make_kernel_var_matrix: Make a quadratic form matrix for the kernel-based variance...
View source: R/quadratic_forms.R
| make_kernel_var_matrix | R Documentation |
Make a quadratic form matrix for the kernel-based variance estimator of Breidt, Opsomer, and Sanchez-Borrego (2016)
Description
Constructs the quadratic form matrix for the kernel-based variance estimator of Breidt, Opsomer, and Sanchez-Borrego (2016). The bandwidth is automatically chosen to result in the smallest possible nonempty kernel window.
Usage
make_kernel_var_matrix(x, kernel = "Epanechnikov", bandwidth = "auto")
Arguments
x |
A numeric vector, giving the values of an auxiliary variable. |
kernel |
The name of a kernel function. Currently only "Epanechnikov" is supported. |
bandwidth |
The bandwidth to use for the kernel.
The default value is |
Details
This kernel-based variance estimator was proposed by Breidt, Opsomer, and Sanchez-Borrego (2016), for use with samples selected using systematic sampling or where only a single sampling unit is selected from each stratum (sometimes referred to as "fine stratification").
Suppose there are n sampled units, and
for each unit i there is a numeric population characteristic x_i
and there is a weighted total \hat{Y}_i, where
\hat{Y}_i is only observed in the selected sample but x_i
is known prior to sampling.
The variance estimator has the following form:
\hat{V}_{ker}=\frac{1}{C_d} \sum_{i=1}^n (\hat{Y}_i-\sum_{j=1}^n d_j(i) \hat{Y}_j)^2
The terms d_j(i) are kernel weights given by
d_j(i)=\frac{K(\frac{x_i-x_j}{h})}{\sum_{j=1}^n K(\frac{x_i-x_j}{h})}
where K(\cdot) is a symmetric, bounded kernel function
and h is a bandwidth parameter. The normalizing constant C_d
is computed as:
C_d=\frac{1}{n} \sum_{i=1}^n(1-2 d_i(i)+\sum_{j=1}^H d_j^2(i))
If n=2, then the estimator is simply the estimator
used for simple random sampling without replacement.
If n=1, then the matrix simply has an entry equal to 0.
Value
The quadratic form matrix for the variance estimator,
with dimension equal to the length of x. The resulting
object has an attribute bandwidth that can be retrieved
using attr(Q, 'bandwidth')
References
Breidt, F. J., Opsomer, J. D., & Sanchez-Borrego, I. (2016). "Nonparametric Variance Estimation Under Fine Stratification: An Alternative to Collapsed Strata." Journal of the American Statistical Association, 111(514), 822–833. https://doi.org/10.1080/01621459.2015.1058264
Examples
# The auxiliary variable has the same value for all units
make_kernel_var_matrix(c(1, 1, 1))
# The auxiliary variable differs across units
make_kernel_var_matrix(c(1, 2, 3))
# View the bandwidth that was automatically selected
Q <- make_kernel_var_matrix(c(1, 2, 4))
attr(Q, 'bandwidth')
