rQCC (version 1.20.7)

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

Arguments

n

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

method

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.

Value

It returns a numeric value.

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), method="sd")

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

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

Using the Shamos estimator (shamos{rQCC}), an unbiased estimator of \(\sigma\) is calculated by shamos(x)/c4.factor(length(x), method="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), method="mad2")

Using the squared Shamos estimator, an unbiased estimator of \(\sigma^2\) is calculated by shamos(x)^2/w4.factor(length(x), method="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 (2020). Investigation of finite-sample properties of robust location and scale estimators. Communications in Statistics - Simulation and Computation, To appear. https://doi.org/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

# NOT RUN {
# unbiasing factor for estimating the standard deviation
c4.factor(n=10, method="sd")
c4.factor(n=10, method="mad")
c4.factor(n=10, method="shamos")

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


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