scholarly journals Invariant Measure for Continuous Open Chain of Contours with Discrete Time

2021 ◽  
Author(s):  
Marina V. Yashina ◽  
Alexander G. Tatashev
Keyword(s):  
Invariant Measure ◽  
Discrete Time ◽  
Open Chain

On Null-Recurrent Markov Chains

1960 ◽  
Vol 12 ◽  
pp. 278-288 ◽  
Author(s):  
John Lamperti
Keyword(s):  
Markov Chain ◽  
Markov Process ◽  
Invariant Measure ◽  
Discrete Time ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Positive Vector ◽  
Scalar Multiple ◽  
Basic Facts ◽  
Image Position

Throughout this paper, the symbol P = [Pij] will represent the transition probability matrix of an irreducible, null-recurrent Markov process in discrete time. Explanation of this terminology and basic facts about such chains may be found in (6, ch. 15). It is known (3) that for each such matrix P there is a unique (except for a positive scalar multiple) positive vector Q = {qi} such that QP = Q, or1this vector is often called the "invariant measure" of the Markov chain.The first problem to be considered in this paper is that of determining for which vectors U(0) = {μi(0)} the vectors U(n) converge, or are summable, to the invariant measure Q, where U(n) = U(0)Pn has components2In § 2, this problem is attacked for general P. The main result is a negative one, and shows how to form U(0) for which U(n) will not be (termwise) Abel summable.

On branching processes allowing immigration

1972 ◽  
Vol 9 (01) ◽  
pp. 24-31 ◽  
Author(s):  
Y. S. Yang
Keyword(s):  
Invariant Measure ◽  
Discrete Time ◽  
Continuous Time ◽  
Branching Processes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Positive Recurrence ◽  
Discrete Time Case ◽  
Necessary And Sufficient

Continuous time one-type branching processes allowing immigration are considered. The invariant measure, which is shown to be unique, is exhibited. From this, a condition for positive recurrence similar to that of Heathcote's in the discrete time case is obtained. For the critical discrete time case, Seneta's sufficient condition for positive recurrence is improved to give a necessary and sufficient condition.

On branching processes allowing immigration

1972 ◽  
Vol 9 (1) ◽  
pp. 24-31 ◽  
Author(s):  
Y. S. Yang
Keyword(s):  
Invariant Measure ◽  
Discrete Time ◽  
Continuous Time ◽  
Branching Processes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Positive Recurrence ◽  
Discrete Time Case ◽  
Necessary And Sufficient

Continuous time one-type branching processes allowing immigration are considered. The invariant measure, which is shown to be unique, is exhibited. From this, a condition for positive recurrence similar to that of Heathcote's in the discrete time case is obtained. For the critical discrete time case, Seneta's sufficient condition for positive recurrence is improved to give a necessary and sufficient condition.

The Effect of Partly Missing Covariates on Statistical Power in Randomized Controlled Trials With Discrete-Time Survival Endpoints

Methodology ◽  
2017 ◽  
Vol 13 (2) ◽  
pp. 41-60
Author(s):  
Shahab Jolani ◽  
Maryam Safarkhani
Keyword(s):  
Randomized Controlled Trials ◽  
Discrete Time ◽  
Treatment Effect ◽  
Survival Data ◽  
Controlled Trials ◽  
Missing Covariates ◽  
Indicator Method ◽  
Imputation Methods ◽  
Randomized Controlled ◽  
Baseline Covariates

Abstract. In randomized controlled trials (RCTs), a common strategy to increase power to detect a treatment effect is adjustment for baseline covariates. However, adjustment with partly missing covariates, where complete cases are only used, is inefficient. We consider different alternatives in trials with discrete-time survival data, where subjects are measured in discrete-time intervals while they may experience an event at any point in time. The results of a Monte Carlo simulation study, as well as a case study of randomized trials in smokers with attention deficit hyperactivity disorder (ADHD), indicated that single and multiple imputation methods outperform the other methods and increase precision in estimating the treatment effect. Missing indicator method, which uses a dummy variable in the statistical model to indicate whether the value for that variable is missing and sets the same value to all missing values, is comparable to imputation methods. Nevertheless, the power level to detect the treatment effect based on missing indicator method is marginally lower than the imputation methods, particularly when the missingness depends on the outcome. In conclusion, it appears that imputation of partly missing (baseline) covariates should be preferred in the analysis of discrete-time survival data.

Sign in / Sign up

Export Citation Format

Share Document