For a given mixture weight alpha, use importance sample with or withour control variates for estimation
1 2 3 | mixture.is.estimation(seed, N, mixture.param, alpha, fname = "f",
rpname = "rp", rqname = "rq", dpname = "dp", dqname = "dq",
cv = TRUE)
|
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})}. |
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