The asymptotic efficiency constant \(\sigma_1\) of the t-MLE for scatter
find_sigma1(df_data, df_est, p)A real value. Returns the constant \(\sigma_1\) (cf. Vogel and Tyler 2014, p. 870, Example 2). This first appeared in Tyler (1982, p. 432, Example 3).
A positive real number or Inf. The degrees of freedom of the
data-generationg t-distribution. Inf means normal distribution.
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.
An integer, at least 2.
Daniel Vogel
Let \(X_1,...,X_n\) be an i.i.d. sample from \(t_{\nu,p}(\mu, S)\), i.e.,
a p-variate t-distribution with \(\nu\) degrees of freedom, location parameter \(\mu\)
and shape matrix \(S\). The limit case \(\nu = \infty\) is allowed, where \(t_{\infty,p}(\mu,S)\) is
\(N_p(\mu,S)\).
Let \(\hat{S}_n\) be the \(t_m\) MLE for scatter.
Also here, \(m=\infty\) is allowed: This is the sample covariance matrix.
If \(\hat{S}_n\) is applied to \(X_1,...,X_n\), then, as \(n \to \infty\),
\(\hat{S}_n\) converges in probability to \(\eta S\).
The function find_sigma1() returns a scalar appearing in the asymptotic
covariance matrix of \(\hat{S}_n\).
The scalar \(\sigma_1\) is defined as
$$ \sigma_1 = \frac{(p+2)^2 \gamma_1}{(2\gamma_2 + p)^2}, $$
where
$$\gamma_1 = \frac{E\{\phi^2(R/\eta)\}}{p(p+2)} \quad \mbox{ and } \quad
\gamma_2 = \frac{1}{p} E\left\{\frac{R}{\eta}\phi'\left(\frac{R}{\eta}\right)\right\},$$
furthermore
\(\phi(y) = y(m+p)/(m+y)\) and \(R = (X - \mu)^\top S^{-1} (X-\mu)\) for
\(X \sim t_{\nu,p}(\mu,S)\), and \(\eta\) 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 \(\sigma_1\) value is
\(1 + 2/p\) regardless of the underlying elliptical distribution. However, since
the Tyler estimator provides shape information only, but none on scale,
\(\eta\) is irrelevant in this case.
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
find_sigma1(df_data = Inf, df_est = 3, p = 10)
find_sigma1(df_data = 4.5, df_est = 4.5, p = 2)
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