Quantile Densities Ratio

• Quantile densities ratio is a function that compares the local sparsity of two distributions by analyzing their quantile density functions.
• It plays a key role in assessing stochastic orders and enables shape analysis via pdQ transforms, even when explicit quantile formulas are unavailable.
• The QDR framework facilitates efficient inference under density ratio models and guides optimal nonparametric estimation of quantile functions.

The quantile densities ratio is a comparative function that quantifies the local behavior of two probability distributions at corresponding quantiles of their respective laws. Formally, for two continuous random variables with well-defined quantile functions and positive densities, the quantile densities ratio (QDR) is defined as the ratio of their quantile density (sparsity) functions across the unit interval. QDRs are central in stochastic order theory, statistical inference under density ratio models, shape analysis via probability-density-quantile (pdQ) transforms, and robust distributional comparisons, especially when comparing random variables where explicit quantile expressions are unavailable or computationally intractable (Staudte, 2016, Arriaza et al., 24 Jan 2026, Chen et al., 2013).

1. Formal Definitions and Foundational Properties

Let $F_X$ and $F_Y$ denote the cumulative distribution functions (CDFs) of two continuous random variables $X$ and $Y$, with corresponding positive densities $f_X$ and $f_Y$. Their quantile functions are $Q_X(p) = F_X^{-1}(p)$ and $Q_Y(p) = F_Y^{-1}(p)$ for $p \in (0,1)$. The quantile density (also called the sparsity function) of $X$ is

$$q_X(p) = \frac{d}{dp} Q_X(p) = \frac{1}{f_X(Q_X(p))}.$$

The quantile densities ratio is then given by

$$\mathrm{QDR}(p) = \frac{q_X(p)}{q_Y(p)} = \frac{f_Y(Q_Y(p))}{f_X(Q_X(p))}, \quad 0 < p < 1.$$

QDR characterizes the local relative sparsity of the two distributions at corresponding quantile levels. Its behavior—specifically, monotonicity, unimodality, or multimodality—encodes important stochastic orders and has direct implications for comparative inference and model assessment (Arriaza et al., 24 Jan 2026).

2. Stochastic Orders and QDR Monotonicity

Convex transform order, star-shaped order, and other related stochastic orders are described directly via the monotonicity properties of the QDR:

• Convex Transform Order ($X \leq_c Y$): $\mathrm{QDR}(p)$ is non-decreasing in $p$ (Arriaza et al., 24 Jan 2026).
• Star-Shaped Order ($X \leq_\star Y$): $G^{-1}(p)/F^{-1}(p)$ is non-decreasing on $(0,1)$. This is equivalently determined by sign properties of specific boundary expressions involving the QDR and the quantile functions.
• If QDR is unimodal (increasing then decreasing), at most two turning points can exist in the corresponding quantile function ratio. Sufficient (and often necessary) conditions for such structures are provided in (Arriaza et al., 24 Jan 2026) via sign-change analysis of

$\delta(p) = Q_X(p)\,\mathrm{QDR}(p) - Q_Y(p).$

The framework allows comparison of random variables in the absence of explicit quantile formulas and also enables the derivation of stochastic order relationships for distributions with non-monotonic hazard or mean residual life functions.

An illustrative application is to Tukey-generalized distributions, with closed-form conditions for when the QDR is unimodal and subsequent monotonicity properties of the quantile function ratios. This structural theory is further related to classical aging notions, such as IFRA and DMRL, via corresponding characterizations of the QDR at the distribution’s boundaries.

3. Probability Density Quantile (pdQ) Transform and Ratio

A complementary approach to distributional comparison is provided by the pdQ transform (Staudte, 2016). For $F$ with density $f$ and quantile function $Q(u)$,

$$\text{pdQ:}\quad f^*(u) = \frac{f(Q(u))}{\int_{-\infty}^{\infty} f^2(x)\,dx},\quad u \in (0,1).$$

This is a probability density on $(0,1)$, invariant to location and scale, and encodes the “shape” of the original distribution.

Given two laws with pdQs $f_1^*, f_2^*$, the pdQ ratio

$R(u) = \frac{f_1^*(u)}{f_2^*(u)}, \quad 0 < u < 1,$

serves as a unit-interval function comparing shapes. It feeds directly into measures such as the Hellinger distance and the (symmetrized) Kullback-Leibler divergence,

$$I^*(1:2) = \int_0^1 f_1^*(u)\,\ln R(u)\,du, \quad J^*(1,2) = I^*(1:2) + I^*(2:1),$$

thus enabling global distributional divergence on the unit interval, independent of alignment issues or explicit consideration of location/scale differences (Staudte, 2016).

Table: Key QDR-related objects in the pdQ framework

Notation	Definition	Role
$f^*(u)$	pdQ: $f(Q(u))/\kappa$	Shape density on $(0,1)$
$R(u)$	pdQ ratio	Local shape comparison
$I^*$	KL divergence on $(0,1)$	Global shape dissimilarity

The explicit boundary behavior of $R(u)$ is determined via tail classification of pdQs, governed by the order of the first nonzero derivative at $u=1$; different tail weights yield divergence of the pdQ ratio, which directly impacts the divergence metrics.

