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

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 A^ε _n . . 39

A.2.2. Asymptotic distribution of B^ε _n . . 42

A.2.3. Asymptotic distribution of ALR_n . . 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

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for independently distributed random variables Y_i and noise components Z_i^iiid N(0,1) for i = 1,...,n. For the moment, the signal f_n belongs to the class of parametric functions

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and both the amplitude μ_n and the length I_n 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 I_n 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

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that calculates local likelihood ratios on each interval

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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

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that averages local likelihood ratios over the set I_n. 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 ALR_n, M _n and a modification of the latter called penalized scan statistic

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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 f_n in Model 1.1, whereas now we consider

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We are interested in denoising the data, which means finding an estimator f^ = fˆ₁,..., fˆ_n belong to F, s.t. the residuals e = Y − fˆ for the data set Y =Y₁,...,Y_n "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

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where q is the (1 - α)-quantile of the ALR null distribution. It is clear that the set C_n(Y, q) contains arbitrary many estimators. Thus it is necessary to develop an optimality criterion to pick the "best" fˆ (symbol cannot be displayed) C_n(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

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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.

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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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