Parameter Estimation and Hypothesis Testing in Spectral Analysis of Stationary Time Series

330 Pages·1986·24.263 MB·English

by  K. Dzhaparidze (auth.)

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Spril1ger Series in Statistics
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K. Dzhaparidze
Parameter Estitnation
and Hypothesis Testing
in Spectral Analysis
of Stationary Titne Series
Translated by Samuel Kotz
Springer-Verlag
New York Berlin Heidelberg Tokyo
K. Dzhaparidze Samuel Kotz (TransLator)
Mathematisch Centrum Department of Management Science
Kruislaan 413 and Statistics
Postbus 4079 University of Maryland
1098 SJ Amsterdam College Park, Maryland 20742
The Netherlands U.S.A.
AMS Classification: 62MlO, 62F99
Library of Congress Cataloging-in-Publication Data
Dzhaparidze, K. O.
Parameter estimation and hypothesis testing in
spectral analysis of stationary time series.
(Springer series in statistics)
Translation of: Asimptoticheski effektivnoe
ofsenivanie parametrov spektra gaussovskogo vremennogo
riada.
Bibliography: p.
Includes index.
1. Time-series analysis. 2. Spectral theory
(Mathematics) 3. Parameter estimation. 4. Statistical
hypothesis testing. I. Title. II. Series.
QA280.D9313 1985 519.5'5 85-22207
The original Russian edition was published by the Publishing House of the University
of Tiblissi in 1981 "Schiitzung von Parametern und Priifung von Hypothesen in der
Spektralanalyse von stationiiren vorliiufigen Reihen" .
© 1986 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover I st edition 1986
All rights reserved. No part of this book may be translated or reproduced in any form
without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New
York 10010, U.S.A.
9 8 7 654 3 2 1
ISBN-13:978-1-4612-932S-S e-ISBN-13:978-1-4612-4842-2
DOl: 10.1007/978-1-4612-4842-2
CONTENTS
Introduction
CHAPTER I
Properties of Maximum Likelihood Function for a Gaussian Time Series 35
1. General Expression for the log Likelihood 35
2. Asymptotic Expression for the "Principal Part" of the log Likelihood 50
3. The Asymptotic Differentiability of Gaussian Distributions with Spectral
Densities Separated from Zero 59
4. The Asymptotic Differentiability of Gaussian Distributions with Spectral
Densities Possessing Fixed Zeros 67
Appendix 1 73
Appendix 2 78
Appendix 3. Remarks and Bibliography 93
CHAPTER II
Estimation of Parameters by Means of P. Whittle's Method 102
1. Asymptotic Maximum Likelihood Estimators 102
2. Properties of Asymptotic Maximum Likelihood Estimators in the Case of
Strictly Positive Spectral Density 104
3. Consistency, Asymptotic Normality, and Asymptotic Efficiency of the
Estimator 6 in the Case of Spectral Density Possessing Fixed Zeros 110
4. Examples of Determination of Asymptotic Maximum Likelihood Estimators 115
5. Asymptotic Maximum Likelihood Estimator of the Spectrum of Processes
Distorted by "White Noise" 128
6. Least-Squares Estimation of Parameters of a Spectrum of a Linear Process 139
7. Estimation by Means of the Whittle Method of Spectrum Parameters of
General Processes Satisfying the Strong Mixing Condition 149
Appendix 1 151
Appendix 2 166
Appendix 3. Remarks and Bibliography 182
Vi Contents
CHAPTER III
Simplified Estimators Possessing "Nice" Asymptotic Properties 198
1. Asymptotic Properties of Simplified Estimators 198
2. Examples of Preliminary Consistent Estimators 210
3. Examples of Constructing Simplified Estimators 222
Appendix 1. Remarks and Bibliography 234
CHAPTER IV
Testing Hypotheses on Spectrum Parameters of a Gaussian Time Series 236
1. Testing Simple Hypotheses 236
2. Testing Composite Hypotheses (The Case of a Sequence of General
"Asymptotically Differentiable Experiments") 247
3. Testing of Composite Hypothesis about a Parameter of a Spectrum of a
Gaussian Time Series 258
Appendix 1. Remarks and Bibliography 265
Chapter V
Goodness-of-Fit Tests for Testing the Hypothesis about the Spectrum of
Linear Processes 273
1. A Class of Goodness-of-Fit Tests for Testing a Simple Hypothesis about the
Spectrum of Linear Processes 273
