On the distributions of the ratios of the extreme roots to the trace of the Wishart matrix

Schuurmann, F. J.; Krishnaiah, P. R.; Chattopadhyay, Arup

doi:10.1016/0047-259x(73)90033-x

Cited by 50 publications

(25 citation statements)

References 6 publications

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“…Various authors derived exact formulas for U in terms of high dimensional integrals or as inverses of certain Laplace transforms, which could then be evaluated numerically for small values of p, see [5,18]. More recently, [12] developed asymptotic expansions for tail probabilities of U by considering the extrema of certain random fields.…”

Section: Introductionmentioning

confidence: 99%

On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart matrix

Nadler

2011

Journal of Multivariate Analysis

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The ratio of the largest eigenvalue divided by the trace of a p×p random Wishart matrix with n degrees of freedom and identity covariance matrix plays an important role in various hypothesis testing problems, both in statistics and in signal processing. In this paper we derive an approximate explicit expression for the distribution of this ratio, by considering the joint limit as both p, n → ∞ with p/n → c. Our analysis reveals that even though asymptotically in this limit the ratio follows a Tracy-Widom (TW) distribution, one of the leading error terms depends on the second derivative of the TW distribution, and is non-negligible for practical values of p, in particular for determining tail probabilities. We thus propose to explicitly include this term in the approximate distribution for the ratio. We illustrate empirically using simulations that adding this term to the TW distribution yields a quite accurate expression to the empirical distribution of the ratio, even for small values of p, n.

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Section: Introductionmentioning

confidence: 99%

On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart matrix

Nadler

2011

Journal of Multivariate Analysis

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“…Indeed, comparing the detection performance of the modified AIC estimator, (19), with that of the RMT estimator, (25) in [17], no estimator dominates the other over all parameter ranges. In particular, the modified AIC algorithm may be useful in legacy systems where only minimal changes to an existing code can be made.…”

Section: Discussionmentioning

confidence: 99%

“…Then, asymptotically in , overestimation occurs when (40) Thus, asymptotically, the overestimation probability of the AIC estimator depends on the tail statistics of the ratio of the largest eigenvalue of a Wishart matrix with identity covariance and parameters and to the mean of all its eigenvalues. For small values of (say ), the tail distribution of this random variable was tabulated, see [25], and in principle could be numerically calculated for larger values of with the aid of a computer.…”

Section: Analysis Of MDL and Aic Detection Performancementioning

confidence: 99%

Nonparametric Detection of Signals by Information Theoretic Criteria: Performance Analysis and an Improved Estimator

Nadler

2010

IEEE Trans. Signal Process.

146

116

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Abstract-Determining the number of sources from observed data is a fundamental problem in many scientific fields. In this paper we consider the nonparametric setting, and focus on the detection performance of two popular estimators based on information theoretic criteria, the Akaike information criterion (AIC) and minimum description length (MDL). We present three contributions on this subject. First, we derive a new expression for the detection performance of the MDL estimator, which exhibits a much closer fit to simulations in comparison to previous formulas. Second, we present a random matrix theory viewpoint of the performance of the AIC estimator, including approximate analytical formulas for its overestimation probability. Finally, we show that a small increase in the penalty term of AIC leads to an estimator with a very good detection performance and a negligible overestimation probability.

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“…For further details on the likelihood ratio test for the rank of , the reader is referred to Kshirsagar (1972) and Rao (1973). Anderson (1951b) were given in Hsu (1941b Krishnaiah and Chattopadhyay (1973) and Krishnaiah and Schuurmann (1974).…”

Section: Tests On Discriminant Functions For Several Populations Usinmentioning

confidence: 99%

Selection of Variables in Discriminant Analysis.

Krishnaiah¹

1982

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On the distributions of the ratios of the extreme roots to the trace of the Wishart matrix

Cited by 50 publications

References 6 publications

On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart matrix

On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart matrix

Nonparametric Detection of Signals by Information Theoretic Criteria: Performance Analysis and an Improved Estimator

Selection of Variables in Discriminant Analysis.

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