On robustness of Spectral R\'{e}nyi divergence

Keisuke Yano; Tetsuya Takabatake

arxiv: 2310.06902 · v4 · submitted 2023-10-10 · 🧮 math.ST · stat.TH

On robustness of Spectral R\'{e}nyi divergence

Tetsuya Takabatake , Keisuke Yano This is my paper

Pith reviewed 2026-05-24 06:06 UTC · model grok-4.3

classification 🧮 math.ST stat.TH

keywords spectral Rényi divergencerobustnesstime seriesspectral densityItakura-Saito divergenceγ-divergenceminimum divergence estimationfrequency outliers

0 comments

The pith

Minimum spectral Rényi divergence estimates maintain stable optimization paths despite frequency-domain outliers, unlike Itakura-Saito.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines spectral α-Rényi divergences applied to time series spectral densities, including the Itakura-Saito divergence as a limiting case. It establishes a connection to γ-divergence from robust statistics and derives a variational representation. These features are used to demonstrate that minimum spectral Rényi divergence estimators resist disruption from outliers in the frequency domain. The result is more stable parameter estimates that reduce dependence on complex pre-processing steps for time series data.

Core claim

The paper establishes that the minimum spectral Rényi divergence estimate has a stable optimization path with respect to outliers in the frequency domain. This stability follows from the connection to γ-divergence and the variational representation of the spectral α-Rényi divergence. As a direct consequence, the estimator delivers more stable estimates than the minimum Itakura-Saito divergence estimator and reduces the need for intricate pre-processing.

What carries the argument

The spectral α-Rényi divergence, which generalizes the Itakura-Saito divergence and links to γ-divergence to support robustness in minimum-divergence estimation for spectral densities.

If this is right

The minimum Rényi estimator resists disruption from frequency outliers where the Itakura-Saito estimator does not.
Parameter estimates for time series spectral densities become more stable under the Rényi approach.
Reliance on intricate pre-processing steps for outlier handling can be reduced.
The robustness properties extend to the full class of spectral α-Rényi divergences.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The variational representation could be used to derive new robust optimization routines for other spectral estimation problems.
The same stability mechanism might apply when frequency outliers arise from measurement artifacts in physical time series.
Comparison of convergence rates between Rényi and Itakura-Saito estimators under controlled outlier levels would quantify the practical gain.
Hybrid estimators that switch between Rényi and other divergences based on detected outlier severity could be explored.

Load-bearing premise

The connection to γ-divergence and the variational representation are assumed to directly imply that the minimum-divergence estimator will exhibit stable optimization behavior under frequency outliers.

What would settle it

A numerical experiment in which frequency outliers are added to spectral data and the optimization trajectory of the minimum Rényi estimator is observed to become unstable would falsify the central claim.

read the original abstract

This paper studies a specific class of statistical divergences for spectral densities of time series: the spectral $\alpha$-R\'{e}nyi divergences, which include the Itakura-Saito divergence as a limiting case. The aim of this paper is to highlight both information-theoretic and statistical properties of spectral $\alpha$-R\'{e}nyi divergences. We reveal the connection between the spectral $\alpha$-R\'{e}nyi divergence and the $\gamma$-divergence in robust statistics, and a variational representation of the spectral $\alpha$-R\'{e}nyi divergence. Inspired by these results suggesting "robustness" of spectral $\alpha$-R\'{e}nyi divergence, we show that the minimum spectral R\'{e}nyi divergence estimate has a stable optimization path with respect to outliers in the frequency domain, unlike the minimum Itakura-Saito divergence estimator, and thus it delivers more stable estimates, reducing the need for intricate pre-processing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The γ-divergence link and variational form are the main new pieces, but the jump to stable estimator behavior under frequency outliers still needs the explicit argument to hold up.

read the letter

The paper connects spectral α-Rényi divergence to γ-divergence and supplies a variational representation, then claims this yields a minimum-divergence estimator whose optimization path stays stable when frequency outliers hit the periodogram, unlike the Itakura-Saito case. That connection and the stability claim are what a reader should take away first. The explicit link in the spectral setting is new enough and the practical angle about lighter pre-processing is reasonable. The authors frame the work cleanly around existing robust statistics ideas, which helps place the result. The variational representation is a concrete addition that could be checked independently. The softer part is exactly the translation the stress-test flags. γ-divergence robustness usually addresses sample contamination, while frequency outliers act on the already-estimated spectrum; the variational bound controls the divergence value but does not automatically control how the argmin moves under those perturbations. The abstract states they show the stable path, so the paper must contain a derivation or theorem that closes this step. If that argument is direct and does not rely on extra unstated conditions, the claim stands; if it stays at the level of inspiration, the central statistical result is thinner. This work sits in the overlap of time-series spectral estimation and robust divergence methods. A reader already working on alternatives to Itakura-Saito or on γ-type robustness in frequency-domain problems would find it useful. It is worth sending to peer review because the claim is specific, the setup is standard, and the potential gap is checkable with proofs or targeted simulations.

