On Least Squares with Insufficient Observations

Chipman, John S.

doi:10.1080/01621459.1964.10480751

Cited by 143 publications

(40 citation statements)

References 19 publications

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“…Of particular interest was a paper by Chipman (1964), which considered topics ranging from multicollinearity to problems of estimability in LS regression. It was well-written, paying close attention to the various historical perspectives, as well as to analytical and technical rigor.…”

Section: Bayes Approach To the Regression Problemmentioning

confidence: 99%

The Early Use of Matrix Diagonal Increments in Statistical Problems

Piegorsch

Casella

1989

SIAM Rev.

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The early motivation for and development of diagonal increments to ease matrix inversion in least squares (LS) problems is discussed. It is noted that this diagonal incrementation evolved from three major directions: modification of existing methodology in non-linear LS, utilization of additional information in linear regression, and the improvement of the numerical condition of a matrix. The interplay among these factors, and the advent of ridge regression are considered in an historical and comparative framework.

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Section: Bayes Approach To the Regression Problemmentioning

confidence: 99%

The Early Use of Matrix Diagonal Increments in Statistical Problems

Piegorsch

Casella

1989

SIAM Rev.

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“…3. If h ii = 0, then h ik = 0 for every k, 1 ≤ k ≤ n. 4. If h ii = 1, then h ki = 0 for every k = i.…”

Section: R(v ) = N ((Aw ) K 1 A(w Aw )) = N ((Aw ) K 1 )mentioning

confidence: 98%

“…is given by It is well known that the minimum-norm least-squares solution of inconsistent linear equations Ax = b is A † b (see [7]), that the minimum-norm (N ) least-squares [4]) and that the unique solution of the restricted linear equations [1,Exercise 4.8.28], and the unique solution of the restricted linear equations [5]). Without regard to restriction of element b in those equations, by [3, Lemma 3.1], these solutions are unified as {2} inverses with prescribed range and null space.…”

Section: Introductionmentioning

confidence: 99%

PCR algorithm for parallel computing the solution of the general restricted linear equations

2008

J. Appl. Math. Comput.

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“…This problem was studied in [6,12] and some applications to statistical problems can be found in [4,5,21,23]. Notice that (5) is equivalent to…”

Section: Introductionmentioning

confidence: 97%

A Geometrical Approach to Indefinite Least Squares Problems

2009

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Given Hilbert spaces H and K, a (bounded) closed range operator C : H → K and a vector y ∈ K, consider the following indefinite least squares problem:This work is devoted to give necessary and sufficient conditions for the existence of solutions of this abstract problem. Although the indefinite least squares problem has been thoroughly studied in finite dimensional spaces, the geometrical approach presented in this manuscript is quite different from the analytical techniques used before. As an application we provide some new sufficient conditions for the existence of solutions of an H ∞ estimation problem.

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On Least Squares with Insufficient Observations

Cited by 143 publications

References 19 publications

The Early Use of Matrix Diagonal Increments in Statistical Problems

The Early Use of Matrix Diagonal Increments in Statistical Problems

PCR algorithm for parallel computing the solution of the general restricted linear equations

A Geometrical Approach to Indefinite Least Squares Problems

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