Central Limit Theorem Approximation Research Articles

Data sharpening for improving central limit theorem approximations for data envelopment analysis–type efficiency estimators

Asymptotic statistical inference on productivity and production efficiency, using nonparametric envelopment estimators, is now available thanks to the basic central limit theorems (CLTs) developed in Kneip, Simar, and Wilson (2015). They provide asymptotic distributions of averages of Data Envelopment Analysis (DEA) and Free Disposal Hull (FDH) estimators of production efficiency. As shown in their Monte-Carlo experiments, due to the curse of dimensionality, the accuracy of the normal approximation is disappointing when the sample size is not large enough. Simar and Zelenyuk (2020) have suggested a simple way to improve the approximation by using a more appropriate estimator of the variances. In this paper we suggest a novel way to improve the approximation, by smoothing out the spurious values of efficiency estimates when they are in a neighborhood of 1. This results in sharpening the data for observations near the estimated efficient frontier. The method is very easy to implement and does not require more computations than the original method. We compare our approach using Monte-Carlo experiments, both with the basic method and with the improved method suggested in Simar & Zelenyuk (2020) and in both cases we observe significant improvements. We show also that the Simar & Zelenyuk (2020) idea of the variance correction can also be adapted to our sharpening method, bringing additional improvements. We illustrate the method with some real data sets.

Efficient Bayesian inference of fully stochastic epidemiological models with applications to COVID-19.

Epidemiological forecasts are beset by uncertainties about the underlying epidemiological processes, and the surveillance process through which data are acquired. We present a Bayesian inference methodology that quantifies these uncertainties, for epidemics that are modelled by (possibly) non-stationary, continuous-time, Markov population processes. The efficiency of the method derives from a functional central limit theorem approximation of the likelihood, valid for large populations. We demonstrate the methodology by analysing the early stages of the COVID-19 pandemic in the UK, based on age-structured data for the number of deaths. This includes maximum a posteriori estimates, Markov chain Monte Carlo sampling of the posterior, computation of the model evidence, and the determination of parameter sensitivities via the Fisher information matrix. Our methodology is implemented in PyRoss, an open-source platform for analysis of epidemiological compartment models.

Refining the central limit theorem approximation via extreme value theory

We suggest approximating the distribution of the sum of independent and identically distributed random variables with a Pareto-like tail by combining extreme value approximations for the largest summands with a normal approximation for the sum of the smaller summands. If the tail is well approximated by a Pareto density, then this new approximation has substantially smaller error rates compared to the usual normal approximation for underlying distributions with finite variance and less than three moments. It can also provide an accurate approximation for some infinite variance distributions.

Central Limit Theorem Approximations for the Number of Runs in Markov-dependent Multi-type Sequences

We consider Markov-dependent multi-type sequences and study various kinds of runs (overlapping, non-overlapping, exact, etc.) by examining additive functionals based on state visits and transitions in an appropriately constructed Markov chain. We establish multivariate Central Limit Theorems for the number of these runs and obtain the covariance matrix of the limiting multivariate normal distribution in closed form using the potential matrix. An application in reliability theory is discussed.

Central Limit Theorem approximations for the number of runs in Markov-dependent binary sequences

We consider Markov-dependent binary sequences and study various types of success runs (overlapping, non-overlapping, exact, etc.) by examining additive functionals based on state visits and transitions in an appropriate Markov chain. We establish a multivariate Central Limit Theorem for the number of these types of runs and obtain its covariance matrix by means of the recurrent potential matrix of the Markov chain. Explicit expressions for the covariance matrix are given in the Bernoulli and a simple Markov-dependent case by expressing the recurrent potential matrix in terms of the stationary distribution and the mean transition times in the chain. We also obtain a multivariate Central Limit Theorem for the joint number of non-overlapping runs of various sizes and give its covariance matrix in explicit form for Markov dependent trials.

Functional Central Limit Theorem approximations and the distribution of the Dickey–Fuller test with strongly heteroskedastic data

We simulate the small-sample distribution of the Dickey–Fuller (DF) test with data generated from various GARCH(1,1) processes where the parameters α and β are close to the boundary of integration. As the length of the sample increases, the small-sample distributions of the DF test converge slowly to the asymptotic one, and the convergence is even slower as α+ β approaches unity. This suggests that, with strongly heteroskedastic data, we must use caution when relying on asymptotic tools that use the Functional Central Limit Theorem (FCLT). Indeed, with close-to-integrated GARCH(1,1) data, the asymptotic DF critical values lead to grossly oversized tests.

Space-Time Codes for Wireless Optical Communications

A space-time channel coding technique is presented for overcoming turbulence-induced fading in an atmospheric optical heterodyne communication system that uses multiple transmit and receive apertures. In particular, a design criterion for minimizing the pairwise probability of codeword error in a space-time code (STC) is developed from a central limit theorem approximation. This design criterion maximizes the mean-to-standard-deviation ratio of the received energy difference between codewords. It leads to STCs that are a subset of the previously reported STCs for Rayleigh channels, namely those created from orthogonal designs. This approach also extends to other fading channels with independent, zero-mean path gains. Consequently, for large numbers of transmit and receive antennas, STCs created from orthogonal designs minimize the pairwise codeword error probability for this larger class of fading channels.

On the Uniformity of Sequential Procedures

An extension of a central limit theorem for a mean of a random number of observations is given. A natural application occurs in the area of fixed-width confidence intervals. We provide an example which shows that the standard procedure does not preserve the intended coverage probability uniformly over nontrivial sets of distribution functions. The major weak convergence result is used to provide conditions for and a simple proof of such uniformity. The results are also shown to hold for M-estimates of location. 1. Introduction. This paper has evolved from a study of the uniformity of central limit theorem approximations for sequential fixed-width confidence intervals and related ranking and selection procedures. The main weak convergence result concerns a process W#(s, t) = n-I Etn8] {Gt(Xi) -EGt(Xl)X

A Taylor's Theorem—Central Limit Theorem Approximation: Its Use in Obtaining the Probability Distribution of Long-Range Profit

Although the functional relationship between long-range profit (LRP) and its various components is usually well defined, the difficulty of determining the probability distribution of LRP prevents an immediate answer to a question such as “What are the chances that the LRP associated with decision D will exceed X dollars?” In a situation such as this, the first thought is to use Monte Carlo (i.e., random sampling) in order to provide an approximation to the distribution of LRP. An alternative approach, the use of error propagation formulas based on a Taylor series expansion coupled with a Central Limit Theorem approximation, is discussed.