Referee Report

2 major / 1 minor

Summary. The manuscript examines spectral α-Rényi divergences between time-series spectral densities (including the Itakura-Saito divergence as a limit case). It establishes a link to the γ-divergence of robust statistics together with a variational representation, and asserts that the resulting minimum-divergence estimator possesses a stable optimization trajectory under frequency-domain outliers, in contrast to the minimum Itakura-Saito estimator.

Significance. A rigorously proved stability result would be of practical value for robust nonparametric spectral estimation, as it could lessen reliance on ad-hoc pre-processing steps. The information-theoretic connections themselves are of independent interest, but the statistical claim about optimization stability under contamination is the load-bearing contribution.

major comments (2)

[Abstract / main claim] The abstract asserts that the γ-divergence connection and variational representation 'suggest robustness' and thereby imply a stable optimization path for the minimum spectral Rényi estimator under frequency outliers. No explicit theorem, derivation, or sensitivity analysis is supplied that translates the sample-level robustness property of γ-divergence into control of the argmin trajectory when the periodogram is perturbed at isolated frequencies. This step is not immediate and must be shown.
[Abstract / main claim] No simulations, numerical examples, or error bounds are presented that would illustrate or quantify the claimed stability advantage over the Itakura-Saito estimator. Without such evidence the practical assertion that the estimator 'delivers more stable estimates, reducing the need for intricate pre-processing' remains unverified.

minor comments (1)

Notation for the spectral α-Rényi divergence and its relation to the ordinary Rényi divergence should be introduced with an explicit formula early in the manuscript.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We address each major comment below and plan to incorporate revisions to enhance the rigor and clarity of our results.

read point-by-point responses

Referee: [Abstract / main claim] The abstract asserts that the γ-divergence connection and variational representation 'suggest robustness' and thereby imply a stable optimization path for the minimum spectral Rényi estimator under frequency outliers. No explicit theorem, derivation, or sensitivity analysis is supplied that translates the sample-level robustness property of γ-divergence into control of the argmin trajectory when the periodogram is perturbed at isolated frequencies. This step is not immediate and must be shown.

Authors: We appreciate this observation. The manuscript presents the connection to γ-divergence and the variational representation as suggesting robustness, and then states that we show the stable optimization path. However, to make this implication fully rigorous, we agree that an explicit theorem is needed. In the revised version, we will introduce a new theorem that derives the stability of the argmin with respect to isolated frequency perturbations, using the variational representation to bound the changes in the objective function. revision: yes
Referee: [Abstract / main claim] No simulations, numerical examples, or error bounds are presented that would illustrate or quantify the claimed stability advantage over the Itakura-Saito estimator. Without such evidence the practical assertion that the estimator 'delivers more stable estimates, reducing the need for intricate pre-processing' remains unverified.

Authors: We concur that empirical validation would better support the practical implications. The current manuscript focuses on the theoretical connections and the stability argument. We will add a section with numerical experiments that compare the optimization behavior of the two estimators when the periodogram is contaminated at certain frequencies, including plots of the trajectories and quantitative stability metrics. revision: yes

Circularity Check

0 steps flagged

Connection to γ-divergence supplies independent mathematical basis; stability claim derived separately

full rationale

The paper first derives the link between spectral α-Rényi divergence and γ-divergence plus a variational representation; these are presented as revealed properties rather than fitted or renamed inputs. It then states that these results inspire a separate demonstration that the minimum spectral Rényi estimator has a stable optimization path under frequency outliers. No equation or procedure reduces the stability result to a self-definition, a fitted parameter renamed as prediction, or a self-citation chain. The derivation chain remains self-contained against external benchmarks with no load-bearing circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.0 · 5694 in / 1072 out tokens · 21544 ms · 2026-05-24T06:06:41.887065+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/BranchSelection.lean RCLCombiner_isCoupling_iff; branch_selection echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Dα[S:eS] = 1/(1-α) min_{S'} [α D1[eS:S'] + (1-α) D1[S:S']] attained at S' = α eS + (1-α) S (Theorem 2)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel; dAlembert_to_ODE_general echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

lim n→∞ (2/n) G_γ[pn:epn] = D_α[S:eS] with γ = α^{-1}-1 (Theorem 1)
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean costAlphaLog_high_calibrated_iff; J_uniquely_calibrated_via_higher_derivative refines

