---
title: "Conditional likelihood scores and variance-accumulation profiles"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Conditional likelihood scores and variance-accumulation profiles}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>", fig.width = 6,
                      fig.height = 3.5)
```

This vignette (synthetic data) shows the conditional-likelihood interface
and the optional variance-accumulation profile of the manuscript's
Section 4.

## 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 $\tau(r) = \int_0^r \sigma(u)^{-2}du / \int_0^1 \sigma(u)^{-2}du$.

```{r}
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").

```{r}
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
```

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

```{r}
all.equal(fit_cal$path$G, fit_opg$path$G)
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))
aersn_test(fit_opg, null = 1, reference = ref_opg)
```

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").