U. Grenander and M. Rosenblatt :
“On spectral analysis of stationary time series ,”
Proc. Natl. Acad. Sci. U.S.A.
38 : 6
(June 1952 ),
pp. 519–521 .
MR
48737
Zbl
0047.12503
article

Abstract
People
BibTeX

The present statistical theory of analysis of stationary time series (e.g., extrapolation) has assumed complete knowledge of the covariance sequence or equivalently of the spectrum of the process. It is, therefore, of great importance to be able to estimate one of these. However, knowledge of the spectrum seems to yield greater immediate insight into the structure of the process. This seems to have first been noted in a fundamental paper by Bartlett [1950]. An unpublished paper by Tukey [1949] deals with some aspects of the problem of estimating the spectrum.

@article {key48737m,
AUTHOR = {Grenander, Ulf and Rosenblatt, Murray},
TITLE = {On spectral analysis of stationary time
series},
JOURNAL = {Proc. Natl. Acad. Sci. U.S.A.},
FJOURNAL = {Proceedings of the National Academy
of Sciences of the United States of
America},
VOLUME = {38},
NUMBER = {6},
MONTH = {June},
YEAR = {1952},
PAGES = {519--521},
DOI = {10.1073/pnas.38.6.519},
NOTE = {MR:48737. Zbl:0047.12503.},
ISSN = {0027-8424},
}
U. Grenander and M. Rosenblatt :
“Statistical spectral analysis of time series arising from stationary stochastic processes ,”
Ann. Math. Stat.
24 : 4
(December 1953 ),
pp. 537–558 .
MR
58901
Zbl
0053.41005
article

Abstract
People
BibTeX
@article {key58901m,
AUTHOR = {Grenander, Ulf and Rosenblatt, Murray},
TITLE = {Statistical spectral analysis of time
series arising from stationary stochastic
processes},
JOURNAL = {Ann. Math. Stat.},
FJOURNAL = {Annals of Mathematical Statistics},
VOLUME = {24},
NUMBER = {4},
MONTH = {December},
YEAR = {1953},
PAGES = {537--558},
DOI = {10.1214/aoms/1177728913},
NOTE = {MR:58901. Zbl:0053.41005.},
ISSN = {0003-4851},
}
U. Grenander and M. Rosenblatt :
“Comments on statistical spectral analysis ,”
Skand. Aktuarietidskr.
36 : Suppl. 1
(1953 ),
pp. 182–202 .
MR
60796
Zbl
0053.41101
article

Abstract
People
BibTeX

One of the objects of this paper is to give a heuristic derivation of the main result in [Grenander and Rosenblatt 1953] and to camment on its practical application in time series analysis. The rigorous proofs in [Grenander and Rosenblatt 1953] were rather detailed and laborious. We feel that this paper will be useful to the reader whose main interest is not in mathematical niceties. Time series analysis is of great practical importance and has grown rapidly in the past few years. Much work is yet to be done. The comments in this paper should not he interpreted as recommendations for optimal procedures. Tables and graphs of relevant limiting distribution functions are given. Although this paper deals with the distribution theory of a class of estimates of the spectral distribution function of a real stationary time series, we shall first consider some nonstatistical aspects of time series analysis.

@article {key60796m,
AUTHOR = {Grenander, Ulf and Rosenblatt, Murray},
TITLE = {Comments on statistical spectral analysis},
JOURNAL = {Skand. Aktuarietidskr.},
FJOURNAL = {Skandinavisk Aktuarietidskrift. Scandinavian
Actuarial Journal},
VOLUME = {36},
NUMBER = {Suppl. 1},
YEAR = {1953},
PAGES = {182--202},
DOI = {10.1080/03461238.1953.10419471},
NOTE = {MR:60796. Zbl:0053.41101.},
ISSN = {0037-606X},
}
U. Grenander and M. Rosenblatt :
“An extension of a theorem of G. Szegő and its application to the study of stochastic processes ,”
Trans. Am. Math. Soc.
76 : 1
(January 1954 ),
pp. 112–126 .
MR
58902
Zbl
0059.11804
article

Abstract
People
BibTeX

In this paper we study minimum problems associated with quadratic forms
\[ Q_n = c^{\prime}M^{(n)}c \]
where \( c \) is a column vector with components \( c_0 \) , \( c_1, \dots \) , \( c_n \) and \( M^{(n)} \) is a Hermitian matrix with the elements
\[ m_{p,q}^{(n)} = \int_{-\pi}^{\pi} e^{i(p-q)\lambda}f(\lambda)\,d\lambda, \quad p,q = 0,1,\dots,n. \]
We denote the conjugate of the transpose of a matrix \( A \) by \( A^{\prime} \) . Here \( f(\lambda) \) is a nonnegative integrable function in \( (-\pi,\pi] \) . We shall define \( f(\lambda) \) with period \( 2\pi \) on the real axis. Some of these minimum problems arise in the theory of stationary stochastic processes. These applications will be discussed in §5 [Grenander 1951].

