Detection with the scan and the average likelihood ratio

Chan, Hock Peng; Walther, Guenther

doi:10.5705/ss.2011.169

Cited by 55 publications

(101 citation statements)

References 16 publications

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“…We will show that for

(j, k) \in I_{a p p} (l)

double-struckE ((), L R_{n} ((), U_{(k)} MathClass-bin- U_{(j)} MathClass-punc, \frac{k MathClass-bin- j}{n}) 1_{{scriptB}_{m MathClass-punc, n}}) MathClass-rel\leq 14 ((), \sqrt{2 ℓ} MathClass-bin+ m) eventually, uniformly in

( j , k ) and ℓ . Then

A_{n}^{cond} MathClass-rel= O_{p} (1)

can be shown as in the proof of Theorem 3 in Chan & Walther () because

\underset{normallim}{msub} {normalliminf}_{n MathClass-rel\to MathClass-rel\infty} double-struckP ({scriptB}_{m MathClass-punc, n}) MathClass-rel= 1

by Theorem , which is readily seen to hold also with

\frac{k MathClass-bin- j}{n}

in place of

\frac{k MathClass-bin- j MathClass-bin+ 1}{n}

in the definition of

P_{n}^{all}

.…”

Section: Proofsmentioning

confidence: 75%

“…As mentioned previously one can show that ℓ ∈ {2, … , ℓ max } with probability converging to one. As in Lemma 2 of Chan & Walther ) one finds

\frac{# A (I)}{# I_{a p p}} \geq C \frac{η_{n}^{2} F_{n} (I)}{{((), {log}_{2} e ∕ F_{n} (I))}^{8 ∕ 5}}

Standard considerations using Lemma and ( show that the event

({}, \underset{normalinf}{msub} 1 ((), F_{n} (Ĩ) MathClass-rel> F_{0} (Ĩ)) MathClass-rel= 1)

has probability converging to one, hence, on this event

\begin{matrix} left \\ inf_{\tilde{I} \in A (I)} \sqrt{2 l o g L R_{n} (F_{0} (\overset{I}{true}), F_{n} (\overset{I}{true}))} \geq inf_{\tilde{I} \in A (I)} \sqrt{n} \frac{F_{n} (\tilde{I}) - F_{0} (\tilde{I})}{\sqrt{F_{0} (\tilde{I}) \lor F_{n} (\tilde{I})}} by (6) \\ \geq inf_{\tilde{I} \in A (I)} \sqrt{n} ... \end{matrix}

…”

Section: Proofsmentioning

confidence: 79%

“…The density setting investigated here requires a proof that is more involved than the one in the regression setting considered in Chan & Walther (). Further, in the density setting there is no need to consider small intervals with empirical measure less than about log n ∕ n , and

I_{a p p}

is defined accordingly.…”

Section: The Condensed Average Likelihood Ratiomentioning

confidence: 99%

“…In contrast, one sees that the scan M n is competitive only for signals on the smallest scales and it is inferior to P n and

A_{n}^{cond}

otherwise. In the context of regression, the inferiority of the scan at larger scales was expounded theoretically by Chan & Walther (). Note that unlike in the regression context, ‘scale’ is not given by the length | I | but by F r , I ( I ), which is of the order rF 0 ( I ) as long as the latter quantity stays bounded.…”

Section: A Simulation Studymentioning

confidence: 99%

“…Hence,

\underset{normalinf}{msub} L R_{n} ((), F_{0} (Ĩ) MathClass-punc, F_{n} (Ĩ)) MathClass-rel\geq \frac{1}{F_{r MathClass-punc, I} (I) (1 MathClass-bin- F_{r MathClass-punc, I} (I))} normalexp ({}, B_{n} ((), B_{n} MathClass-bin∕ 2 MathClass-bin+ \sqrt{2 normallog \frac{e}{F_{r MathClass-punc, I} (I)}}))

and so

A_{n}^{cond} \overset{MathClass-rel\to}{stackrel} MathClass-rel\infty

as in the proof of Theorem 3 in Chan & Walther ), using (. Because the critical value of

A_{n}^{cond}

stays bounded by Proposition , the claim follows.…”

Section: Proofsmentioning

confidence: 99%

See 4 more Smart Citations

Optimal detection of a jump in the intensity of a Poisson process or in a density with likelihood ratio statistics

Rivera

Walther

2013

Scandinavian J Statistics

Self Cite

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We consider the problem of detecting a ‘bump’ in the intensity of a Poisson process or in a density. We analyze two types of likelihood ratio‐based statistics, which allow for exact finite sample inference and asymptotically optimal detection: The maximum of the penalized square root of log likelihood ratios (‘penalized scan’) evaluated over a certain sparse set of intervals and a certain average of log likelihood ratios (‘condensed average likelihood ratio’). We show that penalizing the square root of the log likelihood ratio — rather than the log likelihood ratio itself — leads to a simple penalty term that yields optimal power. The thus derived penalty may prove useful for other problems that involve a Brownian bridge in the limit. The second key tool is an approximating set of intervals that is rich enough to allow for optimal detection, but which is also sparse enough to allow justifying the validity of the penalization scheme simply via the union bound. This results in a considerable simplification in the theoretical treatment compared with the usual approach for this type of penalization technique, which requires establishing an exponential inequality for the variation of the test statistic. Another advantage of using the sparse approximating set is that it allows fast computation in nearly linear time. We present a simulation study that illustrates the superior performance of the penalized scan and of the condensed average likelihood ratio compared with the standard scan statistic.

