beran.est: The ocation estimator of Beran (1974)
In nilanjanalaha/log.location: What the Package Does (One Line, Title Case)

Description Usage Arguments Details Value Author(s) References Examples

Suppose the observations are sampled from a density f(x-m) where m is the location parameter and f is an unknown density. This function, based on Beran (1974), gives a nonparametric estimator of m. This estimator uses Fourier basis expansion to estimate the score function corresponding to the location model, which has the form

φ_F(t)=f'(F^{-1}(t))/f(F^{-1}(t)).

The Fourier coefficients of the score function are computed by score.coeff.

1	beran.est(x, theta, M, init, alpha)

`x`	An array of length n; represents the data.
`theta`	A small number, should be of order O_p(n^{-1/2}). The default is 4n^{-1/2}. This is the tuning parameter for Fourier coefficient estimation.
`M`	The number of Fourier basis to be used to approximate φ_F.
`init`	Optional. A vector of initial estimators of m. The default value is the sample median.
`alpha`	The confidence level for the confidence bands. An (1-alpha) percent confidence interval is constructed. Alpha should lie in the interval (0, 0.50). The default value is 0.05.

theta: theta do not need to depend on the range of the data because the estimators depend only on the rank of the dataponts. If theta=z_n which equals z n^{-1/2}, then the estimated coefficients are root-n consistent by Theorem 2.1 of Beran (1974).

M: A higher value of M decreases bias, but increases the variance. Theorem 4.1 of Beran (1974) states that if M\to∞ as the sample size n grows and \lim_{n\to ∞} M^6/n=0, then this estimator is asymptotically efficient. A larger M always gives a more conservative confidence interval.

init: Beran (1974) recommends using sample median and warns that the method will be sensitive to the choice of the preliminary estimator. This should be a root-n consistent estimator of m. Some other choices are the mean, or the trimmed mean.

A list of length two.

estimate: An array of same length as init , giving the estimators of m based on the corresponding initial estimators in init. If init is missing, only one estimate is produced, which uses the sample median as the initial estimator.
CI: A matrix giving the (1-alpha) percent confidence intervals (CI). Each row corresponds to an initial estimator in init (if missing, the sample median is used). The first column corresponds to the lower CI, and the second column corresponds to the upper CI.

Nilanjana Laha (maintainer), nlaha@hsph.harvard.edu.

Beran, R. (1974). Asymptotically efficient adaptive rank estimates in location models. Ann. Statist., 2, 63-74.