find_sigma1: The asymptotic efficiency constant sigma_1 of the t-MLE for...

In robFitConGraph: Graph-Constrained Robust Covariance Estimation

The asymptotic efficiency constant σ_1 of the t-MLE for scatter

Description

The asymptotic efficiency constant σ_1 of the t-MLE for scatter

Usage

find_sigma1(df_data, df_est, p)

Arguments

df_data	A positive real number or Inf. The degrees of freedom of the data-generationg t-distribution. Inf means normal distribution.
df_est	A non-negative real number or Inf. The degrees of freedom of the t-distribution the M-estimator is derived from. Inf is the usual sample covariance, 0 is Tyler's M-estimator.
p	An integer, at least 2.

Details

Let X_1,...,X_n be an i.i.d. sample from t_{ν,p}(μ, S), i.e., a p-variate t-distribution with ν degrees of freedom, location parameter μ and shape matrix S. The limit case ν = ∞ is allowed, where t_{∞,p}(μ,S) is N_p(μ,S).

Let \hat{S}_n be the t_m MLE for scatter. Also here, m=∞ is allowed: This is the sample covariance matrix. If \hat{S}_n is applied to X_1,...,X_n, then, as n \to ∞, \hat{S}_n converges in probability to η S. The function find_sigma1() returns a scalar appearing in the asymptotic covariance matrix of \hat{S}_n.

The scalar σ_1 is defined as

σ_1 = \frac{(p+2)^2 γ_1}{(2γ_2 + p)^2},

where

γ_1 = \frac{E\{φ^2(R/η)\}}{p(p+2)} \quad \mbox{ and } \quad γ_2 = \frac{1}{p} E≤ft\{\frac{R}{η}φ'≤ft(\frac{R}{η}\right)\right\},

furthermore φ(y) = y(m+p)/(m+y) and R = (X - μ)^\top S^{-1} (X-μ) for X \sim t_{ν,p}(μ,S), and η is defined in the help page of find_eta.

A noteworthy difference between find_sigma1 and find_eta is that the argument df_est may be 0 for find_sigma1, but must strictly positive for find_eta. For both functions, df_data must be strictly positive. There is no such thing as a t-distribution with zero degrees of freedom. There is such a thing as a t-MLE with zero degrees of freedom: the Tyler estimator. Its σ_1 value is 1 + 2/p regardless of the underlying elliptical distribution. However, since the Tyler estimator provides shape information only, but none on scale, η is irrelevant in this case.

Value

A real value. Returns the constant σ_1 (cf. Vogel and Tyler 2014, p. 870, Example 2). This first appeared in Tyler (1982, p. 432, Example 3).

Author(s)

Daniel Vogel

References

Vogel, D., Tyler, D. E. (2014): Robust estimators for nondecomposable elliptical graphical models, Biometrika, 101, 865-882

Tyler, D. E. (1982): Radial estimates and the test for sphericity, Biometrika, 69, 2, pp. 429-36

Examples

find_sigma1(df_data = Inf, df_est = 3,   p = 10)
find_sigma1(df_data = 4.5, df_est = 4.5, p = 2)

robFitConGraph documentation built on Dec. 1, 2022, 1:21 a.m.