Two-dimensional confidence regions

Calculates the (profile, if necessary) loglikelihood for a pair of parameters from which confidence regions can be plotted using plot.confreg.

conf_region(
  object,
  which_pars = NULL,
  range1 = c(NA, NA),
  range2 = c(NA, NA),
  conf = 95,
  mult = 2,
  num = c(10, 10),
  type = c("vertical", "cholesky", "spectral", "none"),
  ...
)

Arguments

object

An object of class "chandwich" returned by adjust_loglik.

which_pars

A vector of length 2 specifying the 2 (unfixed) parameters for which confidence region is required. Can be either a numeric vector, specifying indices of the components of the full parameter vector, or a character vector of parameter names, which must be a subset of those supplied in par_names in the call to adjust_loglik that produced object.

which_pars must not have any parameters in common with attr(object, "fixed_pars"). which_pars must not contain all of the unfixed parameters, i.e. there is no point in profiling over all the unfixed parameters.

If which_pars is not supplied but the current model has exactly two free parameters, i.e. attr(object, "p_current") = 2 then which_pars is set to attr(object, "free_pars") = 2.

range1, range2

Numeric vectors of length 2. Respective ranges (of the form c(lower, upper)) of values of which_pars[1] and which_pars[2] over which to profile. Missing values in range1 and/or range2 are filled in using conf and mult. See below for details.

conf

A numeric scalar in (0, 100). The highest confidence level of interest. This is only relevant if range1 and/or range2 are not completely specified. In that event conf is used, in combination with mult, to try to set up the grid of parameter values to include the largest confidence region of interest.

mult

A numeric vector of length 1 or the same length as which_pars. The search for the profile loglikelihood-based confidence limits is conducted over the corresponding symmetric confidence intervals (based on approximate normal theory), extended by a factor of the corresponding component of mult.

num

A numeric vector of length 1 or 2. The numbers of values at which to evaluate the profile loglikelihood either side of the MLE. num[i] relates to which_pars[i]. If num has length 1 then num is replicated to have length 2.

type

A character scalar. The argument type to the function returned by adjust_loglik, that is, the type of adjustment made to the independence loglikelihood function.

...

Further arguments to be passed to optim. These may include gr, method, lower, upper or control. Any arguments that are not appropriate for optim, i.e. not in methods::formalArgs(stats::optim), will be removed without warning.

Value

An object of class "confreg", a list with components

grid1, grid2: Numeric vectors. Respective values of which_pars[1] and which_pars[2] in the grid over which the (profile) loglikelihood is evaluated.
max_loglik: A numeric scalar. The value value of the loglikelihood at its maximum.
prof_loglik: An 2 num + 1 by 2 num + 1 numeric matrix containing the values of the (profile) loglikelihood.
type: A character scalar. The input type.
which_pars: A numeric or character vector. The input which_pars. If the which_pars was numeric then it is supplemented by the parameter names, if these are available in object.
name: A character scalar. The name of the model, stored in attr(object, "name").

Examples

# -------------------------- GEV model, owtemps data -----------------------
# ------------ following Section 5.2 of Chandler and Bate (2007) -----------

gev_loglik <- function(pars, data) {
  o_pars <- pars[c(1, 3, 5)] + pars[c(2, 4, 6)]
  w_pars <- pars[c(1, 3, 5)] - pars[c(2, 4, 6)]
  if (isTRUE(o_pars[2] <= 0 | w_pars[2] <= 0)) return(-Inf)
  o_data <- data[, "Oxford"]
  w_data <- data[, "Worthing"]
  check <- 1 + o_pars[3] * (o_data - o_pars[1]) / o_pars[2]
  if (isTRUE(any(check <= 0))) return(-Inf)
  check <- 1 + w_pars[3] * (w_data - w_pars[1]) / w_pars[2]
  if (isTRUE(any(check <= 0))) return(-Inf)
  o_loglik <- log_gev(o_data, o_pars[1], o_pars[2], o_pars[3])
  w_loglik <- log_gev(w_data, w_pars[1], w_pars[2], w_pars[3])
  return(o_loglik + w_loglik)
}

# Initial estimates (method of moments for the Gumbel case)
sigma <- as.numeric(sqrt(6 * diag(var(owtemps))) / pi)
mu <- as.numeric(colMeans(owtemps) - 0.57722 * sigma)
init <- c(mean(mu), -diff(mu) / 2, mean(sigma), -diff(sigma) / 2, 0, 0)

# Log-likelihood adjustment of the full model
par_names <- c("mu[0]", "mu[1]", "sigma[0]", "sigma[1]", "xi[0]", "xi[1]")
large <- adjust_loglik(gev_loglik, data = owtemps, init = init,
                       par_names = par_names)

# \donttest{
# Plots like those in Figure 4 of Chandler and Bate (2007)
# (a)
which_pars <- c("mu[0]", "mu[1]")
reg_1 <- conf_region(large, which_pars = which_pars)
#> Waiting for profiling to be done...
reg_none_1 <- conf_region(large, which_pars = which_pars, type = "none")
#> Waiting for profiling to be done...
plot(reg_1, reg_none_1)

# (b)
which_pars <- c("sigma[0]", "sigma[1]")
reg_2 <- conf_region(large, which_pars = which_pars)
#> Waiting for profiling to be done...
reg_none_2 <- conf_region(large, which_pars = which_pars, type = "none")
#> Waiting for profiling to be done...
plot(reg_2, reg_none_2)

# (c)
# Note: the naive and bivariate model contours are the reversed in the paper
which_pars <- c("sigma[0]", "xi[0]")
reg_3 <- conf_region(large, which_pars = which_pars)
#> Waiting for profiling to be done...
reg_none_3 <- conf_region(large, which_pars = which_pars, type = "none")
#> Waiting for profiling to be done...
plot(reg_3, reg_none_3)

# }

# --------- Misspecified Poisson model for negative binomial data ----------

# ... following Section 5.1 of the "Object-Oriented Computation of Sandwich
# Estimators" vignette of the sandwich package
# https://cran.r-project.org/web/packages/sandwich/vignettes/sandwich-OOP.pdf

# Simulate data
set.seed(123)
x <- rnorm(250)
y <- rnbinom(250, mu = exp(1 + x), size = 1)
# Fit misspecified Poisson model
fm_pois <- glm(y ~ x + I(x^2), family = poisson)
summary(fm_pois)$coefficients
#>                Estimate Std. Error   z value      Pr(>|z|)
#> (Intercept)  1.06326821 0.04135723 25.709367 9.184267e-146
#> x            0.99607219 0.05353446 18.606186  2.862861e-77
#> I(x^2)      -0.04912373 0.02314608 -2.122335  3.380961e-02

# Contributions to the independence loglikelihood
pois_glm_loglik <- function(pars, y, x) {
  log_mu <- pars[1] + pars[2] * x + pars[3] * x ^ 2
  return(dpois(y, lambda = exp(log_mu), log = TRUE))
}
pars <- c("alpha", "beta", "gamma")
# Linear model (gamma fixed at 0)
pois_lin <- adjust_loglik(pois_glm_loglik, y = y, x = x, par_names = pars,
                          fixed_pars = "gamma")
pois_vertical <- conf_region(pois_lin)
#> Waiting for profiling to be done...
pois_none <- conf_region(pois_lin, type = "none")
#> Waiting for profiling to be done...
plot(pois_none, pois_vertical, conf = c(50, 75, 95, 99), col = 2:1, lwd = 2,
     lty = 1)

Arguments

Value

See also

Examples