A Comparison of Probability Density Functions Fitted by Moments and Maximum Likelihood Estimation Methods Used for Diameter Distribution Estimation

Abstract

Modeling diameter distribution is a crucial aspect of forest management, requiring the selection of an appropriate probability density function or cumulative distribution function along with a fitting method. This study compared the suitability of eight probability density functions—A Charlier, beta, generalized beta, gamma, Gumbel, Johnson’s SB, and Weibull (two- and three-parameter)—fitted using both derivative methods (Moments) fitted in SAS/STAT^TM and optimization methods (MLE) fitted with the ‘optim’ function in R for diameter distribution estimation in forest stands. The A Charlier and Gumbel functions were used for the first time in this type of comparison. The data were derived from 167 permanent sample plots in an Atlantic forest (Quercus robur) and 59 temporary sample plots in tropical forests (Tectona grandis). Fit quality was assessed using various indices, including Kolmogorov–Smirnov, Cramér–von Mises, mean absolute error, bias, and mean squared error. The results indicated that Johnson’s SB function was more suitable for describing the diameter distribution of the stands. Johnson’s SB, three-parameter Weibull, and generalized beta consistently performed well across different fitting methods, while the fits produced by gamma, Gumbel, and two-parameter Weibull were of poor quality.

1. Introduction

Tree diameter distribution is an important component of forest management which is used for decision making, such as what and when to harvest and what silvicultural treatment to apply [1]. Researchers can deduce details about the dynamics of forests, including recruitment rates, mortality patterns, and growth rates, by examining changes in diameter distributions across time [2]. Based on the diameter class model, we can assess a stand’s stability, volume production, assortment structure, and maturity [3,4], thus making it an important component in decision-support systems.

To estimate stand characteristics and their structures with a diameter class model, researchers rely on the use of a probability density function (PDF) or its cumulative form (CDF). Various probability density functions have been adopted in describing stem frequency in both natural forest stands and forest plantations, such as beta [5,6], generalized beta [6,7], gamma [8,9], Johnson’s SB [10,11], Weibull [6,12,13,14], etc., with different levels of success. To date, the Weibull, gamma, beta, and Johnson’s SB functions are the traditional functions commonly used in quantitative forestry. Other important functions like the Charlier (type A) have gained less importance. Besides the earlier work of Schnur [15] on the diameter distribution of Loblolly pine (Pinus taeda L), few and old published studies have been documented in the forestry literature on the Charlier function to the best of our knowledge.

In forestry, functions can be fitted using either the derivative methods (Moments, percentiles, conditional maximum likelihood, mode, etc.) and/or numerical optimization (Maximum Likelihood Estimation) [6]. One advantage of the derivative methods is their simplicity and ease of estimation, obtained from stand characteristics [10,16,17,18]. However, it is difficult to quantify the uncertainty in parameter estimates with derivative methods [6]. On the other hand, the uncertainty in parameter estimates from optimization can be determined through the Hessian matrix [19]. In this research, we compared both methods. The Weibull function has been fitted by derivative methods such as Moments, percentiles, and optimization to mixed spruce–fir stands [20] and tropical and Atlantic forests [6]. Recently, the same methods were used for the Weibull function to characterize the diameter distribution of European beech (Fagus sylvatica L.) [21]. The gamma function has been fitted by Moments [9] and optimization [6]. The beta function has been fitted by the method of Moments [6,22]. The generalized beta function has been fitted by Moments [6,7] and optimization [6]. The traditional methods for fitting Johnson’s SB function also include conditional maximum likelihood, Moments, mode, percentiles, and the optimization method by maximum likelihood [6].

Therefore, the main objective of this study was to compare the precision of A Charlier, Gumbel, Gamma-2P, Weibull (2P and 3P), beta, generalized beta, and Johnson’s SB functions fitted by a derivative method (Moments) and a numerical optimization method (MLE). We used data from two broadleaf species of even-aged stands in Atlantic and tropical forests.

2. Materials and Methods

2.1. Data Set

Data for this study were sourced from 167 permanent research plots (PRPs) in even-aged pedunculate oak (Quercus robur L.) stands across Galicia (northwest Spain) and 59 temporary sample plots (TSPs) in teak (Tectona grandis L. f.) plantations in Omo and Gambari (south Nigeria). Plot sizes varied from 225 m² to 1345 m² in pedunculate oak stands and from 400 m² to 625 m² in teak plantations (depending on stand density), ensuring a minimum of 30 trees per plot. Pedunculate oak plots were situated in dominant stands (constituting more than 85% of species standing basal area), while plots for both species covered a diverse range of combinations in terms of age, tree density, and site conditions.

