giveth: Stone (1975)'s estimator

In nilanjanalaha/log.location: What the Package Does (One Line, Title Case)

Description

Gives a truncated one step estimator for the location parameter in a symmetric location family. This estimator is constructed using Stone (1975)'s methods. This estimator uses the kernel density estimator (KDE) to estimate the scores. Similar to all other one-step estimators, this estimator requires a preliminary estimator.

Usage

giveth(x, inth, tn = 0.6, dn = 30, alpha = 0.5)

Arguments

x	A vector of length n; the dataset
inth	A preliminary estimator for θ. The default is the median.
tn	A parameter associated with the bandwidth of the kernel density estimator. The default value is 0.60.
dn	A parameter for the truncation. The default value is 20.
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.

Details

Stone (1975) uses r_n=\hatσ t_n as the bandwidth of the Gaussian kernel, where \hatσ is the median of the X_i-inth's. For asymptotic efficiency of the estimators, a) dn\to∞ and tn\to 0 b)

\frac{(dn)^2}{n^{1-ε}(tn)^6}=O(1)

for some ε>0. A rule of thumb plug-in estimate for the optimal kernel width is 1.059\hatσ n^{-1/5} where \hatσ is an estimator of the standard deviation, which does not satisfy the above relation. Stone(1975) takes inth to be the median in his simulations. The default choices of dn and tn are taken from Stone (1975), who considered a sample of size 40. Stone(1975)'s estimator uses Gaussian Kernels to estimate the unknown symmetric density.

Value

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.

Author(s)

Nilanjana Laha (maintainer), [e-mail].

References

Laha, N. Location estimation fr symmetric and log-concave densities. Submitted.

Stone, C. (1975). Adaptive maximum likelihood estimators of a location parameter, Ann. Statist., 3, 267-284.

Examples

x <- rnorm(100); inth <- mean(x);
 giveth(x, inth=inth)
 giveth(x)

nilanjanalaha/log.location documentation built on Dec. 31, 2020, 12:07 a.m.