| relative.efficiency {rQCC} | R Documentation |
Relative efficiency (RE)
Description
RE calculates the relative efficiency value
of a location estimator ("median", "HL1", "HL2", "HL3")
with respect to the sample mean and
that of a scale estimator ("range", "mad", "shamos")
with respect to the sample standard deviation.
Usage
RE(n, estimator=c("mean", "median", "HL1", "HL2", "HL3", "sd", "range", "mad", "shamos"),
poolType =c("A", "B", "C"),
baseEstimator, basePoolType, correction=TRUE, correctionBase)
Arguments
n |
a vector of sample sizes ( |
estimator |
a character string specifying the estimator, must be
one of |
poolType |
Type for how to pool estimators, must be
one of |
baseEstimator |
a character string specifying the baseline estimator
on the numerator of the relative efficiency, must be
one of |
basePoolType |
Type for how to pool baseline estimator, must be
one of |
correction |
logical. A finite-sample bias correction for the estimator with a single sample.
|
correctionBase |
logical.
A finite-sample bias correction for the baseline estimator with a single sample.
If missing, |
Details
Under the assumption of the normal distribution, the function calculates the relative efficiency value of the mean, median and Hodges-Lehmann (HL1, HL2, HL3) estimators with respect to the selected baseline estimator (default is the sample mean) and that of the standard deviation, range, median absolute deviation (MAD) and Shamos estimators with respect to the selected baseline estimator (default is the sample standard deviation).
For the case of the sample mean, standard deviation and range,
it is possible to derive their variances in analytic form,
but, for the other case, it may be impossible.
In this case,
the variances with n=1,2,\ldots,100 are obtained
using the extensive Monte Carlo simulation with 1E07 replicates.
For n > 100, the variances are approximated based on the method of Hayes (2014).
To obtain the relative efficiency value of the unbiased scale estimators,
use correction=TRUE option.
Note that the location estimators
("mean", "median", "HL1", "HL2", "HL3") are unbiased.
If n is a vector of multiple values (multiple samples), the RE function calculates
the relative efficiency value of the pooled estimator. In this case, only unbiased pooled estimator
and baseline estimator are considered.
That is, we use correction=TURE and correctionBase=TURE for multiple samples.
Note that the relative efficiency (RE) of \hat{\theta}_2 with respect to
\hat{\theta}_1 is defined as
\mathrm{RE}(\hat{\theta}_2 | \hat{\theta}_1)
=\frac{\mathrm{Var}(\hat{\theta}_1)}{\mathrm{Var}(\hat{\theta}_2)}.
Value
It returns a numeric value.
Author(s)
Chanseok Park
References
Park, C., H. Kim, and M. Wang (2022).
Investigation of finite-sample properties of robust location and scale estimators.
Communications in Statistics - Simulation and Computation,
51, 2619-2645.
doi: 10.1080/03610918.2019.1699114
Hayes, K. (2014). Finite-sample bias-correction factors for the median absolute deviation. Communications in Statistics: Simulation and Computation, 43, 2205–2212.
See Also
n.times.eVar.of.HL1{rQCC}
for the empirical variance of the HL1 estimator (times n).
n.times.eVar.of.HL2{rQCC}
for the empirical variance of the HL2 estimator (times n).
n.times.eVar.of.HL3{rQCC}
for the empirical variance of the HL3 estimator (times n).
n.times.eVar.of.mad{rQCC}
for the empirical variance of the MAD estimator (times n).
n.times.eVar.of.median{rQCC}
for the empirical variance of the median estimator (times n).
n.times.eVar.of.shamos{rQCC}
for the empirical variance of the Shamos estimator (times n).
Examples
#################
# Single sample #
#################
# RE of the Hodges-Lehmann (HL2) estimator
# with respect to the sample standard deviation under the normal distribution.
RE(n=5, estimator="HL2")
# RE of the unbiased Shamos estimator
# with respect to the unbiased sample standard deviation under the normal distribution.
RE(n=5, estimator="shamos")
# RE of the original Shamos estimator
# with respect to the sample standard deviation under the normal distribution.
RE(n=5, estimator="shamos", correction=FALSE)
# RE of the unbiased range ( (maximum - minimum) / d2 )
# with respect to the unibased sample standard deviation under the normal distribution.
RE(n=6, estimator="range")
# RE of the original range (maximum minus minimum)
# with respect to the sample standard deviation under the normal distribution.
RE(n=6, estimator="range", correction=FALSE)
####################
# Multiple samples #
####################
# With multiple samples, only the unbiased pooled estimators are considered.
# RE of the pooled median (pooling type A) with respect to the mean (pooling type A)
RE( n=c(4,5), estimator="median" )
# RE of the pooled median (pooling type A) with respect to the median (pooling type C)
RE( n=c(4,5), estimator="median", baseEstimator="median", basePoolType="C")
# RE of the pooled mad (pooling type A) with respect to the standard deviation (pooling type A)
RE( n=c(4,5), estimator="mad")
# RE of the pooled mad (pooling type A) with respect to the standard deviation (pooling type C)
RE( n=c(4,5), estimator="mad", basePoolType="C")
# RE of the pooled standard deviation (pooling type A) with respect to the sd (pooling type C)
RE( n=c(4,5), estimator="sd", baseEstimator="sd", basePoolType="C" )