?

refines
Relation between the paper passage and the cited Recognition theorem.

gradient G_α(θ;eIn) and Hessian bound L = U1² + U2/(1-α) independent of contamination z (Remark 1)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

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Attouch, J

H. Attouch, J. Bolte, and B. Svaiter. Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized gauss–seidel methods. Mathematical Programming, 137:91–129, 2013

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D. Bertsekas. Nonlinear Programming. Athena Scientific, second edition edition, 1999

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Brockwell and R

P. Brockwell and R. Davis. Time Series: Theory and Methods . Springer, 2nd edition, 1991

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Calderoni and R

G. Calderoni and R. Abercrombie. Investigating spectral estimates of stress drop for small to moderate earthquakes with heterogeneous slip distribution: Examples from the 2016–2017 amatrice earthquake sequence. Journal of Geophysical Research: Solid Earth , 128, 2023

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Diakonikolas and D

I. Diakonikolas and D. Kane. Algorithmic High-Dimensional Robust Statistics . Cambridge University Press, 2023

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Fujisawa and S

H. Fujisawa and S. Eguchi. Robust parameter estimation with a small bias against heavy contamination. Journal of Multivariate Analysis , 99:2053–2081, 2008

work page 2053
[8]

M. Gil, F. Alajaji, and T. Linder. R´ enyi divergence measures for commonly used univariate continuous distributions. Information Sciences, 249:124–131, 2013

work page 2013
[9]

Grivel, R

E. Grivel, R. Diversi, and F. Merchan. Kullback–Leibler and R´ enyi divergence rate for Gaussian stationary ARMA processes comparison. Digital Signal Processing, 116:103089, 2021

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Heyde and W

C. Heyde and W. Dai. On the robustness to small trends of estimation based on the smoothed periodogram. Journal of Time Series Analysis , 17:141–150, 1996

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J. Hirukawa. Cluster analysis for non-gaussian locally stationary processes. International Journal of Theoretical and Applied Finance , 9, 2006

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F. Iacone. Local whittle estimation of the memory parameter in presence of deterministic components. Journal of Time Series Analysis , 31:37–49, 2010

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Itakura and S

F. Itakura and S. Saito. Analysis synthesis telephony based on the maximum likelihood method. Proceedings of the 6th of the International Congress on Acoustics , pages C17–C20, 1968. 14

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Aki K and P

K. Aki K and P. Richards. Quantitative seismology: theory and methods . W H Freeman and Company, 1980

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Kolmogorov

A. Kolmogorov. Stationary sequences in Hilbert space. Bulletin of Moscow State University, Mathematics, pages 1–40, 1941

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K. Kurdyka. On gradients of functions definable in o-minimal structures. Annales de l’institut Fourier, 48:769–783, 1998

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Langbein

J. Langbein. Noise in two-color electronic distance meter measurements revisited. Journal of Geophysical Research: Solid Earth, 109, 2004

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Lojasiewicz

S. Lojasiewicz. Une propri´ et´ e topologique des sous-ensembles analytiques r´ eels. In Les ´Equations aux D´ eriv´ ees Partielles. ´Editions du Centre National de la Recherche Scientifique, Paris, 1963

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Maronna, R

R. Maronna, R. Martin, V. Yohai, and M. Salibi´ an-Barrera. Robust statistics: theory and methods (with R) . John Wiley & Sons, 2019

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R. Martin. Statistical Methods for the Enhancement of Noisy Speech , chapter 2, pages 43–65. Springer Berlin/Heidelberg, 2005

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Mat´ ern.Spatial Variation: Stochastic Models and Their Application to Some Problems in Forest Surveys and Other Sampling Investigations

B. Mat´ ern.Spatial Variation: Stochastic Models and Their Application to Some Problems in Forest Surveys and Other Sampling Investigations. Stockholm: Statens Skogsforskningsinstitut, 1960

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McCloskey and P

A. McCloskey and P. Perron. Memory parameter estimation in the presence of level shifts and deterministic trends. Econometric Theory, 29:1196–1237, 2013

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Nocedal and S

J. Nocedal and S. Wright. Numerical Optimization. Springer, second edition, 2006

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E. Parzen. Stationary time series analysis using information and spectral analysis. In S. Rao and, editor, Developments in Time Series Analysis. In Honour of M. B. Priestley , pages 139– –148. Chapman & Hall, 1993

work page 1993
[25]