Two perpendicular diameters at breast height were measured using calipers, recorded to the nearest 0.1 cm, and the arithmetic average was computed. The empirical data reflected left-truncated distributions, as the smallest measured diameter in the field was 5 cm. A total of 9845 diameter measurements were available for pedunculate oak stands and 2812 for teak plantations. Stand variables, including the number of trees per hectare (N), basal area (G), quadratic mean diameter (d_g), dominant height (Ho, and the relative spacing index (RS), were calculated from the inventory data. Tabulated summary statistics, encompassing mean, maximum and minimum values, and standard deviation of the primary stand variables, are presented in

Table 1. Summary of descriptive statistics of stand variables.

	Variable	Mean	Max	Min	SD
Quercus robur (N = 167 plots)	N	854.6	3022.2	302.2	452.5
	G	28.6	72.9	7.7	9.1
	dg	22.2	40.0	8.6	5.9
	HO	17.1	25.6	7.2	3.2
	RS	22.0	39.8	12.7	4.8
Tectona grandis (N = 59 plots)	N	899.3	1744.0	624.0	181.6
	G	26.9	50.8	11.5	8.9
	dg	19.3	23.5	14.3	2.3
	HO	21.9	28.4	17.7	2.2
	RS	15.6	20.1	9.9	2.4

N: density (trees·ha⁻¹); G: basal area (m²·ha⁻¹); d_g: quadratic mean diameter (cm); Ho: dominant height (m); RS: relative spacing index.

.

2.2. Distributions and Fitting Methods

2.2.1. The Charlier Distribution

The Charlier distribution (type A) [15] has the following expression for a random variable x:

$$f\left(x\right)=\frac{1}{\sqrt{2\pi \sigma }}exp\left[-\frac{{(x-\mu )}^2}{2{\sigma }^2}\right]\left[1+\frac{k_3}{3!{\sigma }^3}{H}_3\left(\frac{x-\mu }{\sigma }\right)+\frac{k_4}{4!{\sigma }^4}{H}_4\left(\frac{x-\mu }{\sigma }\right)\right]$$

(1)

where µ is the mean (first moment of the distribution); ${\sigma }^2$ is the variance (second moment of the distribution); H₃(x) = x³ − 3x and H₄(x) = x⁴ − 6x² + 3 are Hermite polynomials; $k_3/{\sigma }^3$ is the asymmetry with $k_3$, the third moment of the distribution; and $k_4/{\sigma }^4$ is the kurtosis with $k_4$, the fourth moment of the distribution.

2.2.2. The Beta Function

The model of the beta distribution for a random variable x [5] is

$$f\left(x\right)=c·{\left(x-L\right)}^{\propto }·{\left(U-x\right)}^{\gamma }$$

(2)

$$\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}c=\frac{1}{\frac{{\left(-L+U\right)}^{1+\gamma }·\Gamma \left(1+\alpha \right)·\Gamma \left(1+\gamma \right)}{{\left(\frac{1}{-L+U}\right)}^{\alpha }·\Gamma \left(2+\alpha +\gamma \right)}}$$

(3)

for the interval L ≤ x ≤ U and f(x) = 0 elsewhere, where x is the diameter at breast height and is assumed to be continuous; f(x) is the density associated with diameter x; L is the lower limit or the diameter with the minimum value of the beta distribution; U is the upper limit or the diameter with the maximum value; c is the scaling factor of the function; α and γ are, respectively, the first and the second exponents that determine the shape of the distribution; and $\Gamma (•)$ is the gamma function.

The method of Moments for the beta function [6,12,22] is computed by the following expressions:

$$\gamma =\frac{\frac{Z}{s_{rel}^2·{(Z+1)}^2}-1}{Z+1}-1$$

(4)

$$\alpha =Z·\left(\gamma +1\right)-1$$

(5)

where

$$Z=\frac{x_{rel}}{1-x_{rel}}$$

(6)

$$x_{rel}=\frac{\overline{d}-L}{U-L}$$

(7)

$$s_{rel}^2=\frac{s^2}{{(U-L)}^2}$$

(8)

and where $\overline{d}$ is the arithmetic mean diameter of the sample distribution and $s^2$ is the variance.

