Periodic Splines and Spectral Estimation

Cogburn, Robert; Davis, Herbert

doi:10.1214/aos/1176342868

Cited by 71 publications

(34 citation statements)

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“…In any event, the slow convergence of nonparametric estimates of f is of concern because even the refinement of Assumption 3 (ii) mentioned in the discussion of that assumption requires a convergence rate that approaches arbitrarily close to n −1/2 as β → 1/2. In principle n κ−1/2 −consistent nonparametric spectral estimates can be found, for any κ > 0 (where, for example, κ depends on kernel order, see eg Cogburn and Davis, 1974), though, as β is unknown, one can never be sure that the κ achieved is sufficient.…”

Section: Final Commentsmentioning

confidence: 99%

Cointegration in Fractional Systems with Unknown Integration Orders

2003

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Cointegration of nonstationary time series is considered in a fractional context. Both the observable series and the cointegrating error can be fractional processes. The familiar situation in which the respective integration orders are 1 and 0 is nested, but these values have typically been assumed known. We allow one or more of them to be unknown real values, in which case Robinson and Marinucci (1997,2001) have justified least squares estimates of the cointegrating vector, as well as narrow-band frequencydomain estimates, which may be less biased. While consistent, these estimates do not always have optimal convergence rates, and they have non-standard limit distributional behaviour. We consider estimates formulated in the frequency domain, that consequently allow for a wide variety of (parametric) autocorrelation in the short memory input series, as well as time-domain estimates based on autoregressive transformation. Both can be interpreted as approximating generalized least squares and Gaussian maximum likelihood estimates. The estimates share the same limiting distribution, having mixed normal asymptotics (yielding Wald test statistics with 2 c null limit distributions), irrespective of whether the integration orders are known or unknown, subject in the latter case to their estimation with adequate rates of convergence. The parameters describing the short memory stationary input series are n -consistently estimable, but the assumptions imposed on these series are much more general than ones of autoregressive moving average type. A Monte Carlo study of finite-sample performance and an empirical application to testing the PPP hypothesis are included. ACKNOWLEDGMENTSWe are grateful for the comments of a Co-Editor and referees, which have led to significant improvements.

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Section: Final Commentsmentioning

confidence: 99%

Cointegration in Fractional Systems with Unknown Integration Orders

2003

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“…Early smoothing techniques were based on kernel smoothers. Smoothing splines for this purpose were suggested by Cogburn and Davis (1974). Wahba (1980) method in which the estimator was obtained by spline smoothing the log-periodogram with a data-driven smoothing parameter.…”

Section: Introductionmentioning

confidence: 99%

Bayesian Estimation of the Spectral Density of a Time Series

Choudhuri

Ghosal

Roy

2004

Journal of the American Statistical Association

108

182

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This article describes a Bayesian approach to estimating the spectral density of a stationary time series. A nonparametric prior on the spectral density is described through Bernstein polynomials. Because the actual likelihood is very complicated, a pseudoposterior distribution is obtained by updating the prior using the Whittle likelihood. A Markov chain Monte Carlo algorithm for sampling from this posterior distribution is described that is used for computing the posterior mean, variance, and other statistics. A consistency result is established for this pseudoposterior distribution that holds for a short-memory Gaussian time series and under some conditions on the prior. To prove this asymptotic result, a general consistency theorem of Schwartz is extended for a triangular array of independent, nonidentically distributed observations. This extension is also of independent interest. A simulation study is conducted to compare the proposed method with some existing methods. The method is illustrated with the well-studied sunspot dataset.

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“…, }, with = (2 + 1) × data values, and even for small (e.g., = 1 or 2) this gives a very good approximation to the full periodic spline. Cogburn and Davis [23] present the theory of periodic smoothing spline with application to the estimation of periodic functions and the R function periodicSpline from the package splines might provide an alternative computational approach. Figure 2 which shows fitted curves equivalent to those in Figure 1 but with = 1.…”

Section: Nonparametric Curve Estimation and Periodic Splinesmentioning

confidence: 99%

Exploratory Methods for the Study of Incomplete and Intersecting Shape Boundaries from Landmark Data

Hamed

Aykroyd

2016

Journal of Probability and Statistics

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Structured spatial point patterns appear in many applications within the natural sciences. The points often record the location of key features, called landmarks, on continuous object boundaries, such as anatomical features on a human face. In other situations, the points may simply be arbitrarily spaced marks along a smooth curve, such as on handwritten numbers. This paper proposes novel exploratory methods for the identification of structure within point datasets. In particular, points are linked together to form curves which estimate the original shape from which the points are the only recorded information. Nonparametric regression methods are applied to polar coordinate variables obtained from the point locations and periodic modelling allows closed curves to be fitted even when data are available on only part of the boundary. Further, the model allows discontinuities to be identified to describe rapid changes in the curves. These generalizations are particularly important when the points represent shapes which are occluded or are intersecting. A range of real-data examples is used to motivate the modelling and to illustrate the flexibility of the approach. The method successfully identifies the underlying structure and its output could also be used as the basis for further analysis.

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Periodic Splines and Spectral Estimation

Cited by 71 publications

References 0 publications

Cointegration in Fractional Systems with Unknown Integration Orders

Cointegration in Fractional Systems with Unknown Integration Orders

Bayesian Estimation of the Spectral Density of a Time Series

Exploratory Methods for the Study of Incomplete and Intersecting Shape Boundaries from Landmark Data

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