Least Product Relative Error Estimator

• Least Product Relative Error (LPRE) estimator is a robust method defined for multiplicative regression models by minimizing the product of two relative errors.
• It ensures a unique, convex solution in parametric, semiparametric single-index, and functional settings, while maintaining strict scale invariance.
• LPRE offers strong large-sample properties and improved performance under skewed or heavy-tailed noise, with efficient gradient-based optimization algorithms.

The Least Product Relative Error (LPRE) estimator is a statistical estimation method for multiplicative regression models, defined by the minimization of the product of two relative errors: one relative to the observed response and one relative to the model-predicted value. LPRE estimation is strictly scale-invariant, admits a uniquely defined solution due to the convexity of its loss function, and has been established for standard parametric, semiparametric single-index, and functional regression settings. LPRE demonstrates superior robustness compared to least-squares-type estimators, especially under skewed or heavy-tailed noise, while achieving strong large-sample properties including asymptotic normality and root-n consistency (Wang et al., 2016, Chen et al., 2013, Yan et al., 2023).

1. Model Formulations and Scope

The canonical LPRE estimator is applicable to the multiplicative regression model:

$Y_i = \exp(X_i^T \beta_0)\, \varepsilon_i,$

where $Y_i > 0$ denotes the response, $X_i \in \mathbb R^p$ is a (possibly augmented) covariate vector, $\beta_0 \in \mathbb R^p$ is the true parameter, and $\varepsilon_i > 0$ is an error variable with $$E(\varepsilon_i|X_i) = E(\varepsilon_i^{-1}|X_i) = 1$$ (Chen et al., 2013). In the semiparametric single-index extension,

$$y_i = \exp\big\{\,g(x_i^T\beta_0)\,\big\}\, \varepsilon_i,$$

$\beta_0$ (with $\|\beta_0\| = 1$ for identifiability) appears only through $x_i^T\beta_0$ and $Y_i > 0$0 is an unknown smooth link (Wang et al., 2016). Functional LPRE generalizes the model to scalar-on-function regression,

$Y_i > 0$1

where $Y_i > 0$2 and $Y_i > 0$3 are square-integrable functions on $Y_i > 0$4 (Yan et al., 2023).

LPRE estimates parameters by focusing on errors expressed as ratios, thereby providing a natural analysis of scale-invariant models.

2. LPRE Criterion and Core Properties

For data $Y_i > 0$5, define for each $Y_i > 0$6:

$Y_i > 0$7

with $Y_i > 0$8. The LPRE loss is

$Y_i > 0$9

The aggregate criterion for $X_i \in \mathbb R^p$0 observations is then

$X_i \in \mathbb R^p$1

This criterion is strictly convex, continuous, and smooth in $X_i \in \mathbb R^p$2 under standard moment conditions (Chen et al., 2013). For single-index and functional models, analogous forms express $X_i \in \mathbb R^p$3 or functional equivalents via the index $X_i \in \mathbb R^p$4 or integral $X_i \in \mathbb R^p$5 (Wang et al., 2016, Yan et al., 2023).

The scale-invariant property is immediate, since rescaling $X_i \in \mathbb R^p$6 or any covariate leaves $X_i \in \mathbb R^p$7 unchanged.

3. Estimation Algorithms and Computational Aspects

For linear models, parameter estimation proceeds by direction minimization of $X_i \in \mathbb R^p$8. Gradient and Hessian are available in closed form:

$X_i \in \mathbb R^p$9

$\beta_0 \in \mathbb R^p$0

Newton–Raphson steps ensure rapid and globally convergent optimization due to strict convexity. The unique minimizer exists under positive-definiteness of $\beta_0 \in \mathbb R^p$1 (Chen et al., 2013).

Single-index and functional models require more elaborate schemes:

• Single-index LPRE: Employs a two-stage procedure. First, for given $\beta_0 \in \mathbb R^p$2, $\beta_0 \in \mathbb R^p$3 is estimated by local linear smoothing of $\beta_0 \in \mathbb R^p$4 against $\beta_0 \in \mathbb R^p$5 (using bandwidth $\beta_0 \in \mathbb R^p$6). Then, $\beta_0 \in \mathbb R^p$7 is updated (subject to $\beta_0 \in \mathbb R^p$8) via Newton–Raphson on the profile LPRE criterion, leveraging the derivative $\beta_0 \in \mathbb R^p$9 estimated with a bandwidth $\varepsilon_i > 0$0 (Wang et al., 2016).
• Functional LPRE: Expands $\varepsilon_i > 0$1 in a B-spline basis, applies penalized LPRE least squares, and solves the resulting convex problem via Newton–Raphson. Spline penalty order and knot selection control smoothness (Yan et al., 2023).

For massive data, optimal subsampling strategies (A-optimal or Hessian-free L-optimal) further accelerate functional LPRE, with rigorous guarantees on the quality of the estimator from subsamples.

