Implied density for pairwise differences given HMRE prior.
Computes the implied densities of random effect differences given HMRE prior.
dhmre_pairwise(x, mu = 0, sigma = 1)
x |
Numeric. Difference in random effects. |
mu |
Numeric. HMRE Prior location. |
sigma |
Numeric. (Default: 1; must be > 0). HMRE prior scale. |
The HMRE prior for the RE-SD is \int N^+(σ_p | exp(h_p))LN(h_p | 4μ, √{4}σ)dh_p. The random effects are distributed as u_{k,p} \sim N(0, σ_p). The implied prior is therefore u_{k,p} - u_{\lnot k, p} \sim N(0, √{2}σ). Note that there is a singularity at 0, because the integrand at sigma = 0 is an infinite spike. We currently integrate (using a change of variables) starting at machine precision-zero. Consider this the approximation of the limit as we approach 0 positively. This is therefore divergent when assessed at a difference of zero, due to the RESD taking on a zero value (and an infinite function value). This is expected, as the limit of a Gaussian as sigma -> 0 is the Dirac delta function.
Numeric vector.
Stephen R. Martin
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