drmdel (version 1.0)

quantileDRM: Estimate the quantiles of the populations under DRM

Description

Suppose we have m+1 samples, labeled as $0, \, 1, \, \ldots, \, m$, whose population distributions satisfy the density ratio model (DRM) (see drmdel for the definition of DRM). The quantileDRM function estimates the quantiles of the population distributions.

Usage

quantileDRM(k, p, drmfit, cov=TRUE, interpolation=TRUE,
            adjust=FALSE, adj_factor=NULL, bw=NULL, show_bw=FALSE)

Arguments

k

a vector of labels of populations whose quantiles are to be estimated, with k[i] = 0, 1, ..., m. It could also be a single integer value (in the set of {0, 1, ..., m}), in which case, it means that we estimate the quantile of the same popu

p

a vector of probabilities (the same length as argument "k") at which the quantiles are estimated; It could also be a single value, in which case, it means that for each population k, we estimate the quantile at a same probability value.

drmfit

a fitted DRM object (an output from the drmdel function). See drmdel for details.

cov

a logical variable specifying whether to estimate the covariance matrix of the quantile estimators. The default is TRUE.

interpolation

an argument passed to the function cdfDRM for estimating cumulative distribution functions (CDFs). It is a logical variable specifying whether to linearly interpolate the estimated cumulative dis

adjust

a logical variable specifying whether to adjust the CDF estimation by a factor when estimating quantiles; The default is FALSE. See Details section.

adj_factor

the adjustment factor (a single value) for the estimated CDF for quantile estimation, if adjust=TRUE. The default, NULL, uses $-1/(2n_{total})$, where $n_{total}$ is the total sample size. See Details section.

bw

a vector of bandwidths (the same length as argument "k") for kernel density estimation required for estimating the covariance matrix of the quantile estimators; It could also be a single value, in which case, it means that for each populat

show_bw

a logical variable specifying whether to output bandwidths when argument cov=TRUE. The default is FALSE.

Value

• est
  quantile estimates.
• cov
  estimated covariance matrix of the quantile estimators, available only if argument cov=TRUE.
• bw
  bandwidths used for kernel density estimation required for estimating the covariance matrix of the quantile estimators, available only if argument cov=TRUE and show_bw=TRUE.

Details

Denote the estimated CDF of the k$\textsuperscript{th}$ population as $\hat{F}_k(x)$. The p$\textsuperscript{th}$ quantile of $F_k(x)$ then is estimated as $$\inf{ x: \, \hat{F}_k(x) \ge p}.$$

The estimated CDF $\hat{F}_k(x)$ reaches its maximum value, 1, at the largest observed data point. If the true CDF $F_k(x)$ is continuous, $F_k(x)$ tends to 1 when x tends to infinity. Hence, when estimate an upper quanitle, say 0.95$\textsuperscript{th}$ quantile, of $F_k$, the quantile estimator is likely to under estimate the ture quantile, especially when sample size is not too large. To adjust an upper quantile estimator for possible under-estimation, one may want to adjust the estimated CDF as $$\hat{F}_k(x) + \mbox{adj\_factor}.$$ and use the adjusted CDF to estimate quantiles. To make an upper quantile estimator larger, adj_factor should be a negative value. Similarly, to adjust lower quantile estimates for possible over-estimation, adj_factor should be a positive value.

The quantileDRM function, by default, does not adjust CDF estimators (adjust=FALSE). When adjust=TRUE, the default adj_factor is set to $-1/(2n_{total})$, where $n_{total}$ is the total sample size.

References

J. Chen and Y. Liu (2013), Quantile and quantile-function estimations under density ratio model. To appear in The Annals of Statistics, 2013.

Examples

# Data generation
set.seed(25)
n_samples <- c(100, 200, 180, 150, 175)  # sample sizes
x0 <- rgamma(n_samples[1], shape=5, rate=1.8)
x1 <- rgamma(n_samples[2], shape=12, rate=1.2)
x2 <- rgamma(n_samples[3], shape=12, rate=1.2)
x3 <- rgamma(n_samples[4], shape=18, rate=5)
x4 <- rgamma(n_samples[5], shape=25, rate=2.6)
x <- c(x0, x1, x2, x3, x4)

# Fit a DRM with the basis function q(x) = (x, log(abs(x))), which
# is the basis function for gamma family. This basis function is
# the built-in basis function 6.
drmfit <- drmdel(x=x, n_samples=n_samples, basis_func=6)

# Quantile estimation
# Denote the p^th quantile of the k^th, k=0, 1, ..., 4, population
# as q_{k,p}.

# Estimate q_{0,0.25}, q_{0,0.6}, q_{1,0.1} and q_{2,0.1}.
(qe <- quantileDRM(k=c(0, 0, 1, 2), p=c(0.25, 0.6, 0.1, 0.1),
                  drmfit=drmfit))

# Estimate the 0.05^th, 0.2^th and 0.8^th quantiles of F_3
(qe1 <- quantileDRM(k=3, p=c(0.05, 0.2, 0.8), drmfit=drmfit))
 
# Estimate the 0.05^th quantiles of F_1, F_3 and F_4
(qe2 <- quantileDRM(k=c(1 , 3, 4), p=0.05, drmfit=drmfit))