Newton's predictive recursion — morie_predictive

\(\hat f_i(\theta) = (1-w_i)\hat f_{i-1}(\theta) + w_i \psi(X_i;\theta)\hat f_{i-1}(\theta) / \int \psi(X_i;t)\hat f_{i-1}(t)dt\) (Sec. 5.4), estimating the MIXING density in \(p_F(x) = \int \psi(x;\theta)dF\) – a deconvolution.

Usage

morie_predictive_recursion(
  x,
  theta_grid = NULL,
  sigma = 1,
  weights = NULL,
  f0 = NULL
)

Arguments

x: numeric observations from the mixture.
theta_grid: support of the mixing density.
sigma: positive Gaussian kernel scale.
weights: the w_i, strictly inside (0, 1).
f0: initial mixing density; uniform when NULL.

Value

list: theta_grid, f_mixing, mixed_density, order_dependent, single_pass, weight_rule, n, method.

Details

A single sweep, no MCMC, which is what makes it cheap. The price is that the estimate DEPENDS ON THE ORDER of the observations, since each step conditions on the running estimate. Weights need \(\sum w_i = \infty\) and \(\sum w_i^2 < \infty\); \((i+2)^{-2/3}\) satisfies both and stays inside (0, 1), which \((i+1)^{-2/3}\) does not at the first step. Mirrors morie.fn.gh_c5_7.

References

Ghosal and van der Vaart, Sec. 5.4; Newton (2002).

Examples

set.seed(1)
morie_predictive_recursion(rnorm(50), sigma = 1)$order_dependent
#> [1] TRUE