Can distributional approximations give exact answers?

2013 ◽  
Vol 32 (1) ◽  
pp. 47-54
Author(s):  
M. Griffiths
Keyword(s):  
Distributional Approximations

Some distributional approximations in Markovian queueing networks

1982 ◽  
Vol 14 (03) ◽  
pp. 654-671 ◽  
Author(s):  
T. C. Brown ◽  
P. K. Pollett
Keyword(s):  
Queueing Networks ◽  
Poisson Processes ◽  
Single Class ◽  
Open Network ◽  
State Dependent ◽  
Service Rates ◽  
Distributional Approximations ◽  
Closed Network ◽  
Poisson Approximations ◽  
Markovian Queueing Networks

We consider single-class Markovian queueing networks with state-dependent service rates (the immigration processes of Whittle (1968)). The distance of customer flows from Poisson processes is estimated in both the open and closed cases. The bounds on distances lead to simple criteria for good Poisson approximations. Using the bounds, we give an asymptotic, closed network version of the ‘loop criterion' of Melamed (1979) for an open network. Approximation of two or more flows by independent Poisson processes is also studied.

Some distributional approximations in Markovian queueing networks

1982 ◽  
Vol 14 (3) ◽  
pp. 654-671 ◽  
Author(s):  
T. C. Brown ◽  
P. K. Pollett
Keyword(s):  
Queueing Networks ◽  
Poisson Processes ◽  
Single Class ◽  
Open Network ◽  
State Dependent ◽  
Service Rates ◽  
Distributional Approximations ◽  
Closed Network ◽  
Poisson Approximations ◽  
Markovian Queueing Networks

We consider single-class Markovian queueing networks with state-dependent service rates (the immigration processes of Whittle (1968)). The distance of customer flows from Poisson processes is estimated in both the open and closed cases. The bounds on distances lead to simple criteria for good Poisson approximations. Using the bounds, we give an asymptotic, closed network version of the ‘loop criterion' of Melamed (1979) for an open network. Approximation of two or more flows by independent Poisson processes is also studied.

scholarly journals Bootstrap Methods in Econometrics

2019 ◽  
Vol 11 (1) ◽  
pp. 193-224 ◽  
Author(s):  
Joel L. Horowitz
Keyword(s):  
Asymptotic Distribution ◽  
Confidence Intervals ◽  
Test Statistic ◽  
Bootstrap Methods ◽  
First Order ◽  
Coverage Probabilities ◽  
Asymptotic Distribution Theory ◽  
Nominal Coverage ◽  
Distributional Approximations ◽  
Approximations To Distributions

The bootstrap is a method for estimating the distribution of an estimator or test statistic by resampling one's data or a model estimated from the data. Under conditions that hold in a wide variety of econometric applications, the bootstrap provides approximations to distributions of statistics, coverage probabilities of confidence intervals, and rejection probabilities of hypothesis tests that are more accurate than the approximations of first-order asymptotic distribution theory. The reductions in the differences between true and nominal coverage or rejection probabilities can be very large. In addition, the bootstrap provides a way to carry out inference in certain settings where obtaining analytic distributional approximations is difficult or impossible. This article explains the usefulness and limitations of the bootstrap in contexts of interest in econometrics. The presentation is informal and expository. It provides an intuitive understanding of how the bootstrap works. Mathematical details are available in the references that are cited.

scholarly journals ALTERNATIVE ASYMPTOTICS AND THE PARTIALLY LINEAR MODEL WITH MANY REGRESSORS

2016 ◽  
Vol 34 (2) ◽  
pp. 277-301 ◽  
Author(s):  
Matias D. Cattaneo ◽  
Michael Jansson ◽  
Whitney K. Newey
Keyword(s):  
Degrees Of Freedom ◽  
Asymptotic Theory ◽  
Empirical Studies ◽  
Structural Effect ◽  
Partially Linear Model ◽  
Partially Linear ◽  
Many Instruments ◽  
Small Bandwidth ◽  
Distributional Approximations ◽  
Inference Methods

Many empirical studies estimate the structural effect of some variable on an outcome of interest while allowing for many covariates. We present inference methods that account for many covariates. The methods are based on asymptotics where the number of covariates grows as fast as the sample size. We find a limiting normal distribution with variance that is larger than the standard one. We also find that with homoskedasticity this larger variance can be accounted for by using degrees-of-freedom-adjusted standard errors. We link this asymptotic theory to previous results for many instruments and for small bandwidth(s) distributional approximations.

Two new proofs of the Erdös–Kac Theorem, with bound on the rate of convergence, by Stein's method for distributional approximations

2009 ◽  
Vol 147 (1) ◽  
pp. 95-114 ◽  
Author(s):  
ADAM J. HARPER
Keyword(s):  
Rate Of Convergence ◽  
Normal Approximation ◽  
Poisson Approximation ◽  
Stein’S Method ◽  
Stein's Method ◽  
Quantitative Form ◽  
Approximation Argument ◽  
Exchangeable Pairs ◽  
Distributional Approximations

AbstractIn this paper, we apply Stein's method for distributional approximations to prove a quantitative form of the Erdös–Kac Theorem. We obtain our best bound on the rate of convergence, on the order of log log log n (log log n)−1/2, by making an intermediate Poisson approximation; we believe that this approach is simpler and more probabilistic than others, and we also obtain an explicit numerical value for the constant implicit in the bound. Different ways of applying Stein's method to prove the Erdös–Kac Theorem are discussed, including a Normal approximation argument via exchangeable pairs, where the suitability of a Poisson approximation naturally suggests itself.

scholarly journals Functionals of clusters of extremes

2003 ◽  
Vol 35 (4) ◽  
pp. 1028-1045 ◽  
Author(s):  
Johan Segers
Keyword(s):  
Markov Chains ◽  
Conditional Distribution ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
High Threshold ◽  
Mixing Condition ◽  
Multivariate Extreme Value Distribution ◽  
Finite Dimensional ◽  
Multivariate Extreme Value ◽  
Distributional Approximations

For arbitrary stationary sequences of random variables satisfying a mild mixing condition, distributional approximations are established for functionals of clusters of exceedances over a high threshold. The approximations are in terms of the distribution of the process conditionally on the event that the first variable exceeds the threshold. This conditional distribution is shown to converge to a nontrivial limit if the finite-dimensional distributions of the process are in the domain of attraction of a multivariate extreme-value distribution. In this case, therefore, limit distributions are obtained for functionals of clusters of extremes, thereby generalizing results for higher-order stationary Markov chains by Yun (2000).

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