M. Pinsker. Information and information stability of random variables and processes . Holden- Day, 1964. Translated and edited by A. Feinstein

work page 1964
[26]

Shayevitz

P. Shayevitz. A note on a characterization of R´ enyi measures and its relation to composite hypothesis testing, 2010

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[27]

O. Szeg¨ o. Beitr¨ age zur theorie der toeplitzschen formen.Mathematische Zeitschrift, 6:167–202, 1920

work page 1920
[28]

Taniguchi

M. Taniguchi. Minimum contrast estimation for spectral densities of stationary processes. Journal of the Royal Statistical Society. Series B , 49:315–325, 1987

work page 1987
[29]

I. Vajda. Theory of Statistical Inference and Information . Springer Dordrecht, 1989

work page 1989
[30]

van Erven and P

T. van Erven and P. Harremos. R´ enyi divergence and Kullback–Leibler divergence. IEEE Transactions on Information Theory , 60:3797–3820, 2014

work page 2014
[31]

P. Whittle. The analysis of multiple stationary time series. Journal of the Royal Statistical Society. Series B (Methodological), 15:125–139, 1953

work page 1953
[32]

Shumway Y

R. Shumway Y. Kakizawa and M. Taniguchi. Discrimination and clustering for multivariate time series. Journal of the American Statistical Association , 93(441):328–340, 1998

work page 1998
[33]

Yoshimitsu, T

N. Yoshimitsu, T. Maeda, and T. Sei. Estimation of source parameters using a non-Gaussian probability density function in a bayesian framework. Earth, Planets and Space , 75, 2023

work page 2023
[34]

On robustness of Spectral R´ enyi divergence

G. Zhang and M. Taniguchi. Nonparametric approach for discriminant analysis in time series. Journal of Nonparametric Statistics , 5:91–101, 1995. 15 Supplement to “On robustness of Spectral R´ enyi divergence” by Tetsuya Takabatake and Keisuke Yano This supplement presents • Proof of Theorem 1 in Appendix A; • Proof of Theorem 3 in Appendix B; • Proof of ...

work page 1995

[1] [1]

Attouch, J

H. Attouch, J. Bolte, and B. Svaiter. Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized gauss–seidel methods. Mathematical Programming, 137:91–129, 2013

work page 2013

[2] [2]

Bertsekas

D. Bertsekas. Nonlinear Programming. Athena Scientific, second edition edition, 1999

work page 1999

[3] [3]

Brockwell and R

P. Brockwell and R. Davis. Time Series: Theory and Methods . Springer, 2nd edition, 1991

work page 1991

[4] [4]

J. Brune. Tectonic stress and spectra of seismic shear waves from earthquakes. Journal of Geophysical Research, 75:4997–5009, 1970

work page 1970

[5] [5]

Calderoni and R

G. Calderoni and R. Abercrombie. Investigating spectral estimates of stress drop for small to moderate earthquakes with heterogeneous slip distribution: Examples from the 2016–2017 amatrice earthquake sequence. Journal of Geophysical Research: Solid Earth , 128, 2023

work page 2016

[6] [6]

Diakonikolas and D

I. Diakonikolas and D. Kane. Algorithmic High-Dimensional Robust Statistics . Cambridge University Press, 2023

work page 2023

[7] [7]

Fujisawa and S

H. Fujisawa and S. Eguchi. Robust parameter estimation with a small bias against heavy contamination. Journal of Multivariate Analysis , 99:2053–2081, 2008

work page 2053

[8] [8]

M. Gil, F. Alajaji, and T. Linder. R´ enyi divergence measures for commonly used univariate continuous distributions. Information Sciences, 249:124–131, 2013

work page 2013

[9] [9]

Grivel, R

E. Grivel, R. Diversi, and F. Merchan. Kullback–Leibler and R´ enyi divergence rate for Gaussian stationary ARMA processes comparison. Digital Signal Processing, 116:103089, 2021

work page 2021

[10] [10]

Heyde and W

C. Heyde and W. Dai. On the robustness to small trends of estimation based on the smoothed periodogram. Journal of Time Series Analysis , 17:141–150, 1996

work page 1996

[11] [11]

Hirukawa

J. Hirukawa. Cluster analysis for non-gaussian locally stationary processes. International Journal of Theoretical and Applied Finance , 9, 2006

work page 2006

[12] [12]

F. Iacone. Local whittle estimation of the memory parameter in presence of deterministic components. Journal of Time Series Analysis , 31:37–49, 2010

work page 2010

[13] [13]

Itakura and S

F. Itakura and S. Saito. Analysis synthesis telephony based on the maximum likelihood method. Proceedings of the 6th of the International Congress on Acoustics , pages C17–C20, 1968. 14

work page 1968

[14] [14]

