New Quantile Regression Model Tames Bounded Data from Body Fat to Sleep

Statisticians have introduced a new regression model designed for data that live strictly between zero and one, such as rates, proportions, and indices, and it outperformed nine established rivals on two real datasets. It builds on a probability distribution that had previously been used mainly for telecommunications data but had never been turned into a regression framework for bounded responses.

The problem the authors tackle is familiar to anyone who has tried to model percentages with ordinary linear regression. A response variable such as the proportion of body fat or the share of sleep spent in rapid eye movement is confined to the unit interval, yet a standard Gaussian regression line has an infinite range and can happily predict values below zero or above one. The conventional workaround is to transform the response onto the whole real line, but that destroys the natural interpretation of the original scale, and the assumptions of normality and constant variance often fail because such data are frequently skewed, sometimes strongly to the left.

The beta distribution has long been the default choice for bounded data, but it comes with a practical handicap: its cumulative distribution and quantile functions have no closed form, which complicates estimation and simulation. A growing family of alternatives has been proposed, including unit versions of the gamma, logistic exponential, Gompertz, inverse exponentiated Lomax, inverse-Gaussian, exponential, Weibull, Lindley, and Birnbaum–Saunders distributions, several of which have been extended into quantile regression frameworks. What ties these quantile-based efforts together is a shared requirement: the quantile function of the underlying distribution must have a closed analytical form, because the regression is built by reparameterizing the distribution directly in terms of that function rather than its mean.

The new model starts from the unit Gumbel–logistic (UGL) distribution, introduced by Stojanović and colleagues, which arises by pushing a zero-centered Gumbel random variable through the general logistic map onto the interval from zero to one. The resulting density has a simple closed expression, and, crucially, its cumulative distribution function is also closed-form, avoiding the incomplete beta functions and other special functions that the beta distribution requires. Inverting the cumulative function yields a closed-form quantile function, and that is the key that unlocks the regression. The authors reparameterize the distribution in terms of a quantile of interest, so that a fixed quantile order p and a quantile parameter mu replace the original shape parameter, while a second parameter beta governs the residual dispersion of the response around that quantile.

Because the moments of the UGL distribution do not admit a simple closed form, relating the mean response to covariates is not straightforward, so the authors instead link the quantile parameter to a linear predictor through a logit link, in the same spirit as logistic regression. Setting the quantile order p to 0.5 yields a median regression model for a unit response, and other quantile levels can be targeted by changing p. Probit and complementary log-log links may also be used within the framework. Parameters are estimated by maximum likelihood, with the score equations solved numerically using the maxLik function in R, and the authors derive the necessary chain-rule derivatives explicitly because the shape of each observation’s distribution depends on its fitted quantile.

Two Monte Carlo simulation studies, each with 5,000 replications, tested the machinery. The first examined randomized quantile residuals and Cox–Snell residuals across sample sizes from 50 to 500 and shape parameters from 0.5 to 3.0. Empirical means of the randomized quantile residuals stayed within plus or minus 0.005 of zero with variances between 0.995 and 1.026, practically indistinguishable from the standard normal, and Anderson–Darling rejection rates sat at or near the nominal 5 percent level. The Kolmogorov–Smirnov test proved overly conservative in small samples and for large shape values, occasionally returning a rejection rate of zero, so the authors recommend the Anderson–Darling test for assessing residual normality in those regimes.

The second simulation evaluated bias, root mean squared error, coverage of 95 percent asymptotic confidence intervals, and average coverage length across five quantile orders from 0.10 to 0.90 and the same range of sample sizes. As expected of consistent estimators, bias and root mean squared error fell monotonically with sample size, coverage probabilities stayed within roughly 0.93 to 0.96 even at n equals 50, and confidence lengths shrank accordingly. One consistent pattern was a sign reversal in the bias of the intercept as the quantile order increased, positive at p equal to 0.10 and increasingly negative toward 0.90, an effect most visible at n equals 50 that faded to negligible levels for sample sizes of 300 or more. The shape parameter beta, rather than the number of covariates, proved the main source of estimation difficulty in small samples: its root mean squared error at n equals 50 rose from about 0.065 when the true beta was 0.5 to roughly 0.24 to 0.26 when the true beta was 2, although by n equals 500 the difference had largely vanished.

