scholarly journals Missing at random: a stochastic process perspective

Biometrika ◽  
2021 ◽  
Author(s):  
D Farewell ◽  
R Daniel ◽  
S Seaman
Keyword(s):  
Stochastic Process ◽  
Missing Data ◽  
Continuous Time ◽  
Missing At Random ◽  
Likelihood Ratios ◽  
Explanatory Variables ◽  
Data Framework ◽  
Stopping Set ◽  
Process Perspective

Abstract We offer a natural and extensible measure-theoretic treatment of missingness at random. Within the standard missing data framework, we give a novel characterization of the observed data as a stopping-set sigma algebra. We demonstrate that the usual missingness at random conditions are equivalent to requiring particular stochastic processes to be adapted to a set-indexed filtration. These measurability conditions ensure the usual factorization of likelihood ratios. We illustrate how the theory extends easily to incorporate explanatory variables, to describe longitudinal data in continuous time, and to admit more general coarsening of observations.

scholarly journals An alternative characterization of MAR in shared parameter models for incomplete longitudinal data and its utilization for sensitivity analysis

2020 ◽  
pp. 1471082X2092711
Author(s):  
Grigorios Papageorgiou ◽  
Dimitris Rizopoulos
Keyword(s):  
Sensitivity Analysis ◽  
Missing Data ◽  
Missing At Random ◽  
Joint Models ◽  
Specific Level ◽  
Conditional Independence Assumption ◽  
Wide Range ◽  
Subject Specific ◽  
Alternative Characterization

Dropout is a common complication in longitudinal studies, especially since the distinction between missing not at random (MNAR) and missing at random (MAR) dropout is intractable. Consequently, one starts with an analysis that is valid under MAR and then performs a sensitivity analysis by considering MNAR departures from it. To this end, specific classes of joint models, such as pattern-mixture models (PMMs) and selection models (SeMs), have been proposed. On the contrary, shared-parameter models (SPMs) have received less attention, possibly because they do not embody a characterization of MAR. A few approaches to achieve MAR in SPMs exist, but are difficult to implement in existing software. In this article, we focus on SPMs for incomplete longitudinal and time-to-dropout data and propose an alternative characterization of MAR by exploiting the conditional independence assumption, under which outcome and missingness are independent given a set of random effects. By doing so, the censoring distribution can be utilized to cover a wide range of assumptions for the missing data mechanism on the subject-specific level. This approach offers substantial advantages over its counterparts and can be easily implemented in existing software. More specifically, it offers flexibility over the assumption for the missing data generating mechanism that governs dropout by allowing subject-specific perturbations of the censoring distribution, whereas in PMMs and SeMs dropout is considered MNAR strictly.

scholarly journals Long-Time Behaviour in a Model of Microtubule Growth

2010 ◽  
Vol 42 (1) ◽  
pp. 268-291
Author(s):  
O. Hryniv ◽  
M. Menshikov
Keyword(s):  
Stochastic Process ◽  
Continuous Time ◽  
Complete Characterization ◽  
Time Behaviour ◽  
Local Growth ◽  
Long Time Behaviour ◽  
Long Time ◽  
Microtubule Growth ◽  
Continuous Time Stochastic Process

We study a continuous-time stochastic process on strings made of two types of particle, whose dynamics mimic the behaviour of microtubules in a living cell; namely, the strings evolve via a competition between (local) growth/shrinking as well as (global) hydrolysis processes. We give a complete characterization of the phase diagram of the model, and derive several criteria of the transient and recurrent regimes for the underlying stochastic process.

scholarly journals Modeling the Energy Harvested by an RF Energy Harvesting System Using Gamma Processes

2019 ◽  
Vol 2019 ◽  
pp. 1-12 ◽  
Author(s):  
Inma T. Castro ◽  
Luis Landesa ◽  
Alberto Serna
Keyword(s):  
Stochastic Process ◽  
Energy Harvesting ◽  
Performance Measures ◽  
Continuous Time ◽  
Gamma Process ◽  
Stochastic Characterization ◽  
Energy Harvesting System ◽  
Gamma Processes ◽  
Continuous Time Stochastic Process

In an Energy Harvesting system (EHS) the gamma process is used to model the electromagnetic energy received from radiofrequency (RF) radiation. The stochastic characterization of the harvested energy as a continuous-time stochastic process, namely, gamma process, is obtained from the Nakagami-m fading model, which describes the signal reception in a large amount of types of radiofrequency channels. Using the gamma process, some performance measures of the EHS system are obtained. Also, a transmission policy subject to different fading conditions is considered.