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“…We will show that for

(j, k) \in I_{a p p} (l)

double-struckE ((), L R_{n} ((), U_{(k)} MathClass-bin- U_{(j)} MathClass-punc, \frac{k MathClass-bin- j}{n}) 1_{{scriptB}_{m MathClass-punc, n}}) MathClass-rel\leq 14 ((), \sqrt{2 ℓ} MathClass-bin+ m) eventually, uniformly in

( j , k ) and ℓ . Then

A_{n}^{cond} MathClass-rel= O_{p} (1)

can be shown as in the proof of Theorem 3 in Chan & Walther () because

\underset{normallim}{msub} {normalliminf}_{n MathClass-rel\to MathClass-rel\infty} double-struckP ({scriptB}_{m MathClass-punc, n}) MathClass-rel= 1

by Theorem , which is readily seen to hold also with

\frac{k MathClass-bin- j}{n}

in place of

\frac{k MathClass-bin- j MathClass-bin+ 1}{n}

in the definition of

P_{n}^{all}

.…”

Section: Proofsmentioning

confidence: 75%

“…As mentioned previously one can show that ℓ ∈ {2, … , ℓ max } with probability converging to one. As in Lemma 2 of Chan & Walther ) one finds

\frac{# A (I)}{# I_{a p p}} \geq C \frac{η_{n}^{2} F_{n} (I)}{{((), {log}_{2} e ∕ F_{n} (I))}^{8 ∕ 5}}

Standard considerations using Lemma and ( show that the event

({}, \underset{normalinf}{msub} 1 ((), F_{n} (Ĩ) MathClass-rel> F_{0} (Ĩ)) MathClass-rel= 1)

has probability converging to one, hence, on this event

\begin{matrix} left \\ inf_{\tilde{I} \in A (I)} \sqrt{2 l o g L R_{n} (F_{0} (\overset{I}{true}), F_{n} (\overset{I}{true}))} \geq inf_{\tilde{I} \in A (I)} \sqrt{n} \frac{F_{n} (\tilde{I}) - F_{0} (\tilde{I})}{\sqrt{F_{0} (\tilde{I}) \lor F_{n} (\tilde{I})}} by (6) \\ \geq inf_{\tilde{I} \in A (I)} \sqrt{n} ... \end{matrix}

…”

Section: Proofsmentioning

confidence: 79%

I_{a p p}

is defined accordingly.…”

Section: The Condensed Average Likelihood Ratiomentioning

confidence: 99%

“…In contrast, one sees that the scan M n is competitive only for signals on the smallest scales and it is inferior to P n and

A_{n}^{cond}

Section: A Simulation Studymentioning

confidence: 99%

“…Hence,

\underset{normalinf}{msub} L R_{n} ((), F_{0} (Ĩ) MathClass-punc, F_{n} (Ĩ)) MathClass-rel\geq \frac{1}{F_{r MathClass-punc, I} (I) (1 MathClass-bin- F_{r MathClass-punc, I} (I))} normalexp ({}, B_{n} ((), B_{n} MathClass-bin∕ 2 MathClass-bin+ \sqrt{2 normallog \frac{e}{F_{r MathClass-punc, I} (I)}}))

and so

A_{n}^{cond} \overset{MathClass-rel\to}{stackrel} MathClass-rel\infty

as in the proof of Theorem 3 in Chan & Walther ), using (. Because the critical value of

A_{n}^{cond}

stays bounded by Proposition , the claim follows.…”

Section: Proofsmentioning

confidence: 99%

See 3 more Smart Citations

Optimal detection of a jump in the intensity of a Poisson process or in a density with likelihood ratio statistics

Rivera

Walther

2013

Scandinavian J Statistics

Self Cite

View full text Add to dashboard Cite

show abstract

Detecting multiple generalized change-points by isolating single ones

Anastasiou

Fryźlewicz

2021

Metrika

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We introduce a new approach, called Isolate-Detect (ID), for the consistent estimation of the number and location of multiple generalized change-points in noisy data sequences. Examples of signal changes that ID can deal with are changes in the mean of a piecewise-constant signal and changes, continuous or not, in the linear trend. The number of change-points can increase with the sample size. Our method is based on an isolation technique, which prevents the consideration of intervals that contain more than one change-point. This isolation enhances ID’s accuracy as it allows for detection in the presence of frequent changes of possibly small magnitudes. In ID, model selection is carried out via thresholding, or an information criterion, or SDLL, or a hybrid involving the former two. The hybrid model selection leads to a general method with very good practical performance and minimal parameter choice. In the scenarios tested, ID is at least as accurate as the state-of-the-art methods; most of the times it outperforms them. ID is implemented in the R packages IDetect and breakfast, available from CRAN.

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Persistence Barcodes Versus Kolmogorov Signatures: Detecting Modes of One-Dimensional Signals

et al. 2015

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We investigate the problem of estimating the number of modes (i.e., local maxima)-a well known question in statistical inference-and we show how to do so without presmoothing the data. To this end, we modify the ideas of persistence barcodes by first relating persistence values in dimension one to distances (with respect to the supremum norm) to the sets of functions with a given number of modes, and subsequently working with norms different from the supremum norm. As a particular case we investigate the Kolmogorov norm. We argue that this modification has certain statistical advantages. We offer confidence bands for the attendant Kolmogorov signatures, thereby allowing for the selection of relevant signatures with a statistically controllable error. As a result of independent interest, we show that taut strings minimize the number of critical points for a very general class of functions. We illustrate our results by several numerical examples.

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Detection with the scan and the average likelihood ratio

Cited by 55 publications

References 16 publications

Optimal detection of a jump in the intensity of a Poisson process or in a density with likelihood ratio statistics

Optimal detection of a jump in the intensity of a Poisson process or in a density with likelihood ratio statistics

Detecting multiple generalized change-points by isolating single ones

Persistence Barcodes Versus Kolmogorov Signatures: Detecting Modes of One-Dimensional Signals

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