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 *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

[Formula is omitted from this preview] (1.1)

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

[Formula is omitted from this preview] (1.1)

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

[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 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

[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 f_{n} 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^
= fˆ_{1},..., fˆ_{n }belong to F, s.t. the residuals e = Y − fˆ for
the data set Y =Y_{1},...,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

[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 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

[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.

[...]

- 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

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