Journal ArticleDOI

# On Darling–Robbins type confidence sequences and sequential tests with power one for parameters of an autoregressive process

15 Nov 1999-Statistics & Probability Letters (Elsevier)-Vol. 45, Iss: 3, pp 205-214

AbstractIn this paper, we have obtained confidence sequences using Chow's generalization of Hajek–Renyi inequality and the law of iterated logarithm for the autoregressive parameter in a stable first order autoregressive model where innovations are assumed to be independently and identically distributed. A sequential testing procedure with power one has also been proposed. We have also indicated a generalization to confidence sequences for several parameters in a pth order autoregressive model.

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Abstract: The plant ‘Heat Rate’ (HR) is a measure of overall efficiency of a thermal power generating system. It depends on a large number of factors, some of which are non-measurable, while data relating to others are seldom available and recorded. However, coal quality (expressed in terms of ‘effective heat value’ (EHV) as kcal/kg) transpires to be one of the important factors that influences HR values and data on EHV are available in any thermal power generating system. In the present work, we propose a prediction interval of the HR values on the basis of only EHV, keeping in mind that coal quality is one of the important (but not the only) factors that have a pronounced effect on the combustion process and hence on HR. The underlying theory borrows the idea of providing simultaneous confidence interval (SCI) to the coefficients of a p-th p(≥1) order autoregressive model (AR(p)). The theory has been substantiated with the help of real life data from a power utility (after suitable base and scale transfo...

### Cites background or methods from "On Darling–Robbins type confidence ..."

• ...The proof of the first one is quite trivial while the second follows as a consequence of Basu & Mukhopadhyay (1998, 1999)....

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• ...Step 2 demands special attention and requires some novel techniques, as in Basu & Mukhopadhyay (1998, 1999), modified accordingly to suit the present problem....

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• ...This difficulty may be overcome by considering the series of HR and EHV for a given plant as a time series (the data are usually collected for each day) and extending the idea of providing confidence interval for the autoregressive parameters as has been done by Basu & Mukhopadhyay (1998, 1999)....

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• ...…Z∗t−1Z ∗′ t−1 )−1 ( n∑ t=1 Z∗t−1εt ) (A6) Write Mn = ∑nt=1 Z∗t−1Z∗′t−1 and (0) = E ( n∑ t=1 Z∗t−1Z ∗′ t−1 ) (A7) It is well known that, as n → ∞ (Mn/n) a.s.−→ (0), and 1 σ 2 (α̂ − α)′Mn(α̂ − α)χ2p (A8) Following Basu & Mukhopadhyay (1998, 1999), a fixed width SCI for l′α, l : l′l = 1, may be given…...

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##### References
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19 Aug 2009
Abstract: 1 Stationary Time Series.- 2 Hilbert Spaces.- 3 Stationary ARMA Processes.- 4 The Spectral Representation of a Stationary Process.- 5 Prediction of Stationary Processes.- 6* Asymptotic Theory.- 7 Estimation of the Mean and the Autocovariance Function.- 8 Estimation for ARMA Models.- 9 Model Building and Forecasting with ARIMA Processes.- 10 Inference for the Spectrum of a Stationary Process.- 11 Multivariate Time Series.- 12 State-Space Models and the Kalman Recursions.- 13 Further Topics.- Appendix: Data Sets.

