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.
Highlights
In plant population and community ecology, self-thinning refers to the progressive decline in stand density through competition-induced mortality as individual plants grow larger and collectively accumulate more biomass while competition intensity increases and site resources become limiting for all to survive (Harper 1977; Begon et al 2006)
A natural question would arise: do distributional assumptions matter in the estimation of self-thinning surface? Despite the vast number of stochastic frontier analysis undertaken in theoretical as well as applied research with empirical data, surprisingly few studies have been devoted to discerning the impact that alternative shapes of the distribution of u can have on the frontier and efficiency estimation and few attempts have been made to explicitly address specification testing in stochastic frontier models (Guo et al 2018; Kumbhakar et al 2018)
The effect of site productivity and the impact of the abundance of woody weeds on the accumulation of stand biomass were not the same across the conditional quantiles. Such differences, when carefully examined, would provide a greater insight into the growth dynamics of the even-aged larch stands growing under the self-thinning surface. This complementarity between stochastic frontier analysis and quantile regression argues for the application of both methods in the estimation of the self-thinning surface in high dimensions, especially when dealing with data that do not conform to the distributional assumptions of stochastic frontier models
Summary
In plant population and community ecology, self-thinning refers to the progressive decline in stand density (i.e. number of plants per unit area) through competition-induced mortality as individual plants grow larger and collectively accumulate more biomass while competition intensity increases and site resources become limiting for all to survive (Harper 1977; Begon et al 2006). For trees in forest stands managed for timber production in particular, the quadratic mean dimeter ( Dq ), an obtainable stand attribute, is often used instead of average tree mass or stem volume as a measure of average tree size in the determination of size-density relationships for stand density management In this case, the maximum size-density relationship of Reineke (1933), which defines a species-specific upper limit of stand density for a given quadratic mean diameter on log scales, has proven to be a special case of the self-thinning boundary line (Pretzsch 2002, 2009; Pretzsch et al 2012)
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