Characterization of the formation of base sheet paper using spectral moments determined from time-series models

1977 ◽  
Vol 82 (2) ◽  
pp. 261-272
Author(s):  
Warren R. Devries ◽  
S. M. Wu
Keyword(s):  
Stochastic Process ◽  
Time Series ◽  
Continuous Time ◽  
Light Transmission ◽  
Time Series Models ◽  
Flock Size ◽  
Spectral Moments ◽  
Fibre Distribution ◽  
Moments Estimates

The fibre distribution in a sheet of paper, referred to as the formation, is largely the result of turbulence, a stochastic process. Continuous time-series models developed from discrete light-transmission profiles are used to characterize formation. The models are used to obtain explicit expressions for the spectral moments of the profiles. From the moments, estimates of two characteristic lengths of the fibre distribution can be obtained and are interpreted as the average and largest flock size. These lengths are used to develop an index for evaluating the formation of four samples of base sheet paper. The results of this characterization agree with other methods, but this technique has the advantage of providing a physical interpretation of the index.

scholarly journals Long-Time Behaviour in a Model of Microtubule Growth

2010 ◽  
Vol 42 (01) ◽  
pp. 268-291 ◽  
Author(s):  
O. Hryniv ◽  
M. Menshikov
Keyword(s):  
Stochastic Process ◽  
Continuous Time ◽  
Complete Characterization ◽  
Time Behaviour ◽  
Local Growth ◽  
Long Time Behaviour ◽  
Long Time ◽  
Microtubule Growth ◽  
Continuous Time Stochastic Process

We study a continuous-time stochastic process on strings made of two types of particle, whose dynamics mimic the behaviour of microtubules in a living cell; namely, the strings evolve via a competition between (local) growth/shrinking as well as (global) hydrolysis processes. We give a complete characterization of the phase diagram of the model, and derive several criteria of the transient and recurrent regimes for the underlying stochastic process.

scholarly journals Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach

2021 ◽  
Author(s):  
Yves Achdou ◽  
Jiequn Han ◽  
Jean-Michel Lasry ◽  
Pierre-Louis Lions ◽  
Benjamin Moll
Keyword(s):  
Continuous Time ◽  
Wealth Distribution ◽  
Closed Form Solution ◽  
Form Solution ◽  
Heterogeneous Agent ◽  
Saving Behavior ◽  
Income And Wealth Distribution ◽  
Heterogeneous Agent Models ◽  
Special Case

Abstract We recast the Aiyagari-Bewley-Huggett model of income and wealth distribution in continuous time. This workhorse model – as well as heterogeneous agent models more generally – then boils down to a system of partial differential equations, a fact we take advantage of to make two types of contributions. First, a number of new theoretical results: (i) an analytic characterization of the consumption and saving behavior of the poor, particularly their marginal propensities to consume; (ii) a closed-form solution for the wealth distribution in a special case with two income types; (iii) a proof that there is a unique stationary equilibrium if the intertemporal elasticity of substitution is weakly greater than one. Second, we develop a simple, efficient and portable algorithm for numerically solving for equilibria in a wide class of heterogeneous agent models, including – but not limited to – the Aiyagari-Bewley-Huggett model.

scholarly journals Evaluation of Cystic and Solid Renal Lesions with Contrast-Enhanced Ultrasound: A Retrospective Study

2021 ◽  
Vol 07 (01) ◽  
pp. E25-E34
Author(s):  
Arash Najafi ◽  
Michael Wildt ◽  
Nicolin Hainc ◽  
Joachim Hohmann
Keyword(s):  
Final Diagnosis ◽  
Cystic Lesions ◽  
Likelihood Ratios ◽  
Renal Lesions ◽  
Contrast Enhanced ◽  
Potentially Malignant ◽  
Sensitivity Specificity ◽  
Solid Lesions ◽  
Conclusive Diagnosis

