RE: Relative efficiency (RE)

In rQCC: Robust Quality Control Chart

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 (ni ≥ 1).
estimator	a character string specifying the estimator, must be one of "mean" (default), "median", "HL1", "HL2", "HL3", "sd", "range", "mad", and "shamos".
poolType	Type for how to pool estimators, must be one of "A" (default), "B", and "C".
baseEstimator	a character string specifying the baseline estimator on the numerator of the relative efficiency, must be one of "mean" (default), "median", "HL1", "HL2", "HL3", "sd", "range", "mad", and "shamos".
basePoolType	Type for how to pool baseline estimator, must be one of "A", "B", and "C". If missing, basePoolType <- poolType is used.
correction	logical. A finite-sample bias correction for the estimator with a single sample. TRUE (default) adjusts a finite-sample bias correction for a scale estimator using c4.factor function.
correctionBase	logical. A finite-sample bias correction for the baseline estimator with a single sample. If missing, correctionBase <- correction is used.

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, ... ,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 θ2_hat with respect to θ1_hat is defined as

RE(θ2_hat|θ1_hat) = Var(θ1_hat) divided by Var(θ2_hat).

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" )

rQCC documentation built on Dec. 28, 2022, 1:49 a.m.