2.2.3. The Gamma Distribution

The equation of the gamma distribution for a continuous random variable x [6,8,23] is

$$f\left(x\right)=\frac{{(x-\gamma )}^{\alpha -1}}{{\beta }^{\alpha }\Gamma \left(\alpha \right)}{·e}^{\left[-\left(\frac{x-\gamma }{\beta }\right)\right]}$$

(9)

where x > $\gamma$ and f(x) = 0 elsewhere, $\alpha$ > 0, and $\beta$ > 0; $\alpha$ is the shape parameter, $\beta$ is an inverse scale parameter, $\gamma$ is the location parameter ($\gamma =0$ for the Gamma-2P distribution), and $\Gamma (•)$ is the gamma function.

The method of Moments for the two-parameter gamma function is computed by the following expressions [24]:

$$\alpha ={\left(\frac{\overline{d}}{s}\right)}^2$$

(10)

$$\beta =\frac{s^2}{\overline{d}}$$

(11)

where $\overline{d}$ is the arithmetic mean diameter and s² is the variance (first and second moments of the distribution, respectively).

2.2.4. The Generalized Beta Distribution (GBD)

The general expression of the generalized beta distribution for a random variable x [7] is

$$f\left(x\right)=\complement ·{\beta }_2^{-\left({\beta }_3+{\beta }_4+1\right)}·{\left(x-{\beta }_1\right)}^{{\beta }_3}·{\left({\beta }_1+{\beta }_2-x\right)}^{{\beta }_4}$$

(12)

$$with:\complement =\frac{\Gamma \left({\beta }_3+{\beta }_4+2\right)}{\Gamma \left({\beta }_3+1\right)·\Gamma \left({\beta }_4+1\right)}$$

(13)

for the interval $({\beta }_1,{\beta }_1+{\beta }_2)$ and 0 otherwise; where x is the diameter at breast height (DBH) and is assumed to be continuous, f(x) is the density associated with diameter x, ${\beta }_1$ is the lower limit of the distribution, ${\beta }_2$ is the upper limit, ${\beta }_3$ and ${\beta }_4$ are exponents that determine the shape of the distribution, $\complement$ is the scaling factor, and $\mathsf{\Gamma }(•)$ is the gamma function.

The method of Moments for the GBD distribution is computed by the following equations [6,7,24]:

$${\beta }_1=x_{min}{\beta }_3=\frac{{\left({\alpha }_1-{\beta }_1\right)}^2\left({\beta }_1+{\beta }_2-{\alpha }_1\right)-{\alpha }_2({\beta }_2-{\beta }_1+{\alpha }_1)}{{\alpha }_2{\beta }_2}$$

(14)

$${\beta }_2=x_{max}-{\beta }_1{\beta }_4=\frac{{\beta }_2({\beta }_3+1)}{{\alpha }_1-{\beta }_1}-{\beta }_3-2$$

(15)

where x is the tree diameter, ${\alpha }_1$ is the mean diameter, and ${\alpha }_2$ is the variance.

2.2.5. The Gumbel Function

The Gumbel PDF [25] is formulated for a random variable, x, as follows:

$$f\left(x\right)=\frac{1}{\beta }exp\left[-\left(\left(\frac{x-\mu }{\beta }\right)+exp\left(-\frac{x-\mu }{\beta }\right)\right)\right]\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}-\mathrm{\infty }<x<\mathrm{\infty }$$

(16)

where μ is the location parameter, β is the scale parameter estimated with Equation (17), and the standard deviation (σ) is calculated as

$$\sigma =\beta \pi /\sqrt{6}$$

(17)

The function was fitted using the location parameter $\left(\widehat{\mu }\right)$ recovered from the experimental mean $\left(\overline{d}\right)$ and standard deviation (σ) of the distributions. Thus, the method of Moments using the first and the second moments of the distributions (mean $\overline{d}$ and variance ${\sigma }^2$) was applied [3,26] by using the following expression (Equation (18)):

$$\overline{d}=\widehat{\mu }+\beta \gamma$$

(18)

where β is the scale parameter obtained from σ by solving Equation (17); $\gamma$ is the Euler–Mascheroni constant, which depends on the sample size; and Equation (18) is solved by $\widehat{\mu }$.

