# Comparison of the scan and the average likelihood ratio in Gaussian mean regression

Excerpt

## Contents

1. Introduction . . 3

2. Comparison of the ALR and the scan statistics in the testing framework . . 6

2.1. Classical theory . . 6

2.2. Introducing the ALR and the scan statistics . . 9

2.3. Asymptotic null distribution of the scan and the ALR statistics . . 10

2.3.1. Asymptotic distribution of the ALR statistic . . 10

2.3.2. Asymptotic distribution of the scan statistic . . 11

2.4. Asymptotic power of the scan and the ALR statistics . . 12

3. Estimation framework: implementation and practical results . . 17

3.1. Estimation procedure . . 17

3.2. SQP method . . 20

3.2.1. Introducing the SQP method . . 20

3.2.2. SQP approach . . 21

3.2.3. Pseudocode . . 22

3.2.4. Damped BFGS method. . . 22

3.3. Computational issues . . 24

3.4. Implementation results . . 26

4. Conclusion . . 31

A. Proofs . . 32

A.1. On the boundedness of the penalized scan statistic . . 32

A.2. Proof of Theorem 2.5 . . 38

A.2.1. Asymptotic distribution of Aε n . . 39

A.2.2. Asymptotic distribution of Bε n . . 42

A.2.3. Asymptotic distribution of ALRn . . 44

A.3. Proof of Theorem 2.9 . . 46

B. Auxiliary proofs . . 51

## 1. Introduction

The present thesis is devoted to the problem of detecting a signal with an unknown spatial extent against a noisy background. This is modelled within the framework of Gaussian mean regression. It has a number of scientiﬁc applications, for example, in epidemiology or astronomy, as stated in Chan and Walther (2011). Apart from that, this is also a challenging statistical issue from a purely theoretical point of view.

This work is divided into two major parts. We begin with considering a model

[Formula is omitted from this preview] (1.1)

for independently distributed random variables Yi and noise components Ziiiid N(0,1) for i = 1,...,n. For the moment, the signal fn belongs to the class of parametric functions

[Formula is omitted from this preview] (1.1)

and both the amplitude μn and the length In of it are unknown.

The detection problem may therefore be equivalently represented by a statistical test, where the null-hypothesis means that no signal is present. According to the Neyman-Pearson lemma, the uniformly most powerful test for the case when In is known is based on the likelihood ratio. In our case, when the signal location In is unknown, we have to analyse all intervals in In, i.e. consider a multiscale problem. There are at least two possible options to propose a test statistic under these circumstances. The ﬁrst one is the maximum likelihood ratio or the scan statistic

[Formula is omitted from this preview]

that calculates local likelihood ratios on each interval

[Interval is omitted from this preview]

and then chooses the maximum over them. This is rather a standard tool in statistical research and considerable amount of literature is available (eg. Glaz et al., 2001, and their references). The second option is the average likelihood ratio (ALR) statistic

[Formula is omitted from this preview]

that averages local likelihood ratios over the set In. Various versions and applications of the ALR statistic were considered, for example, in the works of Chan (2009) and Chan and Walther (2011).

Chapter 2 of the thesis introduces the test statistics ALRn, M n and a modiﬁcation of the latter called penalized scan statistic

[Formula is omitted from this preview]

We provide theoretical analysis of the properties of these statistics, such as the asymptotic null distribution and power for detecting signals.

Another important aspect that we address is the problem of estimating the real signal fn in Model 1.1, whereas now we consider

[Formula and interval are omitted from this preview]

We are interested in denoising the data, which means ﬁnding an estimator f^ = fˆ1,..., fˆn belong to F, s.t. the residuals e = Y − fˆ for the data set Y =Y1,...,Yn "look like" white Gaussian noise. We are using the ALRn to test whether the distribution of residuals is standard normal and therefore deﬁne the feasible set of admissible estimators fˆ as

[Formula is omitted from this preview] (1.2)

where q is the (1 - α)-quantile of the ALR null distribution. It is clear that the set Cn(Y, q) contains arbitrary many estimators. Thus it is necessary to develop an optimality criterion to pick the "best" fˆ (symbol cannot be displayed) Cn(Y, q). We do this by minimizing a cost functional J over C n(Y, q). As such, the denoising problem can be written as a constrained optimization problem

[Formula is omitted from this preview] (1.3)

In Chapter 3 we develop a numerical algorithm for solving (1.3) and illustrate its performance by a numerical example. We also compare these results with the ones obtained for the case when the feasible set (1.2) is deﬁned through the penalized scan, i.e.

[Formula is omitted from this preview] (1.4)

Chapter 4 contains the summary of the thesis results and a brief outlook on possible further research. For convenience, proofs are carried out separately in Appendices A and B.

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Details

Title
Comparison of the scan and the average likelihood ratio in Gaussian mean regression
College
University of Göttingen  (Institut für Mathematische Stochastik)
1.3
Author
Year
2013
Pages
56
Catalog Number
V333977
ISBN (eBook)
9783668240544
ISBN (Book)
9783668240551
File size
1946 KB
Language
English
Notes
This text was written by a non-native English speaker.
Tags
Statistik, statstics, Stochastic, scan, average likelihood ratio, penalized scan
Quote paper
Pavlova Evgenia (Author), 2013, Comparison of the scan and the average likelihood ratio in Gaussian mean regression, Munich, GRIN Verlag, https://www.grin.com/document/333977 