The Annals of Statistics

Random Discriminants

I-Li Lu and Donald Richards

Full-text: Open access

Abstract

Let $X_1, X_2, \cdots, X_n$ be a random sample from a continuous univariate distribution $F$, and let $\Delta = \prod_{1 \leq i < j \leq n}(X_i - X_j)^2$ denote the discriminant, or square of the Vandermonde determinant, constructed from the random sample. The statistic $\Delta$ arises in the study of moment matrices and inference for mixture distributions, the spectral theory of random matrices, control theory and statistical physics. In this paper, we study the probability distribution of $\Delta$. When $X_1, \cdots, X_n$ is a random sample from a normal, gamma or beta population, we use Selberg's beta integral formula to obtain stochastic representations for the exact distribution of $\Delta$. Further, we obtain stochastic bounds for $\Delta$ in the normal and gamma cases. Using the theory of $U$-statistics, we derive the asymptotic distribution of $\Delta$ under certain conditions on $F$.

Article information

Source
Ann. Statist., Volume 21, Number 4 (1993), 1982-2000.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176349406

Digital Object Identifier
doi:10.1214/aos/1176349406

Mathematical Reviews number (MathSciNet)
MR1245777

Zentralblatt MATH identifier
0791.62059

JSTOR
links.jstor.org

Subjects
Primary: 62E15: Exact distribution theory
Secondary: 62H10: Distribution of statistics 60E15: Inequalities; stochastic orderings 62H05: Characterization and structure theory 62G30: Order statistics; empirical distribution functions

Keywords
Moment matrices mixture distributions random discriminants Selberg's beta integral stochastic bounds $U$-statistics Vandermonde determinants

Citation

Lu, I-Li; Richards, Donald. Random Discriminants. Ann. Statist. 21 (1993), no. 4, 1982--2000. doi:10.1214/aos/1176349406. https://projecteuclid.org/euclid.aos/1176349406


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