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Abstract: The Wiley Classics Library consists of selected books that have become recognized classics in their respective fields. With these new unabridged and inexpensive editions, Wiley hopes to extend the life of these important works by making them available to future generations of mathematicians and scientists. Currently available in the Series: T. W. Anderson Statistical Analysis of Time Series T. S. Arthanari & Yadolah Dodge Mathematical Programming in Statistics Emil Artin Geometric Algebra Norman T. J. Bailey The Elements of Stochastic Processes with Applications to the Natural Sciences George E. P. Box & George C. Tiao Bayesian Inference in Statistical Analysis R. W. Carter Simple Groups of Lie Type William G. Cochran & Gertrude M. Cox Experimental Designs, Second Edition Richard Courant Differential and Integral Calculus, Volume I Richard Courant Differential and Integral Calculus, Volume II Richard Courant & D. Hilbert Methods of Mathematical Physics, Volume I Richard Courant & D. Hilbert Methods of Mathematical Physics, Volume II D. R. Cox Planning of Experiments Harold M. S. Coxeter Introduction to Modern Geometry, Second Edition Charles W. Curtis & Irving Reiner Representation Theory of Finite Groups and Associative Algebras Charles W. Curtis & Irving Reiner Methods of Representation Theory with Applications to Finite Groups and Orders, Volume I Charles W. Curtis & Irving Reiner Methods of Representation Theory with Applications to Finite Groups and Orders, Volume II Bruno de Finetti Theory of Probability, Volume 1 Bruno de Finetti Theory of Probability, Volume 2 W. Edwards Deming Sample Design in Business Research Amos de Shalit & Herman Feshbach Theoretical Nuclear Physics, Volume 1 --Nuclear Structure J. L. Doob Stochastic Processes Nelson Dunford & Jacob T. Schwartz Linear Operators, Part One, General Theory Nelson Dunford & Jacob T. Schwartz Linear Operators, Part Two, Spectral Theory--Self Adjoint Operators in Hilbert Space Nelson Dunford & Jacob T. Schwartz Linear Operators, Part Three, Spectral Operators Herman Fsehbach Theoretical Nuclear Physics: Nuclear Reactions Bernard Friedman Lectures on Applications-Oriented Mathematics Gerald d. Hahn & Samuel S. Shapiro Statistical Models in Engineering Morris H. Hansen, William N. Hurwitz & William G. Madow Sample Survey Methods and Theory, Volume I--Methods and Applications Morris H. Hansen, William N. Hurwitz & William G. Madow Sample Survey Methods and Theory, Volume II--Theory Peter Henrici Applied and Computational Complex Analysis, Volume 1--Power Series--lntegration--Conformal Mapping--Location of Zeros Peter Henrici Applied and Computational Complex Analysis, Volume 2--Special Functions--Integral Transforms--Asymptotics--Continued Fractions Peter Henrici Applied and Computational Complex Analysis, Volume 3--Discrete Fourier Analysis--Cauchy Integrals--Construction of Conformal Maps--Univalent Functions Peter Hilton & Yel-Chiang Wu A Course in Modern Algebra Harry Hochetadt Integral Equations Erwin O. Kreyezig Introductory Functional Analysis with Applications William H. Louisell Quantum Statistical Properties of Radiation All Hasan Nayfeh Introduction to Perturbation Techniques Emanuel Parzen Modern Probability Theory and Its Applications P.M. Prenter Splines and Variational Methods Walter Rudin Fourier Analysis on Groups C. L. Siegel Topics in Complex Function Theory, Volume I--Elliptic Functions and Uniformization Theory C. L. Siegel Topics in Complex Function Theory, Volume II--Automorphic and Abelian integrals C. L Siegel Topics in Complex Function Theory, Volume III--Abelian Functions & Modular Functions of Several Variables J. J. Stoker Differential Geometry J. J. Stoker Water Waves: The Mathematical Theory with Applications J. J. Stoker Nonlinear Vibrations in Mechanical and Electrical Systems

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Abstract: 1 Extension and applications of an inequality of Ville and Wald Let x 1… be a sequence of random variables with a specified joint probability distribution P We shall give a method for obtaining probability inequalities and related limit theorems concerning the behavior of the entire sequence of x’s

220 citations

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TL;DR: A sequence of independent, identically distributed random variables with mean 0, variance 1, and moment generating function ϕ(t) = E(etz) finite in some neighborhood of t= 0 is introduced.
Abstract: 1. Introduction—Let x,x 1, x 2 … be a sequence of independent, identically distributed random variables with mean 0, variance 1, and moment generating function ϕ(t) = E(etz) finite in some neighborhood of t= 0, and put S n = x 1, + … + x n, $$\bar x$$ n = S n/n. For any sequence of positive constants a n, n ≥ 1, let P m = P(|$$\bar x$$ n| ≥ a n for some n ≥ m).

57 citations