On the Estimation of Autocorrelation in time Series

Łomnicki, Z. A.; Zaremba, S. K.

doi:10.1214/aoms/1177707042

Cited by 35 publications

(20 citation statements)

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“…The method of statistical differential could be used for estimating the covariances of autocorrelation coefficients [16]. For convenience of analysis, we could treat the LFM as piecewise stationary signal and define it as…”

Section: Variation Analysis For Estimating Autocorrelationmentioning

confidence: 99%

Coprime sampling for nonstationary signal in radar signal processing

Liang

2013

J Wireless Com Network

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Estimating the spectrogram of non-stationary signal relates to many important applications in radar signal processing. In recent years, coprime sampling and array attract attention for their potential of sparse sensing with derivative to estimate autocorrelation coefficients with all lags, which could in turn calculate the power spectrum density. But this theoretical merit is based on the premise that the input signals are wide-sense stationary. In this article, we discuss how to implement coprime sampling for non-stationary signal, especially how to attain the benefits of coprime sampling meanwhile limiting the disadvantages due to lack of observations for estimations. Furthermore, we investigate the usage of coprime sampling for calculating ambiguity function of matched filter in radar system. We also examine the effect of it and conclude several useful guidelines of choosing configuration to conduct the sparse sensing while retain the detection quality.

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Section: Variation Analysis For Estimating Autocorrelationmentioning

confidence: 99%

Coprime sampling for nonstationary signal in radar signal processing

Liang

2013

J Wireless Com Network

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“…Another particular case has been studied by Lomnicki and Zaremba [8]. Both of the authors restricted themselves to the first terms in the expansions.…”

Section: Expressions For a Vector Valued Process {X(t)) T£t Follow Inmentioning

confidence: 99%

Asymptotic Expansions for Moments of Functions of Stochastic Processes and Their Applications

Hurt¹

1986

Statistics &Amp; Risk Modeling

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Let{X(t)> t eT be a stochastic process and g be a real function. Suppose that E|X(t) -0| q+2 -» 0 , q being a natural number and θ being a parameter. Under some smoothing conditions on g asymptotic expansions for Ε g(X(t)) and var g(X(t)) are derived with the remainder terms Ο(E|X(t) -6| g+1 ) and 0(E|x(t) -9| g+2 ) , respectively. The proofs use the idea of the method of statistical differentials. Function g may depend on t explicitely. A generalization to the multi-dimensional case is also given. The asymptotic formulas are applied to the problem of compa-AMS 1980 subject classifications: 62 G 05 , 62 G 20 Key words and phrases: asymptotic expansions, statistical differentials, reliability estimators, deficiency Brought to you by | Nanyang Technological University Authenticated Download Date | 6/10/15 5:10 AM 252 J. Hurt rison of some reliability estimators by deficiency and to investigation the asymptotic behaviour of complicated functions of statistics.

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“…In view of (8.4) and jointly with (8.8), these inequalities, which will be useful also later on, prove our point concerning C~)~),zr and C(k{)~),r if we make # = f. The variance of C(~{)~),~ v can be "computed in the same way as that of C~(~),iv, but with the substitution of the second and fourth eumulants of ~t, r for those of st, so that lim N var C(kd~ = 0 and, therefore, ,dt) of a degree having a given upper bound is superimposed on a process {xt} satisfying the assumptions of the preceding theorem, the trend can be eliminated from a sample by the method of least squares, and the usual asymptotic formula for the covariances of the covariance estimators still applies to the estimators based on the residuals (cf. [13]; the whole paper was written on the assumption that E(et s) was finite, but the proofs of the propositions referred to remain valid when only E(e~) is assumed to be finite). The estimator of R k based on a sample of x t q-]~(t) can be decomposed into two parts: one formed with the "true" process {x~}, and another, denoted by X k and representing the errors in trend elimination.…”

Section: ~ ~Lv--n X Yzr -Z~--n ~=1mentioning

confidence: 99%

“…This still applies when a polynomial trend has to be eliminated. The effects of such a trend elimination were discussed by the authors in the Annals of Mathematical Statistics [13]. In order to treat rigourously the second moments of the estimators of the autoeorrelation coefficients, it was necessary to compute laboriously the fourth moments of the sample covariances; this part of the paper should be regarded as superseded by the more general results published presently.…”

Section: Introductionmentioning

confidence: 99%

“…Higher moments are particularly useful in the investigation of the tails of a probability distribution (and this is why fourth moments were required in [13]), but it is clear that, in general, an exact knowledge of the moments of a distribution is capable of giving precise information on the rapidity with which a distribution tends to its normal limit. The exact formulae (given in Sections 5 and 6) for the moments of the sample covariances are composed of main parts, corresponding to the respective asymptotic values, and of remainder terms.…”

Section: Introductionmentioning

confidence: 99%

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