Abstract Purpose Renal lesions are frequent random findings on CT, MRI, and conventional ultrasound. Since they are usually found accidentally, the respective examinations have not been performed optimally to provide a conclusive diagnosis, making additional multiphase contrast-enhanced examinations necessary. The aim of the study is to correlate CEUS findings with the final diagnosis and to determine whether it is a suitable method for the conclusive characterization of undetermined renal lesions. Materials and Methods All CEUS examinations of focal renal lesions performed at our institute between 2007 and 2014 were retrospectively examined. 437 patients with a total of 491 lesions and 543 examinations were included. 54 patients had bilateral lesions. One patient had three lesions in one kidney. Histology was available in 49 cases and follow-ups in 124 cases. The sensitivity, specificity, positive and negative predictive value as well as positive and negative likelihood ratios were calculated. Results There were 54 malignant and 437 benign lesions. The sensitivity and specificity were 0.981/0.954 overall, 1.000/0.956 for cystic lesions, 0.977/0.906 for solid lesions, and 0.971/0.071 for the histologically confirmed lesions. Bosniak classification was consistent in 289 of 301 lesions (96%). Only 12 lesions (3.9%) were falsely assessed as malignant. Conclusion CEUS is an appropriate method for the clarification of undetermined renal lesions. The characterization of cystic lesions according to Bosniak is adequately possible, especially for potentially malignant lesions (types III and IV).

Characterization of the conditional stationary distribution in Markov chains via systems of linear inequalities

2020 ◽  
Vol 52 (4) ◽  
pp. 1249-1283
Author(s):  
Masatoshi Kimura ◽  
Tetsuya Takine
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Continuous Time ◽  
State Transition ◽  
Convex Polytope ◽  
Linear Inequalities ◽  
Systems Of Linear Inequalities ◽  
Free Sets ◽  
Practical Implications

AbstractThis paper considers ergodic, continuous-time Markov chains $\{X(t)\}_{t \in (\!-\infty,\infty)}$ on $\mathbb{Z}^+=\{0,1,\ldots\}$ . For an arbitrarily fixed $N \in \mathbb{Z}^+$ , we study the conditional stationary distribution $\boldsymbol{\pi}(N)$ given the Markov chain being in $\{0,1,\ldots,N\}$ . We first characterize $\boldsymbol{\pi}(N)$ via systems of linear inequalities and identify simplices that contain $\boldsymbol{\pi}(N)$ , by examining the $(N+1) \times (N+1)$ northwest corner block of the infinitesimal generator $\textbf{\textit{Q}}$ and the subset of the first $N+1$ states whose members are directly reachable from at least one state in $\{N+1,N+2,\ldots\}$ . These results are closely related to the augmented truncation approximation (ATA), and we provide some practical implications for the ATA. Next we consider an extension of the above results, using the $(K+1) \times (K+1)$ ( $K > N$ ) northwest corner block of $\textbf{\textit{Q}}$ and the subset of the first $K+1$ states whose members are directly reachable from at least one state in $\{K+1,K+2,\ldots\}$ . Furthermore, we introduce new state transition structures called (K, N)-skip-free sets, using which we obtain the minimum convex polytope that contains $\boldsymbol{\pi}(N)$ .

scholarly journals Maximum Likelihood Estimation of the VAR(1) Model Parameters with Missing Observations

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Helena Mouriño ◽  
Maria Isabel Barão
Keyword(s):  
Stochastic Process ◽  
Missing Data ◽  
Maximum Likelihood ◽  
Moving Average ◽  
Practical Importance ◽  
Likelihood Estimation ◽  
Model Parameters ◽  
Missing Observations ◽  
Data Set ◽  
The Impact

Missing-data problems are extremely common in practice. To achieve reliable inferential results, we need to take into account this feature of the data. Suppose that the univariate data set under analysis has missing observations. This paper examines the impact of selecting an auxiliary complete data set—whose underlying stochastic process is to some extent interdependent with the former—to improve the efficiency of the estimators for the relevant parameters of the model. The Vector AutoRegressive (VAR) Model has revealed to be an extremely useful tool in capturing the dynamics of bivariate time series. We propose maximum likelihood estimators for the parameters of the VAR(1) Model based on monotone missing data pattern. Estimators’ precision is also derived. Afterwards, we compare the bivariate modelling scheme with its univariate counterpart. More precisely, the univariate data set with missing observations will be modelled by an AutoRegressive Moving Average (ARMA(2,1)) Model. We will also analyse the behaviour of the AutoRegressive Model of order one, AR(1), due to its practical importance. We focus on the mean value of the main stochastic process. By simulation studies, we conclude that the estimator based on the VAR(1) Model is preferable to those derived from the univariate context.

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