maxDiffCDF {logcondens}

Compute maximal difference between CDFs corresponding to log-concave estimates

Description

Compute the maximal difference between two estimated log-concave distribution functions, either the MLEs or the smoothed versions. This function is used to set up a two-sample permutation test that assesses the null hypothesis of equality of distribution functions.

Usage

maxDiffCDF(res1, res2, which = c("MLE", "smooth"), n.grid = 500)

Arguments

res1

An object of class "dlc", usually a result of a call to logConDens for the first sample.

res2

An object of class "dlc", usually a result of a call to logConDens for the second sample.

which

Indicate for which type of estimate the maximal difference should be computed.

n.grid

Number of grid points used to find zeros of \hat f_{n_1}^* - \hat f_{n_2}^* for the smooth estimate.

Details

Given two i.i.d. samples x_1, \ldots, x_{n_1} and y_1, \ldots, y_{n_2} this function computes the maxima of

D_1(t) = \hat F_{n_1}(t) - \hat F_{n_2}(t)

and

D_2(t) = \hat F^*_{n_1}(t) - \hat F^*_{n_2}(t).

Value

test.stat

A two-dimensional vector containing the above maxima.

location

A two-dimensional vector where the maxima occur.

Warning

Note that the algorithm that finds the maximal difference for the smoothed estimate is of approximative nature only. It may fail for very large sample sizes.

Author(s)

Kaspar Rufibach, [e-mail],
http://www.kasparrufibach.ch

Lutz Duembgen, [e-mail],
https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html

References

Duembgen, L. and Rufibach, K. (2009) Maximum likelihood estimation of a log–concave density and its distribution function: basic properties and uniform consistency. Bernoulli, 15(1), 40–68.

Duembgen, L. and Rufibach, K. (2011) logcondens: Computations Related to Univariate Log-Concave Density Estimation. Journal of Statistical Software, 39(6), 1–28. doi:10.18637/jss.v039.i06

Examples

n1 <- 100
n2 <- 120
x <- sort(rgamma(n1, 2, 1))
y <- sort(rgamma(n2, 2, 1))
res1 <- logConDens(x, smoothed = TRUE)
res2 <- logConDens(y, smoothed = TRUE)
d <- maxDiffCDF(res1, res2, n.grid = 200)

## log-concave estimate
xs <- seq(min(res1$xs, res2$xs), max(res1$xs, res2$xs), length = 200)
F1 <- matrix(NA, nrow = length(xs), ncol = 1); F2 <- F1
for (i in 1:length(xs)){
    F1[i] <- evaluateLogConDens(xs[i], res1, which = 3)[, "CDF"]
    F2[i] <- evaluateLogConDens(xs[i], res2, which = 3)[, "CDF"]
    }
par(mfrow = c(1, 2))
plot(xs, abs(F1 - F2), type = "l")
abline(v = d$location[1], lty = 2, col = 3)
abline(h = d$test.stat[1], lty = 2, col = 3)

## smooth estimate
xs <- seq(min(res1$xs, res2$xs), max(res1$xs, res2$xs), length = 200)
F1smooth <- matrix(NA, nrow = length(xs), ncol = 2); F2smooth <- F1smooth
for (i in 1:length(xs)){
    F1smooth[i, ] <- evaluateLogConDens(xs[i], res1, which = 4:5)[, 
        c("smooth.density", "smooth.CDF")]
    F2smooth[i, ] <- evaluateLogConDens(xs[i], res2, which = 4:5)[, 
        c("smooth.density", "smooth.CDF")]
    }
plot(xs, abs(F1smooth[, 2] - F2smooth[, 2]), type = "l")
abline(h = 0)
abline(v = d$location[2], lty = 2, col = c(3, 4))
abline(h = d$test.stat[2], lty = 2, col = c(3, 4))