On the tails of log-concave density estimators

TL;DR

We study the tail behavior of the nonparametric MLE for univariate log-concave densities, showing that with an appropriately shrinking right tail $[b_n,\infty)$, the absolute and relative errors of $\hat f_n$, $\hat\varphi_n$, and $\hat\varphi_n'$ converge uniformly to zero on $[a,b_n]$. The main approach combines structural properties of the log-concave MLE with exponential and maximal inequalities for truncated moments, plus a suite of auxiliary results on empirical processes and mean-excess function bounds. The analysis separates right-tail scenarios where $b_o$ is finite or infinite and provides precise conditions under which tail accuracy holds, contributing both practical tail-approximation guarantees and theoretical insights into log-concave density estimation. The results rely on a detailed toolkit of inequalities, knot-based convex analysis, and mean-excess function bounds that may be of independent interest for log-concave models and tail analysis.

Abstract

It is shown that the nonparametric maximum likelihood estimator of a univariate log-concave probability density satisfies desirable consistency properties in the tail regions. Specifically, let $P$ and $f$ denote the true underlying distribution and density, respectively. If $\hat{f}_n$ is the estimated log-concave density, and $\hat{\varphi}_n = \log \hat{f}_n$, then we specify sequences $(b_n)_{n\in \mathbb{N}}$ such that $P([b_n,\infty)) \to 0$ at a specific speed, ensuring that the absolute errors or absolute relative errors of $\hat{f}_n, \ \hat{\varphi}_n$ and $\hat{\varphi}_n'$ converge to zero uniformly on sets $[a, b_n]$. The main tools, besides characterizations of $\hat{f}_n$, are exponential and maximal inequalities for truncated moments of log-concave distributions, which are of independent interest.

Figures (4)

• Figure 1: The functions $\hat{\varphi}_n$ (left panel) and $\hat{\varphi}_n'(\cdot+)$ (right panel) for one particular sample of size $n = 150$ (top), $n = 500$ (middle) and $n=2000$ (bottom) from $\mathrm{Unif}[0,1]$. The sample is indicated as a rug plot, and the true values $\varphi$ and $\varphi'$ are shown in red.
• Figure 2: Estimated $\gamma$-quantiles of $\hat{\varphi}_n(x)$ (left panel) and $\hat{\varphi}_n'(x+)$ (right panel) for samples of size $n = 150$ (top), $n = 500$ (middle) and $n=2000$ (bottom) from $\mathrm{Unif}[0,1]$. The true values $\varphi(x)$ and $\varphi'(x)$ are shown in red.
• Figure 3: The functions $\hat{\varphi}_n$ (left panel) and $\hat{\varphi}_n'(\cdot+)$ (right panel) for one particular sample of size $n = 150$ (top), $n = 500$ (middle) and $n=2000$ (bottom) from $\mathrm{N}(0,1)$. The sample is indicated as a rug plot, and the true values $\varphi$ and $\varphi'$ are shown in red.
• Figure 4: Estimated $\gamma$-quantiles of $\hat{\varphi}_n(x)$ (left panel) and $\hat{\varphi}_n'(x+)$ (right panel) for samples of size $n = 150$ (top), $n = 500$ (middle) and $n=2000$ (bottom) from $\mathrm{N}(0,1)$. The true values $\varphi(x)$ and $\varphi'(x)$ are shown in red.

Theorems & Definitions (20)

• Theorem 1
• Theorem 2
• Example 1
• Theorem 3
• Example 2
• Proposition 1: Consistency of $\hat{P}_n^{\mathrm{emp}}$
• proof : Proof of Proposition \ref{['prop:Pnhatemp']}
• Proposition 2
• Proposition 3: Properties of $M$ and $\mu$
• proof : Proof of Proposition \ref{['prop:M.and.mu']}