relative.efficiency: 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)

Value

It returns a numeric value.

Arguments

n: a vector of sample sizes ($n_i \ge 1$).
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.

Author

Chanseok Park

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 $$\mathrm{RE}(\hat{\theta}_2 | \hat{\theta}_1) =\frac{\mathrm{Var}(\hat{\theta}_1)}{\mathrm{Var}(\hat{\theta}_2)}.$$

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.
tools:::Rd_expr_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.

Examples

Run this code

#################
# 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" )

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