Documentation of the asymptotic robustness property of the Hotelling T² statistic under non-normal distributions, based on Gneri and Barbosa (2006), and its relationship to the IQCC package implementation.
Theorem 3 (Gneri & Barbosa, 2006, Sección 4.2)
Let \(X\) be a \(p\)-dimensional random vector with continuous distribution, mean vector \(\mu\), positive-definite covariance matrix \(\Sigma\), and finite fourth moments. Let \(X_1, X_2, \dots, X_n\) be an i.i.d. sample, and denote by \(\bar{X}(n)\) and \(S(n)\) the sample mean vector and sample covariance matrix. The Hotelling T² statistic is defined as:
$$T^2 = n (\bar{X}(n) - \mu)' S(n)^{-1} (\bar{X}(n) - \mu).$$
Then, as \(n \to \infty\), \(T^2\) converges in distribution to a chi-squared random variable with \(p\) degrees of freedom.
Proof sketch
The proof proceeds in three steps:
The sample covariance converges in probability: \(S(n)_{ij} \to \Sigma_{ij}\) elementwise (Khinchin's LLN), and by continuity of matrix inversion, \(S(n)^{-1} \to \Sigma^{-1}\) in probability.
By the Central Limit Theorem, \(\sqrt{n}(\bar{X}(n) - \mu) \to N_p(0, \Sigma)\) in distribution.
By Proposition 2 (a Slutsky-type result), \(T^2 \to W' \Sigma^{-1} W\) where \(W \sim N_p(0, \Sigma)\), i.e., \(T^2 \to \chi^2_p\).
Discrepancy in moment conditions
The abstract and introduction of Gneri & Barbosa (2006) state the theorem under the assumption of finite second moments only ("momentos de orden 2 finitos"). However, the formal statement of Theorem 3 (Sección 4.2) requires finite fourth moments. The proof uses:
Khinchin's Law of Large Numbers for the sample covariance \(S(n)\), which requires \(E[|X_i X_j|] < \infty\). By Cauchy-Schwarz, this holds when second moments are finite, so order 2 suffices for this step.
The Central Limit Theorem for \(\sqrt{n}(\bar{X} - \mu)\), which requires finite second moments.
The fourth-moment condition in the theorem statement is therefore conservative. The theorem holds under finite second moments, provided the covariance matrix is finite and positive-definite.
Scope and limitations
The theorem is asymptotic and does not imply:
The exact finite-sample distribution of \(T^2\) under non-normality.
The joint distribution of a sequence of charted points in a control chart.
Valid Average Run Length (ARL), false-alarm risk, or nominal coverage for small samples.
Resolution of parameter estimation uncertainty in Phase I.
Justification of the finite-sample F or beta control limits derived under normality for arbitrary continuous distributions.
Coverage of discrete distributions, singular covariance, increasing dimension with \(n\), temporal dependence, or infinite moments.
Relationship to IQCC functions
Theorem 3 concerns a single-sample T² statistic: $$T^2 = n (\bar{X} - \mu)' S^{-1} (\bar{X} - \mu)$$ where \(\bar{X}\) and \(S\) are the mean and covariance of the same sample, and \(\mu\) is the known null mean.
The IQCC package implements two related but distinct statistics:
| Concept | IQCC function | Notes |
| Theorem 3 statistic | sim_t2_asymptotic | Single sample, known \(\mu = 0\), own \(S\) |
| Phase I T² | T2.1 | Uses grand mean \(\bar{\bar{x}}\) and pooled \(\bar{S}\) from stats |
| Phase II T² | T2.2 | Tests new obs. against Phase I estimates |
| Phase I chart | cchart.T2.1 | Beta (n=1) or F (n>1) limits |
| Phase II chart | cchart.T2.2 | F limits |
| Auxiliary stats | stats | Grand mean, pooled covariance, subgroup means |
Key differences between Theorem 3 and T2.1():
Theorem 3 uses the true null mean \(\mu\);
T2.1()uses the estimated grand mean \(\bar{\bar{x}}\) from Phase I data.Theorem 3 uses the covariance of the same sample;
T2.1()uses the pooled covariance from multiple Phase I subgroups.Theorem 3 is asymptotic as \(n \to \infty\) for a single sample;
T2.1()has \(m\) correlated statistics (one per subgroup) with finite-sample F/beta distributions under normality.
The control limits in cchart.T2.1() and cchart.T2.2() are
based on exact finite-sample distributions under multivariate normality
(F and beta distributions). These limits are not justified by
Theorem 3 for non-normal data in finite samples. The theorem only
guarantees that as the subgroup size \(n \to \infty\), a single
T² statistic approaches a \(\chi^2_p\) distribution regardless of the
underlying continuous distribution (provided the moment conditions hold).