@article {key58902m,
AUTHOR = {Grenander, Ulf and Rosenblatt, Murray},
TITLE = {An extension of a theorem of {G}. {S}zeg\H{o}
and its application to the study of
stochastic processes},
JOURNAL = {Trans. Am. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical
Society},
VOLUME = {76},
NUMBER = {1},
MONTH = {January},
YEAR = {1954},
PAGES = {112--126},
DOI = {10.2307/1990746},
NOTE = {MR:58902. Zbl:0059.11804.},
ISSN = {0002-9947},
}
U. Grenander and M. Rosenblatt :
“Regression analysis of time series with stationary residuals ,”
Proc. Natl. Acad. Sci. U.S.A.
40 : 9
(September 1954 ),
pp. 812–816 .
MR
62403
Zbl
0059.13404
article

People
BibTeX
@article {key62403m,
AUTHOR = {Grenander, Ulf and Rosenblatt, Murray},
TITLE = {Regression analysis of time series with
stationary residuals},
JOURNAL = {Proc. Natl. Acad. Sci. U.S.A.},
FJOURNAL = {Proceedings of the National Academy
of Sciences of the United States of
America},
VOLUME = {40},
NUMBER = {9},
MONTH = {September},
YEAR = {1954},
PAGES = {812--816},
DOI = {10.1073/pnas.40.9.812},
NOTE = {MR:62403. Zbl:0059.13404.},
ISSN = {0027-8424},
}
U. Grenander and M. Rosenblatt :
“Some problems in estimating the spectrum of a time series ,”
pp. 77–93
in
Proceedings of the third Berkeley symposium on mathematical statistics and probability
(Berkeley, CA, 26–31 December 1954 and July–August 1955 ),
vol. 1: Contributions to the theory of statistics .
Edited by J. Nyman .
University of California Press (Berkeley and Los Angeles, CA ),
1956 .
MR
84914
Zbl
0072.36401
incollection

People
BibTeX
@incollection {key84914m,
AUTHOR = {Grenander, Ulf and Rosenblatt, Murray},
TITLE = {Some problems in estimating the spectrum
of a time series},
BOOKTITLE = {Proceedings of the third {B}erkeley
symposium on mathematical statistics
and probability},
EDITOR = {Nyman, Jerzy},
VOLUME = {1: Contributions to the theory of statistics},
PUBLISHER = {University of California Press},
ADDRESS = {Berkeley and Los Angeles, CA},
YEAR = {1956},
PAGES = {77--93},
URL = {https://digitalassets.lib.berkeley.edu/math/ucb/text/math_s3_v1_article-07.pdf},
NOTE = {(Berkeley, CA, 26--31 December 1954
and July--August 1955). MR:84914. Zbl:0072.36401.},
}
U. Grenander and M. Rosenblatt :
Statistical analysis of stationary time series .
Wiley Publications in Mathematical Statistics .
Almqvist & Wiksell (Stockholm ),
1957 .
A 2nd, corrected edition was published in 1984 , then republished in 2008 .
MR
84975
Zbl
0080.12904
book

People
BibTeX
@book {key84975m,
AUTHOR = {Grenander, Ulf and Rosenblatt, Murray},
TITLE = {Statistical analysis of stationary time
series},
SERIES = {Wiley Publications in Mathematical Statistics},
PUBLISHER = {Almqvist \& Wiksell},
ADDRESS = {Stockholm},
YEAR = {1957},
PAGES = {300},
NOTE = {A 2nd, corrected edition was published
in 1984, then republished in 2008. MR:84975.
Zbl:0080.12904.},
}
M. Rosenblatt :
“Statistical analysis of stochastic processes with stationary residuals ,”
pp. 246–275
in
Probability and statistics: The Harald Cramér volume .
Edited by U. Grenander .
Wiley Publications in Statistics .
Almqvist & Wiksell (Stockholm ),
1959 .
MR
107954
Zbl
0201.51701
incollection

People
BibTeX
@incollection {key107954m,
AUTHOR = {Rosenblatt, M.},
TITLE = {Statistical analysis of stochastic processes
with stationary residuals},
BOOKTITLE = {Probability and statistics: {T}he {H}arald
{C}ram\'er volume},
EDITOR = {Grenander, Ulf},
SERIES = {Wiley Publications in Statistics},
PUBLISHER = {Almqvist \& Wiksell},
ADDRESS = {Stockholm},
YEAR = {1959},
PAGES = {246--275},
NOTE = {MR:107954. Zbl:0201.51701.},
}
U. Grenander and M. Rosenblatt :
Statistical analysis of stationary time series ,
2nd, corrected edition.
Chelsea Publishing Company (New York ),
1984 .
2nd, corrected edition of 1957 original . Republished in 2008 .
MR
890514
Zbl
0575.62080
book

People
BibTeX
@book {key890514m,
AUTHOR = {Grenander, Ulf and Rosenblatt, Murray},
TITLE = {Statistical analysis of stationary time
series},
EDITION = {2nd, corrected},
PUBLISHER = {Chelsea Publishing Company},
ADDRESS = {New York},
YEAR = {1984},
PAGES = {308},
NOTE = {2nd, corrected edition of 1957 original.
Republished in 2008. MR:890514. Zbl:0575.62080.},
ISBN = {9780828403207},
}
U. Grenander and M. Rosenblatt :
Statistical analysis of stationary time series ,
2nd, republished edition.
AMS Chelsea Publishing Series 320 .
American Mathematical Society (Providence, RI ),
2008 .
Zbl
0902.62115
book

People
BibTeX
@book {key0902.62115z,
AUTHOR = {Grenander, Ulf and Rosenblatt, Murray},
TITLE = {Statistical analysis of stationary time
series},
EDITION = {2nd, republished},
SERIES = {AMS Chelsea Publishing Series},
NUMBER = {320},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2008},
PAGES = {308},
NOTE = {Zbl:0902.62115.},
ISBN = {9780821844373},
}