confidence Intervals ◽  
Branching Process ◽  
Maximum Likelihood Estimators ◽  
Asymptotic Distributions ◽  
The Mean ◽  
Watson Process

In this article, we consider maximum likelihood estimators of the initial probabilities and the mean of a supercritical Galton-Watson process; we find for these estimators, as well as for the Lotka-Nagaev estimator of the mean, the asymptotic distributions and deduce confidence intervals. As these results hold even if the independence between individuals of the same generation is not satisfied, application to living populations may be considered.

Maximum likelihood estimation for branching processes with immigration

1981 ◽  
Vol 13 (3) ◽  
pp. 498-509 ◽  
Author(s):  
B. R. Bhat ◽  
S. R. Adke
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Continuous Time ◽  
Strong Consistency ◽  
Branching Process ◽  
Branching Processes ◽  
Maximum Likelihood Estimators ◽  
Likelihood Estimation ◽  
Asymptotic Distributions ◽  
Branching Process With Immigration

This paper establishes the strong consistency of the maximum likelihood estimators of the parameters of discrete- and continuous-time Markov branching processes with immigration. The asymptotic distributions of the maximum likelihood estimators of the parameters of a Galton–Watson branching process with immigration are also obtained.

Maximum likelihood estimation for branching processes with immigration

1981 ◽  
Vol 13 (03) ◽  
pp. 498-509 ◽  
Author(s):  
B. R. Bhat ◽  
S. R. Adke
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Continuous Time ◽  
Strong Consistency ◽  
Branching Process ◽  
Branching Processes ◽  
Maximum Likelihood Estimators ◽  
Likelihood Estimation ◽  
Asymptotic Distributions ◽  
Branching Process With Immigration

This paper establishes the strong consistency of the maximum likelihood estimators of the parameters of discrete- and continuous-time Markov branching processes with immigration. The asymptotic distributions of the maximum likelihood estimators of the parameters of a Galton–Watson branching process with immigration are also obtained.

Inference on P(Y

2020 ◽  
Vol 72 (2) ◽  
pp. 89-110
Author(s):  
Manoj Chacko ◽  
Shiny Mathew
Keyword(s):  
Maximum Likelihood ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Maximum Likelihood Estimators ◽  
Record Values ◽  
Asymptotic Distributions ◽  
Bayes Estimators ◽  
Generalized Pareto ◽  
Pareto Distributions ◽  
Generalized Pareto Distributions

In this article, the estimation of [Formula: see text] is considered when [Formula: see text] and [Formula: see text] are two independent generalized Pareto distributions. The maximum likelihood estimators and Bayes estimators of [Formula: see text] are obtained based on record values. The Asymptotic distributions are also obtained together with the corresponding confidence interval of [Formula: see text]. AMS 2000 subject classification: 90B25

On the parity of individuals in a branching process

1976 ◽  
Vol 13 (02) ◽  
pp. 219-230 ◽  
Author(s):  
J. Gani ◽  
I. W. Saunders
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Branching Process ◽  
Yeast Cells ◽  
Death Process ◽  
Second Moments ◽  
The Mean ◽  
Watson Process ◽  
Number Of Cells ◽  
Birth Death

This paper is concerned with the parity of a population of yeast cells, each of which may bud, not bud or die. Two multitype models are considered: a Galton-Watson process in discrete time, and its analogous birth-death process in continuous time. The mean number of cells with parity 0, 1, 2, … is obtained in both cases; some simple results are also derived for the second moments of the two processes.

scholarly journals Causal Inference for a Population of Causally Connected Units

2014 ◽  
Vol 2 (1) ◽  
pp. 13-74 ◽  
Author(s):  
Mark J. van der Laan
Keyword(s):  
Data Structure ◽  
Maximum Likelihood ◽  
Longitudinal Data ◽  
Maximum Likelihood Estimators ◽  
Time Dependent ◽  
Convergence Theory ◽  
Nuisance Parameters ◽  
Target Parameter ◽  
Targeted Maximum Likelihood ◽  
The Mean

AbstractSuppose that we observe a population of causally connected units. On each unit at each time-point on a grid we observe a set of other units the unit is potentially connected with, and a unit-specific longitudinal data structure consisting of baseline and time-dependent covariates, a time-dependent treatment, and a final outcome of interest. The target quantity of interest is defined as the mean outcome for this group of units if the exposures of the units would be probabilistically assigned according to a known specified mechanism, where the latter is called a stochastic intervention. Causal effects of interest are defined as contrasts of the mean of the unit-specific outcomes under different stochastic interventions one wishes to evaluate. This covers a large range of estimation problems from independent units, independent clusters of units, and a single cluster of units in which each unit has a limited number of connections to other units. The allowed dependence includes treatment allocation in response to data on multiple units and so called causal interference as special cases. We present a few motivating classes of examples, propose a structural causal model, define the desired causal quantities, address the identification of these quantities from the observed data, and define maximum likelihood based estimators based on cross-validation. In particular, we present maximum likelihood based super-learning for this network data. Nonetheless, such smoothed/regularized maximum likelihood estimators are not targeted and will thereby be overly bias w.r.t. the target parameter, and, as a consequence, generally not result in asymptotically normally distributed estimators of the statistical target parameter.To formally develop estimation theory, we focus on the simpler case in which the longitudinal data structure is a point-treatment data structure. We formulate a novel targeted maximum likelihood estimator of this estimand and show that the double robustness of the efficient influence curve implies that the bias of the targeted minimum loss-based estimation (TMLE) will be a second-order term involving squared differences of two nuisance parameters. In particular, the TMLE will be consistent if either one of these nuisance parameters is consistently estimated. Due to the causal dependencies between units, the data set may correspond with the realization of a single experiment, so that establishing a (e.g. normal) limit distribution for the targeted maximum likelihood estimators, and corresponding statistical inference, is a challenging topic. We prove two formal theorems establishing the asymptotic normality using advances in weak-convergence theory. We conclude with a discussion and refer to an accompanying technical report for extensions to general longitudinal data structures.

Sign in / Sign up

Export Citation Format

Share Document