rQCC (version 2.22.12)

unbiasing.factor: Finite-sample unbiasing factor

Description

Finite-sample unbiasing factor for estimating the standard deviation (\(\sigma\)) and the variance (\(\sigma^2\)) under the normal distribution.

Usage

c4.factor(n, estimator=c("sd","range", "mad","shamos"))
w4.factor(n, estimator=c("mad2","shamos2"))

Value

It returns a numeric value.

Arguments

n

sample size (\(n \ge 1\)).

estimator

a character string specifying the estimator, must be one of "sd" (default), "range", "mad", "shamos" for c4.factor, and one of "mad2" (default), "shamos2" for w4.factor.

Author

Chanseok Park

Details

The conventional sample standard deviation, range, median absolute deviation (MAD) and Shamos estimators are Fisher-consistent under the normal distribution, but they are not unbiased with a sample of finite size.

Using the sample standard deviation, an unbiased estimator of the standard deviation (\(\sigma\)) is calculated by
sd(x)/c4.factor(length(x), estimator="sd")

Using the range (maximum minus minimum), an unbiased estimator of \(\sigma\) is calculated by
diff(range(x))/c4.factor(length(x), estimator="range")

Using the median absolute deviation (mad{stats}), an unbiased estimator of \(\sigma\) is calculated by
mad(x)/c4.factor(length(x), estimator="mad")

Using the Shamos estimator (shamos{rQCC}), an unbiased estimator of \(\sigma\) is calculated by
shamos(x)/c4.factor(length(x), estimator="shamos")

Note that the formula for the unbiasing factor \(c_4(n)\) is given by $$c_4(n) = \sqrt{\frac{2}{n-1}}\cdot\frac{\Gamma(n/2)}{\Gamma((n-1)/2)}.$$

The squared MAD and squared Shamos are Fisher-consistent for the variance (\(\sigma^2\)) under the normal distribution, but they are not unbiased with a sample of finite size.

An unbiased estimator of the variance (\(\sigma^2\)) is obtained using the finite-sample unbiasing factor (w4.factor).

Using the squared MAD, an unbiased estimator of \(\sigma^2\) is calculated by
mad(x)^2/w4.factor(length(x), estimator="mad2")

Using the squared Shamos estimator, an unbiased estimator of \(\sigma^2\) is calculated by
shamos(x)^2/w4.factor(length(x), estimator="shamos2")

The finite-sample unbiasing factors for the median absolute deviation (MAD) and Shamos estimators are obtained for \(n=1,2,\ldots,100\) using the extensive Monte Carlo simulation with 1E07 replicates. For the case of \(n > 100\), they are obtained using the method of Hayes (2014).

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")

Shamos, M. I. (1976). Geometry and statistics: Problems at the interface. In Traub, J. F., editor, Algorithms and Complexity: New Directions and Recent Results, 251--280. Academic Press, New York.

Hayes, K. (2014). Finite-sample bias-correction factors for the median absolute deviation. Communications in Statistics: Simulation and Computation, 43, 2205--2212.

See Also

mad{stats} for the Fisher-consistent median absolute deviation (MAD) estimator of the standard deviation (\(\sigma\)) under the normal distribution.

mad.unbiased{rQCC} for finite-sample unbiased median absolute deviation (MAD) estimator of the standard deviation (\(\sigma\)) under the normal distribution.

shamos{rQCC} for the Fisher-consistent Shamos estimator of the standard deviation (\(\sigma\)) under the normal distribution.

shamos.unbiased{rQCC} for finite-sample unbiased Shamos estimator of the standard deviation (\(\sigma\)) under the normal distribution.

n.times.eBias.of.mad{rQCC} for the values of the empirical bias of the median absolute deviation (MAD) estimator under the standard normal distribution.

n.times.eBias.of.shamos{rQCC} for the values of the empirical bias of the Shamos estimator under the standard normal distribution.

mad2.unbiased{rQCC} for finite-sample unbiased squared MAD estimator of the variance (\(\sigma^2\)) under the normal distribution.

shamos2.unbiased{rQCC} for finite-sample unbiased squared Shamos estimator of the variance (\(\sigma^2\)) under the normal distribution.

n.times.evar.of.mad{rQCC} for the values of the empirical variance of the median absolute deviation (MAD) estimator under the standard normal distribution.

n.times.evar.of.shamos{rQCC} for the values of the empirical variance of the Shamos estimator under the standard normal distribution.

Examples

# unbiasing factor for estimating the standard deviation
c4.factor(n=10, estimator="sd")
c4.factor(n=10, estimator="mad")
c4.factor(n=10, estimator="shamos")

# Note: d2 notation is widely used for the bias-correction of the range.
d2 = c4.factor(n=10, estimator="range") 
d2

# unbiasing factor for estimating the variance
w4.factor(n=10, "mad2")
w4.factor(n=10, "shamos2")