4. Statistical Inference Using QDR: Density Ratio Models

In semiparametric inference, density ratio models posit

$$\frac{f_k(x)}{f_0(x)} = \exp\{\theta_k^\top q(x)\},$$

for densities $f_k$, $k=0,1,\ldots,m$, a baseline $f_0$, and specified tilting functions $q(x)$ (Chen et al., 2013, Zhang et al., 2020). The estimation and inference of quantiles (and quantile functions) in this context hinges on efficient use of QDR structure. Empirical likelihood techniques under this model yield quantile estimators with Bahadur-type expansions. When the density ratio model holds, information from all samples is efficiently pooled, and the resultant quantile estimators:

• Achieve lower asymptotic variance than naive per-sample estimators,
• Yield shorter, more precise confidence intervals for quantiles and their functions,
• Provide robust inference under mild model misspecification (Chen et al., 2013, Zhang et al., 2020).

For testing hypotheses about quantiles across populations, the empirical likelihood ratio test admits a limiting chi-square distribution for the profile likelihood, reflecting the intrinsic statistical efficiency gained by leveraging the QDR (or, more specifically, the log-density ratio) structure (Zhang et al., 2020).

5. Nonparametric Estimation, Bandwidth Selection, and Confidence Intervals

Kernel-type estimators for quantile density $q(p)$, fundamental for the estimation of QDR, have their performance governed by the local smoothness and curvature of the quantile function. The “quantile optimality ratio”,

$$\mathrm{QOR}(p) = \frac{q(p)}{q''(p)} = \frac{Q'(p)}{Q'''(p)},$$

dictates the optimal bandwidth for kernel-based estimation (Prendergast et al., 2015). Accurate estimation of $q(p)$ (and thus QDR) is achieved by adapting the bandwidth to $\mathrm{QOR}(p)$, zusing either a representative location-scale family or a fitted flexible GLD model.

For two independent samples, inference on the ratio of quantiles,

$R = \frac{Q(p_1)}{Q(p_2)},$

proceeds by estimating $q(p_1), q(p_2)$—typically using kernel or finite-difference methods exploiting $\mathrm{QOR}(p)$—and deriving asymptotic variances via the delta method (Prendergast et al., 2015). Table-based numerical procedures (see (Prendergast et al., 2015)) provide explicit formulas and implementation guidelines.

6. Applications, Robustness, and Connections to Aging Orders

QDR and related constructs are deployed in various domains:

• Income Inequality: Ratios of upper and lower quantiles ($p_1 > p_2$) capture inequality; robust, distribution-free confidence intervals for such ratios are provided by QDR-based inference (Prendergast et al., 2015).
• Transform Orders and Aging: Sufficient conditions on the QDR’s boundary behavior yield membership in stochastic order classes such as IFRA, DMRL, or star-shaped order. QDR structure also informs classification under nonmonotone hazard functions, e.g., bathtub or “upside-down bathtub” (Arriaza et al., 24 Jan 2026).
• Shape Analysis and Model Fitting: pdQ and QDR frameworks enable unified shape comparison and asymmetry assessment, independent of alignment or scale, enhancing shape-based model fitting and hypothesis testing (Staudte, 2016).

Robustness properties are analyzed via influence functions and breakdown points. QDR-based procedures retain efficiency and robustness under moderate contamination and mild model misspecification; care must be taken for quantile ratios with $p$ near zero in the presence of zero-inflation (Prendergast et al., 2015).

7. Practical Computation, Numerical Assessment, and Implementation

Empirical QDR estimation follows these principal steps (Staudte, 2016, Prendergast et al., 2015):

1. Estimate quantile functions $\widehat Q_X, \widehat Q_Y$ on a fine $p$-grid using order statistics.
2. Estimate quantile densities via kernel or spacing methods; bandwidth optimized via $\mathrm{QOR}(p)$ or direct estimation.
3. Form $$\widehat{\mathrm{QDR}}(p) = \widehat q_X(p)/\widehat q_Y(p)$$ across the grid.
4. Assess monotonicity or unimodality numerically; check boundary sign conditions for transform order or order-class membership (Arriaza et al., 24 Jan 2026).
5. For interval estimation for quantile ratios, construct variance estimates via the delta method and form asymptotic normal (typically log-scale) confidence intervals (Prendergast et al., 2015).

In semiparametric DRM contexts, additional steps involve likelihood optimization and root-finding for Lagrange multipliers, as well as profile likelihood and chi-square thresholding for inferential tasks (Zhang et al., 2020).

Numerical and simulation studies consistently show that QDR-guided estimation yields efficient, reliable inference with coverage and width superiority over separate-sample or non-adaptive methods, provided estimation of $q(p)$ is numerically stable and model-based assumptions are not grossly violated (Prendergast et al., 2015, Prendergast et al., 2015, Chen et al., 2013).

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Key References:

• (Arriaza et al., 24 Jan 2026): Sufficient conditions for some transform orders based on the quantile density ratio
• (Staudte, 2016): The Shapes of Things to Come: Probability Density Quantiles
• (Prendergast et al., 2015): Distribution-free Interval Estimators for Ratios of Quantiles
• (Prendergast et al., 2015): Exploiting the Quantile Optimality Ratio to Obtain Better Confidence Intervals for Quantiles
• (Chen et al., 2013): Quantile and quantile-function estimations under density ratio model
• (Zhang et al., 2020): Empirical likelihood ratio test on quantiles under a density ratio model