Limit theory for autoregressive-parameter estimates in an infinite-variance random walk

1989 ◽  
Vol 17 (3) ◽  
pp. 261-278 ◽  
Author(s):  
Keith Knight
Keyword(s):  
Random Walk ◽  
Infinite Variance ◽  
Parameter Estimates ◽  
Limit Theory ◽  
Autoregressive Parameter

Limit Theory for M-Estimates in an Integrated Infinite Variance

1991 ◽  
Vol 7 (2) ◽  
pp. 200-212 ◽  
Author(s):  
Keith Knight
Keyword(s):  
Least Squares ◽  
Rate Of Convergence ◽  
Linear Process ◽  
Least Squares Estimator ◽  
Infinite Variance ◽  
Limiting Distributions ◽  
Limit Theory ◽  
Asymptotically Normal ◽  
Autoregressive Parameter ◽  
M Estimates

We consider the limiting distributions of M-estimates of an “autoregressive” parameter when the observations come from an integrated linear process with infinite variance innovations. It is shown that M-estimates are, asymptotically, infinitely more efficient than the least-squares estimator (in the sense that they have a faster rate of convergence) and are conditionally asymptotically normal.

scholarly journals What can the ideal gas say about global pandemics? Reinterpreting the basic reproduction number

2020 ◽  
Author(s):  
Bradley M Dickson
Keyword(s):  
Random Walk ◽  
Population Density ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
The United States ◽  
Ideal Gas ◽  
Parameter Estimates ◽  
Public Data ◽  
The Basic Reproduction Number ◽  
The Ideal

Through analysis of the ideal gas, we construct a random walk that on average matches the standard susceptible-infective-removed (SIR) model. We show that the most widely referenced parameter, the 'basic reproduction number' (R0), is fundamentally connected to the relative odds of increasing or decreasing the infectives population. As a consequence, for R0 > 1 the probability that no outbreak occurs is 1/R0. In stark contrast to a deterministic SIR, when R0 = 1.5 the random walk has a 67% chance of avoiding outbreak. Thus, an alternative, probabilistic, interpretation of R0 arises, which provides a novel estimate of the critical population density γ/r without fitting SIR models. We demonstrate that SARS-CoV2 in the United States is consistent with our model and attempt an estimate of γ/r. In doing so, we uncover a significant source of bias in public data reporting. Data are aggregated on political boundaries, which bear no concern for dispersion of population density. We show that this introduces bias in fits and parameter estimates, a concern for understanding fundamental virus parameters and for policy making. Anonymized data at the resolution required for contact tracing would afford access to γ/r without fitting. The random walk SIR developed here highlights the intuition that any epidemic is stochastic and recovers all the key parameter values noted by Kermack and McKendrick in 1927.

A Hierarchical Bayesian Model With Correlated Residuals for Investigating Stability and Change in Intensive Longitudinal Data Settings

Methodology ◽  
2014 ◽  
Vol 10 (4) ◽  
pp. 126-137 ◽  
Author(s):  
Fidan Gasimova ◽  
Alexander Robitzsch ◽  
Oliver Wilhelm ◽  
Gizem Hülür
Keyword(s):  
Longitudinal Data ◽  
Bayesian Model ◽  
Working Memory Capacity ◽  
Intraindividual Variability ◽  
Parameter Estimates ◽  
Hierarchical Bayesian ◽  
Hierarchical Bayesian Model ◽  
Intensive Longitudinal Data ◽  
Autoregressive Parameter ◽  
Larger Sample

The present paper’s focus is the modeling of interindividual and intraindividual variability in longitudinal data. We propose a hierarchical Bayesian model with correlated residuals, employing an autoregressive parameter AR(1) for focusing on intraindividual variability. The hierarchical model possesses four individual random effects: intercept, slope, variability, and autocorrelation. The performance of the proposed Bayesian estimation is investigated in simulated longitudinal data with three different sample sizes (N = 100, 200, 500) and three different numbers of measurement points (T = 10, 20, 40). The initial simulation values are selected according to the results of the first 20 measurement occasions from a longitudinal study on working memory capacity in 9th graders. Within this simulation study, we investigate the root mean square error (RMSE), bias, relative percentage bias, and the 90% coverage probability of parameter estimates. Results indicate that more accurate estimates are associated with a larger sample size. One exception to this tendency is the autocorrelation parameter, which shows more sensitivity to an increasing number of time points.

scholarly journals Heavy-tailed branching random walks on multidimensional lattices. A moment approach

2020 ◽  
pp. 1-22
Author(s):  
Anastasiya Rytova ◽  
Elena Yarovaya
Keyword(s):  
Random Walk ◽  
Asymptotic Behaviour ◽  
Continuous Time ◽  
Lattice Point ◽  
Positive Eigenvalue ◽  
Infinite Variance ◽  
Finite Variance ◽  
Branching Random Walks ◽  
Heavy Tailed ◽  
Number Of Particles

We study a continuous-time branching random walk (BRW) on the lattice ℤ d , d ∈ ℕ, with a single source of branching, that is the lattice point where the birth and death of particles can occur. The random walk is assumed to be spatially homogeneous, symmetric and irreducible but, in contrast to the majority of previous investigations, the random walk transition intensities a(x, y) decrease as |y − x|−(d+α) for |y − x| → ∞, where α ∈ (0, 2), that leads to an infinite variance of the random walk jumps. The mechanism of the birth and death of particles at the source is governed by a continuous-time Markov branching process. The source intensity is characterized by a certain parameter β. We calculate the long-time asymptotic behaviour for all integer moments for the number of particles at each lattice point and for the total population size. With respect to the parameter β, a non-trivial critical point β c  > 0 is found for every d ≥ 1. In particular, if β > β c the evolutionary operator generated a behaviour of the first moment for the number of particles has a positive eigenvalue. The existence of a positive eigenvalue yields an exponential growth in t of the particle numbers in the case β > β c called supercritical. Classification of the BRW treated as subcritical (β < β c ) or critical (β = β c ) for the heavy-tailed random walk jumps is more complicated than for a random walk with a finite variance of jumps. We study the asymptotic behaviour of all integer moments of a number of particles at any point y ∈ ℤ d and of the particle population on ℤ d according to the ratio d/α.

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