Description Usage Arguments Details Value Author(s) References Examples

Compute approximate confidence interval for the true log-concave density, on a grid of points. Two main approaches are implemented: In the first, the confidence interval at a fixed point is based on the pointwise asymptotic theory for the log-concave maximum likelihood estimator (MLE) developed in Balabdaoui, Rufibach, and Wellner (2009). In the second, the confidence interval is estimated via the boostrap.

1 2 3 |

`res` |
An object of class |

`xx0` |
Vector of grid points at which to calculate the confidence interval. |

`conf.level` |
Confidence level for the confidence interval(s). The default is 95%. |

`type` |
Vector of strings indicating type of confidence interval to compute. When |

`htype` |
Vector of strings indicating bandwidth selection method if |

`BB` |
number of iterations in the bootstrap if |

In Balabdaoui et al. (2009) it is shown that (if the true density is strictly log-concave) the limiting distribution of the MLE of a log-concave
density *\widehat f_n* at a point *x* is

*n^{2/5}(\widehat f_n(x)-f(x)) \to c_2(x) \bar{C}(0).*

The nuisance parameter *c_2(x)* depends on the true density *f* and the second derivative of its logarithm. The limiting process *\bar{C}(0)*
is found as the second derivative at zero of a particular operator (called the "envelope") of an integrated Brownian motion plus *t^4*.

Three of the confidence intervals are based on inverting the above limit using estimated quantiles of *\bar{C}(0)*, and estimating the nuisance
parameter *c_2(x)*. The options for the function `logConCI`

provide different ways to estimate this nuisance parameter. If `type = "DR"`

,
*c_2(x)* is estimated using derivatives of the smoothed MLE as calculated by the function `logConDens`

(this method does not perform well in
simulations and is therefore not recommended). If `type="ks"`

, *c_2(x)* is estimated using kernel density estimates of the true density and its
first and second derivatives. This is done using the `R`

package ks, and, with this option, a bandwidth selection method `htype`

must also
be chosen. The choices in `htype`

correspond to the various options for bandwidth selection available in ks. If `type = "nrd"`

, the second
derivative of the logarithm of the true density in *c_2(x)* is estimated assuming a normal reference distribution.

Two of the confidence intervals are based on the bootstrap. For `type = "ECDFboot"`

confidence intervals based on re-sampling from the empirical
cumulative distribution function are computed. For `type = "NPMLboot"`

confidence intervals based on re-sampling from the nonparametric maximum
likelihood estimate of log-concave density are computed. Bootstrap confidence intervals take a few minutes to compute!

The default option is `type = "ks"`

with `htype = "hns"`

. Currently available confidence levels are 80%, 90%, 95% and 99%, with a default
of 95%.

Azadbakhsh et al. (2014) provides an empirical study of the relative performance of the various approaches available in this function.

The function returns a list containing the following elements:

`fhat` |
MLE evaluated at grid points. |

`up_DR` |
Upper confidence interval limit when |

`lo_DR` |
Lower confidence interval limit when |

`up_ks_hscv` |
Upper confidence interval limit when |

`lo_ks_hscv` |
Lower confidence interval limit when |

`up_ks_hlscv` |
Upper confidence interval limit when |

`lo_ks_hlscv` |
Lower confidence interval limit when |

`up_ks_hpi` |
Upper confidence interval limit when |

`lo_ks_hpi` |
Lower confidence interval limit when |

`up_ks_hns` |
Upper confidence interval limit when |

`lo_ks_hns` |
Lower confidence interval limit when |

`up_nrd` |
Upper confidence interval limit when |

`lo_nrd` |
Lower confidence interval limit when |

`up_npml` |
Upper confidence interval limit when |

`lo_npml` |
Lower confidence interval limit when |

`up_ecdf` |
Upper confidence interval limit when |

`lo_ecdf` |
Lower confidence interval limit when |

Mahdis Azadbakhsh

Hanna Jankowski, hkj@mathstat.yorku.ca,

http://www.math.yorku.ca/~hkj/

Azadbakhsh, M., Jankowski, H. and Gao, X. (2014).
Computing confidence intervals for log-concave densities.
*Comput. Statist. Data Anal.*, to appear.

Baladbaoui, F., Rufibach, K. and Wellner, J. (2009)
Limit distribution theory for maximum likelihood estimation of a log-concave density.
*Ann. Statist.*, **37(3)**, 1299–1331.

Tarn Duong (2012). ks: Kernel smoothing. R package version 1.8.10. http://CRAN.R-project.org/package=ks

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 | ```
## Not run:
## ===================================================
## Confidence intervals at a fixed point for the density
## ===================================================
data(reliability)
x.rel <- sort(reliability)
# calculate 95
grid <- seq(min(x.rel), max(x.rel), length.out = 200)
res <- logConDens(x.rel)
ci <- logConCI(res, grid, type = c("nrd", "ECDFboot"))
par(las = 1, mar = c(2.5, 3.5, 0.5, 0.5))
hist(x.rel, n = 25, col = gray(0.9), main = "", freq = FALSE,
xlab = "", ylab = "", ylim = c(0, 0.0065), border = gray(0.5))
lines(grid, ci$fhat, col = "black", lwd = 2)
lines(grid, ci$lo_nrd, col = "red", lwd = 2, lty = 2)
lines(grid, ci$up_nrd, col = "red", lwd = 2, lty = 2)
lines(grid, ci$lo_ecdf, col = "blue", lwd = 2, lty = 2)
lines(grid, ci$up_ecdf, col = "blue", lwd = 2, lty = 2)
legend("topleft", col = c("black", "blue", "red"), lwd = 2, lty = c(1, 2, 2), legend =
c("log-concave NPMLE", "CI for type = nrd", "CI for type = ECDFboot"), bty = "n")
## End(Not run)
``` |

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