Interval estimation

Résumé

In statistics, interval estimation is the use of sample data to estimate an interval of possible values of a parameter of interest. This is in contrast to point estimation, which gives a single value. The most prevalent forms of interval estimation are confidence intervals (a frequentist method) and credible intervals (a Bayesian method); less common forms include likelihood intervals and fiducial intervals. Other forms of statistical intervals include tolerance intervals (covering a proportion of a sampled population) and prediction intervals (an estimate of a future observation, used mainly in regression analysis). Non-statistical methods that can lead to interval estimates include fuzzy logic. Discussion Tolerance interval#Relation to other intervalsConfidence interval#Alternatives and critiques and Prediction interval#Contrast with parametric confidence intervals The scientific problems associated with interval estimation may be summarised as follows: :*When interval

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https://en.wikipedia.org/wiki/Interval_estimation

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In statistics, point estimation involves the use of sample data to calculate a single value (known as a point estimate since it identifies a point in some parameter space) which is to serve as a "bes

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Strong convergence of multivariate maxima

Simone Padoan, Stefano Rizzelli

It is well known and readily seen that the maximum of n independent and uniformly on [0, 1] distributed random variables, suitably standardised, converges in total variation distance, as n increases, to the standard negative exponential distribution. We extend this result to higher dimensions by considering copulas. We show that the strong convergence result holds for copulas that are in a differential neighbourhood of a multivariate generalised Pareto copula. Sklar's theorem then implies convergence in variational distance of the maximum of n independent and identically distributed random vectors with arbitrary common distribution function and (under conditions on the marginals) of its appropriately normalised version. We illustrate how these convergence results can be exploited to establish the almost-sure consistency of some estimation procedures for max-stable models, using sample maxima.

CAMBRIDGE UNIV PRESS2020

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A Consistent Confidence Interval for Fuzzy Capability Index

Vahid Partovi Nia

Fuzzy process capability indices are used to determine whether a production process is capable of producing items within specification tolerance, where instead of precise quality we have two membership functions for specification limits. In practice these indices are estimated using sample data and it is of interest to obtain confidence limits for fuzzy capability index given a sample estimate. After introducing fuzzy confidence interval by Parchami et al. in 2005 and 2006, for fuzzy capability index, our observation leads us to propund an open problem about the consistency property for interval estimation of it. In this paper we redefine some concepts about fuzzy confidence interval and then we show the consistency property of the fuzzy confidence interval proposed by Parchami et al. in 2006 holds almost every where.

2008

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Frequentist estimation of an epidemic's spreading potential when observations are scarce

Andrea Kraus, Victor Panaretos

We consider the problem of inferring the potential of an epidemic for escalating into a pandemic on the basis of limited observations in its initial stages. Classical results of Becker & Hasofer (J. R. Statist. Soc. B, 59, 415-29) illustrate that frequentist estimation of the complete set of parameters of an epidemic modelled as a birth and death process remains feasible even when one is able to observe only the deaths and the total number of births. These assumptions on the observation mechanism, however, are too strong to be met in practice. We consider a more realistic scenario where only temporally aggregated random proportions of the deaths are observed over time. We demonstrate that the frequentist estimation of the Malthusian parameter governing the growth of the epidemic is still feasible in this context. We construct explicit straightforwardly calculable estimators motivated heuristically by the martingale dynamics of the process, and show that they admit a rigorous quasilikelihood interpretation. We establish the consistency and asymptotic normality of these estimators, allowing for the construction of approximate confidence intervals that can be used to infer the spreading potential of the epidemic. A simulation study and an application to the initial outbreak data of the 2009 H1N1 influenza pandemic illustrate that the method can be expected to give reasonable results in practice.

Oxford University Press2014

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