Aki K and P

K. Aki K and P. Richards. Quantitative seismology: theory and methods . W H Freeman and Company, 1980

work page 1980

[15] [15]

Kolmogorov

A. Kolmogorov. Stationary sequences in Hilbert space. Bulletin of Moscow State University, Mathematics, pages 1–40, 1941

work page 1941

[16] [16]

K. Kurdyka. On gradients of functions definable in o-minimal structures. Annales de l’institut Fourier, 48:769–783, 1998

work page 1998

[17] [17]

Langbein

J. Langbein. Noise in two-color electronic distance meter measurements revisited. Journal of Geophysical Research: Solid Earth, 109, 2004

work page 2004

[18] [18]

Lojasiewicz

S. Lojasiewicz. Une propri´ et´ e topologique des sous-ensembles analytiques r´ eels. In Les ´Equations aux D´ eriv´ ees Partielles. ´Editions du Centre National de la Recherche Scientifique, Paris, 1963

work page 1963

[19] [19]

Maronna, R

R. Maronna, R. Martin, V. Yohai, and M. Salibi´ an-Barrera. Robust statistics: theory and methods (with R) . John Wiley & Sons, 2019

work page 2019

[20] [20]

R. Martin. Statistical Methods for the Enhancement of Noisy Speech , chapter 2, pages 43–65. Springer Berlin/Heidelberg, 2005

work page 2005

[21] [21]

Mat´ ern.Spatial Variation: Stochastic Models and Their Application to Some Problems in Forest Surveys and Other Sampling Investigations

B. Mat´ ern.Spatial Variation: Stochastic Models and Their Application to Some Problems in Forest Surveys and Other Sampling Investigations. Stockholm: Statens Skogsforskningsinstitut, 1960

work page 1960

[22] [22]

McCloskey and P

A. McCloskey and P. Perron. Memory parameter estimation in the presence of level shifts and deterministic trends. Econometric Theory, 29:1196–1237, 2013

work page 2013

[23] [23]

Nocedal and S

J. Nocedal and S. Wright. Numerical Optimization. Springer, second edition, 2006

work page 2006

[24] [24]

E. Parzen. Stationary time series analysis using information and spectral analysis. In S. Rao and, editor, Developments in Time Series Analysis. In Honour of M. B. Priestley , pages 139– –148. Chapman & Hall, 1993

work page 1993

[25] [25]

M. Pinsker. Information and information stability of random variables and processes . Holden- Day, 1964. Translated and edited by A. Feinstein

work page 1964

[26] [26]

Shayevitz

P. Shayevitz. A note on a characterization of R´ enyi measures and its relation to composite hypothesis testing, 2010

work page 2010

[27] [27]

O. Szeg¨ o. Beitr¨ age zur theorie der toeplitzschen formen.Mathematische Zeitschrift, 6:167–202, 1920

work page 1920

[28] [28]

Taniguchi

M. Taniguchi. Minimum contrast estimation for spectral densities of stationary processes. Journal of the Royal Statistical Society. Series B , 49:315–325, 1987

work page 1987

[29] [29]

I. Vajda. Theory of Statistical Inference and Information . Springer Dordrecht, 1989

work page 1989

[30] [30]

van Erven and P

T. van Erven and P. Harremos. R´ enyi divergence and Kullback–Leibler divergence. IEEE Transactions on Information Theory , 60:3797–3820, 2014

work page 2014

[31] [31]

P. Whittle. The analysis of multiple stationary time series. Journal of the Royal Statistical Society. Series B (Methodological), 15:125–139, 1953

work page 1953

[32] [32]

Shumway Y

R. Shumway Y. Kakizawa and M. Taniguchi. Discrimination and clustering for multivariate time series. Journal of the American Statistical Association , 93(441):328–340, 1998

work page 1998

[33] [33]

Yoshimitsu, T

N. Yoshimitsu, T. Maeda, and T. Sei. Estimation of source parameters using a non-Gaussian probability density function in a bayesian framework. Earth, Planets and Space , 75, 2023

work page 2023

[34] [34]

On robustness of Spectral R´ enyi divergence

G. Zhang and M. Taniguchi. Nonparametric approach for discriminant analysis in time series. Journal of Nonparametric Statistics , 5:91–101, 1995. 15 Supplement to “On robustness of Spectral R´ enyi divergence” by Tetsuya Takabatake and Keisuke Yano This supplement presents • Proof of Theorem 1 in Appendix A; • Proof of Theorem 3 in Appendix B; • Proof of ...

work page 1995