Conditional likelihood scores and variance-accumulation profiles

Fitted scores

For a conditional likelihood with parameter \(\beta\), fitted scores \(s_t(\hat\beta_n)\) and normalized negative Hessian \(\hat J_n\), the influence contributions of a target \(h(\beta)\) are \(\hat{\dot h}_n\hat J_n^{-1}s_t(\hat\beta_n)\). Because fitted scores sum to zero, the centered path is the same under calendar-time and profile centering; a profile changes the grid of the reference distribution.

Nonuniform information accumulation

Consider a Gaussian regression with known, time-varying error scale \(\sigma_t = \sigma(t/n)\) that decreases over the sample, so that information accumulates faster towards the end. The true variance-accumulation profile is \(\tau(r) = \int_0^r \sigma(u)^{-2}du / \int_0^1 \sigma(u)^{-2}du\).

library(aersn)
set.seed(31)
n <- 400
r <- (1:n) / n
sigma <- exp(1 - 2 * r)                    # scale falls from e^1 to e^-1
x <- rnorm(n)
beta0 <- c(0.5, 1)
X <- cbind(1, x)
y <- as.numeric(X %*% beta0) + sigma * rnorm(n)
## weighted least squares = maximum likelihood with known sigma_t
w <- 1 / sigma^2
b <- solve(crossprod(X * w, X), crossprod(X * w, y))
s <- X * as.numeric(y - X %*% b) * w      # conditional scores
J <- crossprod(X * w, X) / n               # normalized negative Hessian
tau_true <- c(0, cumsum(w)) / sum(w)

Under correct specification the conditional score variance \(x_t x_t^\top/\sigma_t^2\) equals the negative expected Hessian, so the two cumulative matrices share the profile and the outer-product estimator of aersn_opg_profile() is justified (Supplement, Proposition “Feasible variance-accumulation profile from score outer products”).

fit_cal <- aersn_mle(s, J, b, target = 2, names = "slope")
fit_opg <- aersn_mle(s, J, b, target = 2, names = "slope", profile = "opg")
plot((0:n) / n, fit_opg$nodes, type = "l", xlab = "sample fraction",
     ylab = "tau", main = "Estimated (solid) and true (dashed) profile")
lines((0:n) / n, tau_true, lty = 2)
abline(0, 1, col = "grey60", lty = 3)

fit_opg$model$proportionality      # departure from a common scalar profile
#> [1] 0.1387254

The path is identical under the two centerings; the reference grid differs, and so does the matched-grid critical value:

all.equal(fit_cal$path$G, fit_opg$path$G)
#> [1] TRUE
ref_cal <- aersn_reference(fit_cal, draws = 20000, seed = 1)
ref_opg <- aersn_reference(fit_opg, draws = 20000, seed = 1)
c(calendar = aersn_critical_value(ref_cal), profile = aersn_critical_value(ref_opg))
#> calendar  profile 
#> 1.829738 1.961995
aersn_test(fit_opg, null = 1, reference = ref_opg)
#> 
#>  Adjusted-range increment hull test, q = 1
#> 
#> data:  fit_opg
#> T = 0.03908, q = 1, n = 400
#> alternative hypothesis: true parameter is not equal to the null value
#> null value: slope = 1.000 
#> estimate:   slope = 0.9982 
#> critical value at level 0.95: 1.962  (do not reject)
#> p-value = 0.966 (Monte Carlo s.e. 0.0013, resolution 0.000050)
#> reference: matched-grid Monte Carlo law for increment-hull gauge: q = 1, profile grid with n = 400 intervals, 20000 draws, seed 1

For the scalar adjusted range the reference law is unchanged by a monotone change of time; only the finite-grid approximation differs between the two grids. A known profile can be supplied instead of "opg" as a numeric vector of length n + 1.

When a profile should not be used

• The outer-product profile relies on martingale-difference scores; for serially correlated influence contributions, lag-zero outer products do not estimate cumulative long-run covariance.
• Consistency of a profile estimate alone does not make the profile-centered path feasible for a general estimator: for the sample mean, the fitted contributions sum to zero and the required path differs from the calendar-centered path by \(\{\tau(r) - r\}S_n(1)\). The manuscript establishes feasibility for fitted conditional scores under a common profile; the package does not estimate a valid profile for arbitrary dependent data.
• The fixed-shape condition \(C(r) = \tau(r)C(1)\) excludes changing covariance orientation (Supplement, Remark “Changing covariance orientation”).