Stochastic frontiers or regression quantiles for estimating the self-thinning surface in higher dimensions?

Abstract

Stochastic frontier analysis and quantile regression are the two econometric approaches that have been commonly adopted in the determination of the self-thinning boundary line or surface in two and higher dimensions since their introduction to the field some 20 years ago. However, the rational for using one method over the other has, in most cases, not been clearly explained perhaps due to a lack of adequate appreciation of differences between the two approaches for delineating the self-thinning surface. Without an adequate understanding of such differences, the most informative analysis may become a missed opportunity, leading to an inefficient use of data, weak statistical inferences and a failure to gain greater insight into the dynamics of plant populations and forest stands that would otherwise be obtained. Using data from 170 plot measurements in even-aged Larix olgensis (A. Henry) plantations across a wide range of site qualities and with different abundances of woody weeds, i.e. naturally regenerated non-crop species, in northeast China, this study compared the two methods in determining the self-thinning surface across eight sample sizes from 30 to 170 with an even interval of 20 observations and also over a range of quantiles through repeated random sampling and estimation. Across all sample sizes and over the quantile range of 0.90 ≤ τ ≤ 0.99, the normal-half normal stochastic frontier estimation proved to be superior to quantile regression in statistical efficiency. Its parameter estimates had lower degrees of variability and correspondingly narrower confidence intervals. This greater efficiency would naturally be conducive to making statistical inferences. The estimated self-thinning surface using all 170 observations enveloped about 96.5% of the data points, a degree of envelopment equivalent to a regression quantile estimation with a τ of 0.965. The stochastic frontier estimation was also more objective because it did not involve the subjective selection of a particular value of τ for the favoured self-thinning surface from several mutually intersecting surfaces as in quantile regression. However, quantile regression could still provide a valuable complement to stochastic frontier analysis in the estimation of the self-thinning surface as it allows the examination of the impact of variables other than stand density on different quantiles of stand biomass.

Estimating the self‐thinning boundary line for oak mixed forests in central China by using stochastic frontier analysis and a proposed variable density model

“…Although quantile regression and stochastic frontier analysis can serve the same purpose of boundary delineation in applied statistics far beyond econometrics, an adequate appreciation of the differences between the two approaches is essential for researchers to select the best method to estimate the self‐thinning surface (Tian et al, 2020). The value of $$$ \tau $$$ chosen for estimating the self‐thinning line mainly ranges from 0.90 to 0.99 in the literature, with 0.95 ≤ $$$ \tau $$$ ≤ 0.99 being the most common choice (Andrews et al, 2018; Condés et al, 2017; Vospernik & Sterba, 2015).…”

“…Furthermore, stochastic frontier analysis may lead to a more objective self‐thinning line because it does not involve the subjective selection of a particular value of $$$ \tau $$$. However, quantile regression can still serve as a valuable complement to stochastic frontier analysis in the estimation of the self‐thinning surface, as it allows the impact of variables other than stand density on different quantiles to be examined (Tian et al, 2020).…”

¿Mínimos cuadrados ordinarios o regresión frontera estocástica para estimar la línea de autoaclareo con la ecuación de Yoda?

“…(2) es un componente del error asociado con la medición de las observaciones individuales, se asume como una perturbación simétrica distribuida independientemente de , recoge las variaciones aleatorias debidas a factores como errores aleatorios, errores en la observación y de medición de los datos, se distribuye como =iid N(0, σ 2 v) (Bi, 2004;Comeau et al, 2010;Salas-Eljatib y Weiskittel, 2018;Tian et al, 2021;Long et al, 2022). (AIC) y criterio de Schwarz (SchC), además de las varianzas de los componentes del error (σ 2 v y σ 2 u), la razón de varianzas de los componentes del error (λ) y la varianza total (σ 2 ) (Bi, 2004;Comeau et al, 2010).…”

“…El propósito de realizar contrastes más objetivos es definir un proceso metodológico preciso y eficiente, tanto para determinar la línea de autoaclareo como para la construcción de GMD (Zhang et al, 2005;VanderSchaaf y Burkhart, 2007; Salas-Eljatib y Weiskittel, 2018;Marchi, 2019;Tian et al, 2021), así como para disponer de estrategias alternativas con eficiencia similar.…”

“…Por su parte, las guías o diagramas para manejar la densidad (GMD) de rodales mediante la prescripción de regímenes de aclareos, se construyen con base en la línea de autoaclareo que reproduce funciones tamaño-densidad (Santiago-García et al, 2013;Brunet-Navarro et al, 2016;Newton, 2021), entre las que sobresale la propuesta por Yoda (Tian et al, 2021).…”