2.2.6. The Johnson’s SB Function

The expression of Johnson’s SB PDF for a continuous random variable x [27] is

$$f\left(x\right)=\frac{\delta }{\sqrt{2\pi }}·\frac{\lambda }{(\epsilon +\lambda -x)(x-\epsilon )}·e^{-\frac{1}{2}{\left[\gamma +\delta ·ln\left(\frac{x-\epsilon }{\epsilon +\lambda -x}\right)\right]}^2}$$

(19)

where f(x) is the probability density function associated with diameter x; $\epsilon$ < x < $\epsilon$ + $\lambda$, with f(x) = 0 elsewhere; −$\infty$ < $\epsilon$ <$\infty$; −$\infty$ <$\gamma$ < $\infty$; $\lambda$ > 0; and $\delta$ > 0.

It is characterized by the location parameter $\epsilon$, the scale parameter $\lambda$, and the shape parameters $\gamma$ and $\delta$ (asymmetry and kurtosis parameters, respectively). The method of Moments used by Cosenza et al. [28] and Gorgoso-Varela et al. [24] is computed as follows:

$$\delta \cong \frac{\mu \left(1-\mu \right)}{Sd\left(x\right)}+\frac{Sd\left(x\right)}{4}\left[\frac{1}{\mu \left(1-\mu \right)}-8\right]$$

(20)

$$\gamma \cong \delta ln\left(\frac{1-\mu }{\mu }\right)+\left(\frac{0.5-\mu }{\delta }\right)$$

(21)

$$\mu =\frac{\overline{d}-\epsilon }{\lambda }$$

(22)

$$Sd\left(x\right)=\frac{{\sigma }_x}{\lambda }$$

(23)

where $\overline{d}$ is the arithmetic mean diameter, $Sd\left(x\right)$ is the modified standard deviation, and ${\sigma }_x$ is each plot diameter’s standard deviation, with the notation used for calculating the $Sd\left(x\right)$.

2.2.7. The Weibull Function (2P and 3P)

The Weibull-3P PDF has the following expression for a continuous random variable x [29]:

$$f\left(x\right)=\left(\frac{c}{b}\right)·{\left(\frac{x-a}{b}\right)}^{c-1}·e^{{-\left(\frac{x-a}{b}\right)}^c}$$

(24)

where f(x) is the probability density of trees with a diameter equal to x, a is the location parameter, b is the scale parameter, and c is the shape parameter. If the location parameter a is equal to zero, the expression corresponds to the Weibull-2P function.

The method of Moments for the Weibull distribution is computed using the following expressions [6,13,14]:

$$b=\frac{\overline{d}-a}{\mathsf{\Gamma }\left(1+\frac{1}{c}\right)}$$

(25)

$${\sigma }^2=\frac{{(\overline{d}-a)}^2}{{\mathsf{\Gamma }}^2\left(1+\frac{1}{c}\right)}\left[\mathsf{\Gamma }\left(1+\frac{2}{c}\right)-{\mathsf{\Gamma }}^2\left(1+\frac{1}{c}\right)\right]$$

(26)

where $\overline{d}$ is the mean diameter of the distribution, ${\sigma }^2$ is the variance, and $\Gamma (•)$ is the gamma function. Equation (26) was resolved by a bisection iterative procedure [30] to obtain the two variables.

2.2.8. Parameters’ Specifications

The size of the DBH classes was considered equal to 1 cm. Location parameters for the beta, generalized beta, Johnson’s SB, and Weibull-3P were considered 0.75·d_min because Gorgoso-Varela et al. [6] achieved good results with this constraint and Parresol [31] also proposed a close value (0.80·d_min). For the Weibull-2P and Gamma-2P functions, location parameters were zero, while the Charlier (type A) and Gumbel functions did not need to set any value. Scale parameters and the upper limit of the four-parameter models (beta, generalized beta, and Johnson’s SB) were considered as the maximum diameter of the distributions (d_max) [6,22].

