find_eta: The proportionality constant eta of the t-MLE for scatter

In robFitConGraph: Graph-Constrained Robust Covariance Estimation

The proportionality constant η of the t-MLE for scatter

Description

The proportionality constant η of the t-MLE for scatter

Usage

find_eta(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 positive real number or Inf. The degrees of freedom of the t-distribution the M-estimator is derived from. Inf is the usual sample covariance (i.e. the MLE under normality).
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_eta() returns the proportionality constant η for inputs ν, m and p. (Note: if ν \neq m, \hat{S}_n is technically not an MLE, but an M-estimator.)

Some specific values:

• If ν = m (also for ∞ = ∞), then η = 1 (i.e., the MLE at the corresponding population distribution consistently estimates its population value).

• If m = ∞ and 2 < ν < ∞, then η = ν/(ν-2).

• If m = ∞ and ν <= 2, then η = ∞. Precisely: \hat{S}_n does not converge in this case.

The general expressions: η is the solution to
F(η) = E(φ(R/η)) - p = 0,
where φ(y) = y(m+p)/(m+y) and R = (X - μ)^\top S^{-1} (X-μ) for X \sim t_{ν,p}(μ,S). For the integral, stats::integrate is used, for finding the root the function stats::uniroot.

In general, ν (df_data) and m (df_est) can take on any positive value, including ∞. The function works well for p <= 100 and ν >= 1. For larger values of p, setting η = 1 provides a good approximation (unless df_data is very small and df_est is rather large). For smaller values of ν, try the following potential remedies:

• Re-consider if ν < 1 is really necessary. This is VERY heavy-tailed.

• Adaptation of the search interval.

• By a suitable substitution, transform the integral to numerically more stable one (bounded support).

F(η) is a decreasing function.

• The larger df_est, the larger η.

• The larger df_data, the smaller η.

• The larger p, the closer η is to 1.

Value

A real value. Returns the constant η (cf. Vogel and Tyler 2014, p. 870, Example 2). This first appeared in Tyler (1982, p. 432, Example 3) as σ^{-1}. It is also stated in Tyler (1983, p. 418) as σ^{-1}_{u,g}.

Author(s)

Daniel Vogel

References

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

Tyler, D. E. (1983): Robustness and efficiency properties of scatter matrices, Biometrika, 70, 2, pp. 411-20

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

Examples

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

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