2. X2 Test for Testing a Simple Hypothesis about the Spectrum of a Linear
Process 277
3. Goodness-of-Fit Test for Testing Composite Hypotheses about the Spectrum
of a Linear Process 284
Appendix 1. Remarks and Bibliography 300
Bibliography 306
Index 321
INTRODUCTION
1. Traditionally the most important problem of mathematical
statistics dealing with random stationary processes Xt, t = ... ,
-1,0,1, '" is the problem of estimating the second order
characteristics, namely the covariance function
or its Fourier transform -- the spectral density I = IC>..) (under
the assumption that the spectral density exists). For this
reason, a vast amount of periodical and monographic
literature is devoted to the nonparametric statistical problem
of estimating the function tJ( T) and especially that of leA)
(see, for example, the books [4,21,22,26,56,77,137,139,140,]).
However, the empirical value t;; of the spectral density I
obtained by applying a certain statistical procedure to the
observed values of the variables Xl' ... , Xn, usually depends in
a complicated manner on the cyclic frequency).. This fact
often presents difficulties in applying the obtained estimate
t;; of the function I to the solution of specific problems
rela ted to the process Xt. Theref ore, in practice, the
obtained values of the estimator t;; (or an estimator of the
covariance function tJ~( T» are almost always "smoothed," i.e.,
are approximated by values of a certain sufficiently simple
1
function = 1<).) (or D(T» of argument). (or of T) which is
then taken as the actual spectral density (or covariance
7
function). Quite often the functions or D(T) -- used as
2 Introduction
approximations -- are chosen in a manner 1 such that they are
defined by an analytic formula involving a finite number of
unknown parameters; estimation of these parameters is a
problem of parametric statistics. In this approach the
non parametric problem of estimating the unknown function f
T»
(or JJ( plays only an auxiliary role: its solution is utilized
solely for choosing a reasonable parametric problem -- under
given conditions. This parametric problem consists of
T»,
stipulating a hypothesis on the form of function f (or JJ(
testing this hypothesis, and next, estimating the unknown
parameters appearing in the expression of the function
corresponding to the accepted hypothesis. This fact is of
crucial importance in real practical situations of estimating
the paraweters of a spectral density (or a covariance
function)" of a random process2 and testing the·. hypothesis
about the form of the spectral density. A basic portion of
this monograph deals with an investigation of these problems.
Chapters II and III are devoted to the problem of
determining estimators of a finite number of unknown
spectral parameters while Chapters IV and V are concerned
with the construction of various criteria for testing
hypotheses concerning the form of the spectral density.
(Below we shall discuss this problem in somewhat more
detail.)
We now note that although the above mentioned problems
are perfectly sensible for a fixed (finite) sample size n and in
applications one will always be dealing with a finite n, the
mathematical results dealing with the case of fixed and not
too large n are very few and usually of little interest. This is
IThe choice o( ( as a ratiollal (unction o( e xp( i}.) is e s pe dally handy since
solutions o( many problems (or random prpcesses with rational spectral
densities were studied in detail (see, (or example [58,150,151)).
2Recently the practical importance of this problelTl has become more evident
for many investigators. One now encounters more orten in the applied
literature that parametric estimators o( spectral density are being utilized
from the very beginning (instead of nonparametric estimators ~); (or
example, the so-called "autoregressive estimators" or similar to them, the
"maximum entropy estimators" are based on the assumption that (().) are o(
a certain special form (ct., e. g., [22,34,77,115,126)).
