Closed-form Strasser-Weber permutation mean and covariance

Computes the exact permutation mean vector and covariance matrix of the representation linear statistics T_c = sum_i z_i * s_{c,i}, from the scores and the design, without re-randomizing. The permutation fixes each block's treated count, so within block z is a simple random sample of size m_b from n_b units. For that sampling, with p_b = m_b / n_b,

• E[T_c] = sum_b p_b * sum_{i in b} s_{c,i}, and

• Cov(T_a, T_b) = sum_b w_b * sum_{i in b} (s_{a,i} - sbar_a)(s_{b,i} - sbar_b), where w_b = m_b (n_b - m_b) / (n_b (n_b - 1)) and sbar is the within-block mean.

The covariance uses within-block-centred cross products, so it is the same whether scores are already centred (as from riposte_score_matrix()) or not; the function centres internally. When scores are centred, the mean is zero, as riposte's combinations assume.

Usage

riposte_sw_moments(scores, z, block)

Arguments

scores

numeric matrix of scores (rows = units, columns = representations), e.g. riposte_score_matrix()$scores.

z

0/1 treatment vector (its per-block sums give the treated counts).

block

factor of block labels.

Value

a list with mu (length = number of representations) and Sigma (representation-by-representation covariance matrix).

Details

Verified against coin's expectation() and covariance() (the same linear-statistic framework) to machine precision; see the tests.