2.2.9. Numerical Optimization or MLE Method: ‘optim’ Function

All eight functions fitted by the derivative method of Moments in SAS/STAT^TM [32] were also compared when fitting them by the Maximum Likelihood Estimation (MLE) numerical optimization method fitted in R software version 4.3.1 through the ‘optim’ function [33]. This function uses the Nelder–Mead optimization algorithm [34]. One advantage of the ‘optim’ method is that it allows for the specification of other types of algorithms (e.g., conjugate gradients, simulated annealing, etc.) [6]. The Charlier (type A) and the beta functions did not obtain results in the adjustment with the ‘optim’ function. We cannot ascertain what could have been the reason for the lack of convergence for some of the distributions. However, we do know that the tendency for such a problem increases with an increase in the number of parameters to be estimated and the complexity of the distribution, which is why this was observed in beta and the A Charlier. The data size of the sample plot could also compound this problem for the MLE method. The parameters’ specifications were the same as those in the method of Moments to facilitate comparisons.

2.3. Goodness of Fit

The consistencies of the functions and the fitting methods were evaluated using the bias, the mean absolute error (MAE), the mean squared error (MSE), the Kolmogorov–Smirnov statistic (D_n), the Cramér-von Mises statistic (W²), and the negative log-likelihood criterion (-ΛΛ) for the fits with the ‘optim’ function.

The bias, the mean absolute error (MAE), and the mean squared error (MSE) were expressed as follows:

$$\mathrm{B}\mathrm{i}\mathrm{a}\mathrm{s}=\frac{{\sum }_{i=1}^N{Y}_i-{\widehat{Y}}_i}{N}$$

(27)

$$\mathrm{M}\mathrm{A}\mathrm{E}=\frac{{\sum }_{i=1}^N\left|Y_i-{\widehat{Y}}_i\right|}{N}$$

(28)

$$\mathrm{M}\mathrm{S}\mathrm{E}=\frac{\sum _{i=1}^N{\left(Y_i-{\widehat{Y}}_i\right)}^2}{N}$$

(29)

where Y_i and $\widehat{Y_i}$ represent the relative frequency of trees observed in diameter class i and the theoretical value predicted by the model, respectively, and N is the number of diameter classes. The bias, mean absolute error (MAE), and mean squared error (MSE) values were computed for each fit, representing the mean relative frequency of trees across total diameter classes and plots.

The Kolmogorov–Smirnov statistic (D_n) was calculated for a given cumulative distribution function (CDF) F(x): $D_n={sup}_x\left|F_n\left(x\right)-F_0\left(x\right)\right|$, where sup x is the supremum of the set of distances. This value was calculated as follows [16]:

$$D_n=max\left\{{max}_{1\le i\le n_i}\left[F_n\left(x_i\right)-F_0\left(x_j\right)\right],{max}_{1\le i\le n_i}\left[F_0\left(x_j\right)-F_n\left(x_{i-1}\right)\right]\right\}$$

(30)

where the cumulative observed frequency F_n(x_i) is compared with the cumulative estimated frequency F₀(x_j).

The Cramér-von Mises statistic (W²) is a measure of the square of the distance between the empirical distribution and the cumulative theoretical distribution [35]:

$$W^2=\sum _{i=1}^n{\left\{\widehat{F}\left(x_i\right)-\frac{\left(i-0.5\right)}{n}\right\}}^2+\frac{1}{12n}$$

(31)

where $\widehat{F}$($x_i)$ is the cumulative theoretical distribution in the diameter class i and n is the number of diameter classes.

For the fits of Maximum Likelihood Estimation using the ‘optim’ function, the negative log-likelihood criterion (-ΛΛ) was also used to compare the results. It is a deviance statistic [36].

Models with low mean values of bias, MAE, MSE, D_n, W², and -ΛΛ were regarded as good models. Based on these indices, the PDFs were ranked [37]. The best model received a score of one, while the worst model received the highest value for each fit index. The scores were then summed for each model, serving as an indicator of the individual function’s performance.

3. Results

The summary statistics of the parameters of A Charlier, beta, generalized beta, gamma, Gumbel, Johnson’s SB, and Weibull (2P and 3P) obtained by derivative and optimization methods for Q. robur and T. grandis stands are presented in

Table 2. Descriptive statistics of the parameters estimated by derivative (Moments) and numerical optimization (MLE) methods for Quercus robur stands.