Likelihood Functions for Gaussian Processes 3
due to the fact that a detailed investigation of the properties
of statistical procedures for a given finite are always very
It
cumbersome and hardly ever lead to explicit and visible
finite results; these investigations can thus be considered
unpromising.
An asymptotic analysis of statistical inference valid in the
limit as is a more satisfying venture. The problem of
It .... CD
the limiting behavior of statistical procedures as n .... are as
CD
a rule mathematically much more tractable than the
corresponding finite sample investigations. At the same time,
this appears to be of sufficient interest from the point of
view of applications as well, since for the values of n which
are not too large, the desired statistical inference (in
particular, the inference concerning the spectral density f)
will usually be quite inaccurate and is thus often considered
practically useless, while for a fixed large n the results
obtained are usually close to those obtained for This is
It .... CD.
the reason that in the work below, we shall be mainly
concerned only with the asymptotic results corresponding to
n ....
CD.
For simplicity, we shall always assume that the process X
t
has a zero expected value E(Xt) = 0. This assumption usually
does not result in a loss of generality since even if E(Xt) is
unknown, in applications of asymptotic results valid for n ....
CD, it is usually sufficient to replace Xt by Xt - (l/It)rj=lXj in
order to assume that the process under consideration possesses
a zero mean value.
We now proceed to a brief description of the content of
this monograph.
Properties of the Likelihood Function for Gaussian Processes
2. Assume that X is a real-valued stationary random process
t
with discrete time t = ... , -1,0,1, ... , zero expected value E(Xt)
= 0, and covariance function P( T) sufficiently rapidly
decreasing at infinity and represented as
J"
E(XtXt-tT) = peT) = ei).T f().)d)..
-Il
When investigating mathematically the problems of
statistical inference concerning a spectral density f of a
stationary process X one should clearly first impose certain
t
4 In trod uction
assumptions on the corresponding probability distributions
which will specify the problem and permit a comparison of
various possible recommendations. Here it is natural to start
with the case of Gaussian processes which represent the
simplest and most important class of random functions, those
most widely studied, and often occurring in applications.3
Assuming that X is a Gaussian process, we can in principle
t
obtain an explicit expression for the logarithm of the
likelihood function
Ln = log Pn(Xl, ..., Xn)
(1) 1
= - ~n log 2" + log det(Bf) + X I BflX}
of random variables Xl' ... , Xn (which are components of the
random column-vector X), where P is an n-dimensional
n
probability density and Bf = [JJ(T-S)], T, S = 1, ... , n, is a
Toeplitz matrix associated with the function f. For this
purpose one is required, however, to solve the complicated
problem of determining the explicit expressions for det(Bf)
and Bfl (particular cases of this are dealt with in [3,87,93,113,
133, and 149]). As it will be seen in Example 1 in Section 1
of Chapter I, even in the simplest case, when X is an
t
autoregressive process, an explicit expression for Ln turns out
to be quite involved even for autoregressive processes of the
first order. Moreover, as the order of autoregression
increases, the formula for Ln becomes more complicated (cf.
[101,111,118]). Formulas contained in papers [3,93,113,133,
and 149] allow us, in principle, to obtain an explicit
expression for Ln also in the case when X is a moving
t
average process or even a mixed autoregressive-moving
average process; however, the formulas involved are
unavoidably, very cumbersome. This is easily seen by
considering the formula (1.1.16) -- (and the results of the
Examples 2-4 iii Section 1 of Chapter I following from it).
This formula is an expression (actually simpler than it was
3We also note that tHe experience accumulated in the course of study of
many other statistical problems (including random processes problems)
gives us the confidence to suppose that methods which are of high
accuracy in the Gaussian case will also be useful when applied to
numerous non-Gaussian probability distributions.