rowRoblox and colRoblox {RobLox}

Optimally robust estimation for location and/or scale

Description

The functions rowRoblox and colRoblox compute optimally robust estimates for normal location und/or scale and (convex) contamination neighborhoods. The definition of these estimators can be found in Rieder (1994) or Kohl (2005), respectively.

Usage

rowRoblox(x, mean, sd, eps, eps.lower, eps.upper, initial.est, k = 1L,
          fsCor = TRUE, mad0 = 1e-4, na.rm = TRUE)
colRoblox(x, mean, sd, eps, eps.lower, eps.upper, initial.est, k = 1L,
          fsCor = TRUE, mad0 = 1e-4, na.rm = TRUE)

Arguments

x

matrix or data.frame of (numeric) data values.

mean

specified mean. See details below.

sd

specified standard deviation which has to be positive. See also details below.

eps

positive real (0 < eps <= 0.5): amount of gross errors. See details below.

eps.lower

positive real (0 <= eps.lower <= eps.upper): lower bound for the amount of gross errors. See details below.

eps.upper

positive real (eps.lower <= eps.upper <= 0.5): upper bound for the amount of gross errors. See details below.

initial.est

initial estimate for mean and/or sd. If missing median and/or MAD are used.

k

positive integer. k-step is used to compute the optimally robust estimator.

fsCor

logical: perform finite-sample correction. See function finiteSampleCorrection.

mad0

scale estimate used if computed MAD is equal to zero

na.rm

logical: if TRUE, the estimator is evaluated at complete.cases(x).

Details

Computes the optimally robust estimator for location with scale specified, scale with location specified, or both if neither is specified. The computation uses a k-step construction with an appropriate initial estimate for location or scale or location and scale, respectively. Valid candidates are e.g. median and/or MAD (default) as well as Kolmogorov(-Smirnov) or Cram\'er von Mises minimum distance estimators; cf. Rieder (1994) and Kohl (2005). In case package Biobase from Bioconductor is installed as is suggested, median and/or MAD are computed using function rowMedians.

These functions are optimized for the situation where one has a matrix and wants to compute the optimally robust estimator for every row, respectively column of this matrix. In particular, the amount of cross errors is assumed to be constant for all rows, respectively columns.

If the amount of gross errors (contamination) is known, it can be specified by eps. The radius of the corresponding infinitesimal contamination neighborhood is obtained by multiplying eps by the square root of the sample size.

If the amount of gross errors (contamination) is unknown, try to find a rough estimate for the amount of gross errors, such that it lies between eps.lower and eps.upper.

In case eps.lower is specified and eps.upper is missing, eps.upper is set to 0.5. In case eps.upper is specified and eps.lower is missing, eps.lower is set to 0.

If neither eps nor eps.lower and/or eps.upper is specified, eps.lower and eps.upper are set to 0 and 0.5, respectively.

If eps is missing, the radius-minimax estimator in sense of Rieder et al. (2008), respectively Section 2.2 of Kohl (2005) is returned.

In case of location, respectively scale one additionally has to specify sd, respectively mean where sd and mean can be a single number, i.e., identical for all rows, respectively columns, or a vector with length identical to the number of rows, respectively columns.

For sample size <= 2, median and/or MAD are used for estimation.

If eps = 0, mean and/or sd are computed.

Value

Object of class "kStepEstimate".

Author(s)

Matthias Kohl [e-mail]

References

M. Kohl (2005). Numerical Contributions to the Asymptotic Theory of Robustness. Dissertation. University of Bayreuth. https://epub.uni-bayreuth.de/id/eprint/839/2/DissMKohl.pdf.

H. Rieder (1994): Robust Asymptotic Statistics. Springer. doi:10.1007/978-1-4684-0624-5

H. Rieder, M. Kohl, and P. Ruckdeschel (2008). The Costs of Not Knowing the Radius. Statistical Methods and Applications 17(1): 13-40. doi:10.1007/s10260-007-0047-7

M. Kohl, P. Ruckdeschel, and H. Rieder (2010). Infinitesimally Robust Estimation in General Smoothly Parametrized Models. Statistical Methods and Applications 19(3): 333-354. doi:10.1007/s10260-010-0133-0.

M. Kohl and H.P. Deigner (2010). Preprocessing of gene expression data by optimally robust estimators. BMC Bioinformatics 11, 583. doi:10.1186/1471-2105-11-583.

See Also

roblox, kStepEstimate-class

Examples

ind <- rbinom(200, size=1, prob=0.05) 
X <- matrix(rnorm(200, mean=ind*3, sd=(1-ind) + ind*9), nrow = 2)
rowRoblox(X)
rowRoblox(X, k = 3)
rowRoblox(X, eps = 0.05)
rowRoblox(X, eps = 0.05, k = 3)

X1 <- t(X)
colRoblox(X1)
colRoblox(X1, k = 3)
colRoblox(X1, eps = 0.05)
colRoblox(X1, eps = 0.05, k = 3)

X2 <- rbind(rnorm(100, mean = -2, sd = 3), rnorm(100, mean = -1, sd = 4))
rowRoblox(X2, sd = c(3, 4))
rowRoblox(X2, eps = 0.03, sd = c(3, 4))
rowRoblox(X2, sd = c(3, 4), k = 4)
rowRoblox(X2, eps = 0.03, sd = c(3, 4), k = 4)

X3 <- cbind(rnorm(100, mean = -2, sd = 3), rnorm(100, mean = 1, sd = 2))
colRoblox(X3, mean = c(-2, 1))
colRoblox(X3, eps = 0.02, mean = c(-2, 1))
colRoblox(X3, mean = c(-2, 1), k = 4)
colRoblox(X3, eps = 0.02, mean = c(-2, 1), k = 4)