For a given mixture weight alpha, use importance sample with or withour control variates for estimation

Description

For a given mixture weight alpha, use importance sample with or withour control variates for estimation

Usage

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mixture.is.estimation(seed, N, mixture.param, alpha, fname = "f",
  rpname = "rp", rqname = "rq", dpname = "dp", dqname = "dq",
  cv = TRUE)

Arguments

seed

seed for sampling

N

sample size

mixture.param

mixture.param = list(p, J, ...), where p is the dimension of the sample, and J is the number of mixture components, including the defensive one. mixture.param should be compatible with user defined functions f(n, j, mixture.param), rp(n, mixture.param), rq(n, j, mixture.param), dp(xmat, mixture.param), dq(xmat, j, mixture.param)

alpha

the mixture weight, sum up to 1

fname

name of user defined function fname(xmat, j, mixture.param). xmat is an n \times p matrix of n samples with p dimensions. fname returns a vector of function values for each row in xmat. fname is defined for j = 1, \cdots, J. j = 1, \cdots, J - 1 corresponds to different proposal mixture components, and j = J corresponds to the defensive mixture component.

rpname

name of user definded function rpname(n, mixture.param). It generates n random samples from target distribution pname. Parameters can be specified in mixture.param. rpname returns an n \times p matrix.

rqname

name of user defined function rqname(n, j, mixture.param). It generate n random samples from the j^{th} mixture component of proposal mixture distribution. rqname returns an n \times p matrix. rqname is defined for j = 1, \cdots, J - 1.

dpname

name of user defined function dpname(xmat, mixture.param). It returns the density of xmat from the target distribution pname as a vector of length nrow(xmat). Note that only the ratio between dpname and dqname matters. So dpname and dqname can return values of C \timesdpname and C \timesdqname respectively.

dqname

name of user defined function dqname(xmat, j, mixture.param). It returns the density of xmat from the proposal distribution q_j as a vector of length nrow(xmat). dqname is defined for j = 1, \cdots, J - 1. Note that only the ratio between dpname and dqname matters. So dpname and dqname can return values of C \timesdpname and C \timesdqname respectively.

cv

TRUE indicates optimizing alpha and beta at the same time, and estimate with the formula

\hat{μ}_{α_{**},β} = \frac{1}{n_2}∑\limits_{i = 1}^{n_2}\frac{f(x_{i})p(x_{i}) - {β}^{\mathsf{T}}\bigl(\boldsymbol{q}(x_{i}) - p(x_{i})\boldsymbol{1}\bigr)}{q_{α_{**}}(x_{i})}.

FALSE indicates optimizing alpha only, and estimate with the formula

\hat{μ}_{α_{*}}= \frac{1}{n_2}∑\limits_{i = 1}^{n_2}\frac{f(x_{i})p(x_{i})}{q_{α_{*}}(x_{i})}.