4. Large-Sample Theory and Statistical Guarantees

For linear multiplicative models, under mild regularity conditions including bounded moments and $\varepsilon_i > 0$2, the LPRE estimator satisfies

$\varepsilon_i > 0$3

with

$\varepsilon_i > 0$4

and a consistent covariance estimator $\varepsilon_i > 0$5 available by plug-in (Chen et al., 2013).

In the single-index semiparametric case, the estimator $\varepsilon_i > 0$6 is root-n consistent and asymptotically normal, with an explicit sandwich covariance formula reflecting the unit-norm constraint. The link estimator $\varepsilon_i > 0$7 achieves uniform error $\varepsilon_i > 0$8 over compact supports (Wang et al., 2016).

For the functional setting, with $\varepsilon_i > 0$9 basis functions and suitably regularized $$E(\varepsilon_i|X_i) = E(\varepsilon_i^{-1}|X_i) = 1$$0, one obtains

$$E(\varepsilon_i|X_i) = E(\varepsilon_i^{-1}|X_i) = 1$$1

and for fixed $$E(\varepsilon_i|X_i) = E(\varepsilon_i^{-1}|X_i) = 1$$2,

$$E(\varepsilon_i|X_i) = E(\varepsilon_i^{-1}|X_i) = 1$$3

where $$E(\varepsilon_i|X_i) = E(\varepsilon_i^{-1}|X_i) = 1$$4 is the number of spline knots and $$E(\varepsilon_i|X_i) = E(\varepsilon_i^{-1}|X_i) = 1$$5 is a model-dependent variance matrix (Yan et al., 2023).

Subsampling versions inherit analogous central limit theorems, with variance inflation reflecting subsample size $$E(\varepsilon_i|X_i) = E(\varepsilon_i^{-1}|X_i) = 1$$6.

5. Empirical Comparisons and Application Results

Simulation studies compare LPRE to least-squares (LS) estimators on transformed (log) data, least absolute deviation (LAD) regression, and least absolute relative error (LARE):

• Under log-uniform and structurally “adversarial” noise (i.e., $$E(\varepsilon_i|X_i) = E(\varepsilon_i^{-1}|X_i) = 1$$7, or distributions with $$E(\varepsilon_i|X_i) = E(\varepsilon_i^{-1}|X_i) = 1$$8), LPRE outperforms LS, LAD, and LARE with much smaller bias and MSE.
• Under normal error distributions ($$E(\varepsilon_i|X_i) = E(\varepsilon_i^{-1}|X_i) = 1$$9), LPRE and LS perform similarly, but LPRE generally yields smaller standard error and improved robustness.
• In a real data application to percent body fat prediction ($$y_i = \exp\big\{\,g(x_i^T\beta_0)\,\big\}\, \varepsilon_i,$$0, 12 predictors), LPRE produces lower values of median absolute error (MPE), product-relative error (MPPE), additive-relative error (MAPE), and squared error (MSPE) than LS regression or LAD, both in- and out-of-sample. LPRE also selects biceps circumference as significant where other methods do not (Chen et al., 2013, Wang et al., 2016).

For the semiparametric body-fat example ($$y_i = \exp\big\{\,g(x_i^T\beta_0)\,\big\}\, \varepsilon_i,$$1), LPRE and LS agree on core predictors, but differ on the sign and significance of certain variables (e.g., knee and ankle), illustrating higher stability and resistance to influential observations for LPRE.

6. Practical Implementation and Robustness

LPRE estimators require no tuning of scale factors and are invariant to units of both response and covariates, necessitating only choice of local smoothing bandwidths or spline penalty parameters. In the single-index model, bandwidths are typically chosen by cross-validation or generalized cross-validation (GCV). Newton–Raphson optimization converges rapidly from any reasonable initialization due to strict convexity.

The underlying product-of-relative-errors loss down-weights large residuals when the magnitude of $$y_i = \exp\big\{\,g(x_i^T\beta_0)\,\big\}\, \varepsilon_i,$$2 or the predicted value is large, imparting natural robustness against outliers or heteroskedasticity. Variance estimation is straightforward either by plug-in or bootstrap, with theoretically established accuracy under model conditions.

For functional and high-dimensional data, optimal and Hessian-free subsampling schemes enable LPRE computation with rigorously controlled loss of precision and scalable complexity.

7. Extensions and Related Methodology

LPRE methodology generalizes readily to semiparametric and functional models, each retaining the core scale-invariance and convexity properties. The framework is fundamentally distinct from least-squares or absolute value regression approaches, which are sensitive to units or may produce non-unique solutions in non-smooth or high-dimensional settings.

LPRE is situated among robust, interpretable, and computation-friendly regression frameworks for multiplicative structures, with particular efficacy in applied problems involving relative, proportional, or scale-dependent data structures. Its empirical and inferential characteristics are governed by explicit moment and regularity assumptions, distinguishing it from less robust or less well-characterized alternatives (Chen et al., 2013, Wang et al., 2016, Yan et al., 2023).