The first real application used the well-known body fat dataset of 252 men, whose body fat proportion measured by underwater weighing was rescaled to the unit interval and modeled against thirteen standardized anthropometric measurements, including age, weight, height, and circumferences of the neck, chest, abdomen, hip, thigh, knee, ankle, biceps, forearm, and wrist. Fitted at the median, the UGLRM achieved the highest log-likelihood of the ten models considered, 418.6683 against 395.7120 for the next-best beta regression model, and the lowest AIC, AICc, and BIC, beating the beta model’s AIC of minus 761.4240 with a value of minus 807.3366. Within the fitted model, abdomen circumference carried the largest positive coefficient at 6.2760, while wrist circumference entered strongly negative at minus 11.0162, consistent with wrist size acting as a proxy for skeletal frame rather than fatness; weight and neck circumference were also negative once abdomen and limb measurements were accounted for. Cox–Snell residual envelopes, QQ- and PP-plots, and Cook’s distance diagnostics supported the fit, and a quantile-process analysis from p equal to 0.10 to 0.90 showed abdomen circumference remaining the dominant driver across the whole response distribution.

The second application drew on a sleep clinic study of 133 patients referred for polysomnography, of whom 106 complete observations remained after excluding missing values. The response was the proportion of total sleep time spent in rapid eye movement, modeled against age and a gender indicator. Again the UGLRM fit best on all four criteria, with a log-likelihood of 135.11 compared with 116.68 for the unit Birnbaum–Saunders model and 116.10 for the beta model, and the lowest AIC, AICc, and BIC. Both covariate effects were negative, with the REM proportion decreasing slightly with age and lower for women, but neither was statistically significant at the 5 percent level, an unsurprising outcome given only two covariates and a modest sample. Residual diagnostics showed reasonable agreement with the reference distributions, with mild tail departures typical of a 106-patient clinical sample, and the quantile process showed a steadily rising intercept with stable slopes and a flat, well-behaved shape parameter estimate.

The authors are careful about scope. Their applications come from medical and health-related data, and they note that outliers and multicollinearity can affect the sensitivity of maximum likelihood estimation and the stability of the estimates, so extending the model to non-medical and higher-dimensional datasets remains future work. Even so, the combination reported here is distinctive: a distribution with closed-form density, cumulative, and quantile functions, a quantile parameterization that connects covariates to any chosen quantile of a bounded response, maximum likelihood estimation without extra computational overhead, and the best information-criterion performance in both benchmark applications against nine competitors, including the beta, unit Kumaraswamy, unit Lindley, and unit log-log regression models. For applied researchers wrestling with proportions, rates, and indices that refuse to behave like Gaussian data, the unit Gumbel–logistic regression model now offers a tractable and competitive option.

Subject of Research: A new quantile regression model based on the unit Gumbel–logistic distribution for bounded response variables

Article Title: A new quantile regression model for bounded responses: mathematical derivation, estimation, and fitting bounded datasets

Article References: Alqasem, O. A., Gemeay, A. M., Alnagar, D. K., Khogeer, H. A., Alrumayh, A., Hussam, E., Gellow, G. T., & Hammad, A. T. (2026). A new quantile regression model for bounded responses: mathematical derivation, estimation, and fitting bounded datasets. Boundary Value Problems, 2026(1), Article 150. https://doi.org/10.1186/s13661-026-02369-8

Image Credits: AI Generated

DOI: 10.1186/s13661-026-02369-8

Keywords: quantile regression, bounded responses, unit Gumbel–logistic distribution, maximum likelihood estimation, beta regression, Monte Carlo simulation, body fat dataset, REM sleep, residual diagnostics, statistics, model comparison, unit interval data