Function	Fit	Param	Mean	St Dev	Min	Max
A Charlier	Derivative (Moments)	μ	21.0193	5.7681	8.3323	37.7631
		σ2	52.7453	32.9372	4.0278	196.5813
		asym	0.3697	0.5164	−0.9847	2.0729
		Kurt	0.0548	1.017	−1.1844	6.1667
Beta	Derivative (Moments)	c	0.0013	0.0030	6.00·Exp−11	0.0218
		L	6.3530	2.5962	3.7500	19.0125
		U	37.9093	9.8654	16.3000	71.6000
		α	0.9685	0.6740	−0.1582	3.2457
		γ	1.3249	0.7836	0.0083	4.4221
Gamma-2P	Derivative (Moments)	α	10.7038	5.5632	2.9692	29.9842
		β	2.3936	1.2093	0.4782	8.0624
	Optimization (MLE)	α	10.1891	5.3742	2.9513	30.8548
		β	2.5451	1.3837	0.4421	10.0525
Generalized beta	Derivative (Moments)	$\complement$	170.4135	560.1196	1.9032	4290.0584
		B1	6.3530	2.5962	3.7500	19.0125
		B2	37.9093	9.8654	16.3000	71.6000
		B3	1.4699	0.9587	0.0084	4.5858
		B4	2.9449	1.4354	0.7609	7.9256
	Optimization (MLE)	$\complement$	209.6410	784.4000	3.0108	7154.8607
		B1	6.3530	2.5962	3.7500	19.0125
		B2	37.9093	9.8654	16.3000	71.6000
		B3	1.5057	0.91418	0.1710	4.0093
		B4	3.0330	1.5694	0.8589	8.4234
Gumbel	Derivative (Moments)	μ	17.8914	5.2504	7.4290	32.9807
		β	5.4191	1.6476	1.5648	10.9319
	Optimization (MLE)	μ	17.7025	5.1462	7.4249	32.8833
		β	6.0555	1.9661	1.5796	12.5816
Johnson’s SB	Derivative (Moments)	ε	6.3530	2.5962	3.7500	19.0125
		λ	37.9093	9.8654	16.3000	71.6000
		γ	0.6328	0.3998	−0.5518	1.6197
		δ	1.1268	0.21093	0.7208	1.6982
	Optimization (MLE)	ε	6.3530	2.5962	3.7500	19.0125
		λ	37.9093	9.8654	16.3000	71.6000
		γ	0.6296	0.3844	−0.4715	1.6283
		δ	1.1226	0.2021	0.7435	1.7484
Weibull-2P	Derivative (Moments)	b	23.3373	6.2908	9.1039	42.2053
		c	3.4887	0.9731	1.7634	6.3410
	Optimization (MLE)	b	23.3570	6.2746	9.1213	42.1990
		c	3.4653	0.9180	1.8427	6.0898
Weibull-3P	Derivative (Moments)	a	6.3530	2.5962	3.7500	19.0125
		b	16.4731	5.1672	5.0848	34.7842
		c	2.2804	0.5399	1.3474	4.3502
	Optimization (MLE)	a	6.3530	2.5962	3.7500	19.01225
		b	16.4569	5.1075	5.0926	34.6642
		c	2.3116	0.54702	1.3650	4.5155

Param: parameter; St Dev: standard deviation; Min: minimum; Max: maximum.

and

Table 3. Summary of descriptive statistics of the parameters estimated by derivative (Moments) and numerical optimization (MLE) methods for Tectona grandis stands.

Function	Fit	Param	Mean	St Dev	Min	Max
A Charlier	Derivative (Moments)	μ	18.6115	2.3531	13.5467	22.8062
		σ2	27.5001	11.2797	12.3188	65.1082
		asym	0.57652	0.4351	−0.4617	1.4558
		Kurt	0.2409	1.1994	−1.1160	5.1092
Beta	Derivative (Moments)	c	0.0016	0.0039	4.95·Exp−11	0.0227
		L	7.5775	1.6863	4.3500	11.1750
		U	31.7254	4.6026	21.3000	39.2000
		α	1.1593	0.7226	0.1094	3.3500
		γ	1.6320	0.9387	0.3156	5.1150
Gamma-2P	Derivative (Moments)	α	14.7649	5.6418	4.3942	28.6115
		β	1.4635	0.6380	0.6929	3.7585
	Optimization (MLE)	α	15.0228	5.6265	4.3999	26.9948
		β	1.4323	0.6207	0.7082	3.6163
Generalized beta	Derivative (Moments)	$\complement$	432.0867	1194.5728	1.9972	7589.0738
		B1	7.5775	1.6863	4.3500	11.1750
		B2	31.7254	4.6026	21.3000	39.2000
		B3	1.8115	0.9587	0.0313	4.4257
		B4	4.2003	1.7369	1.0530	10.3882
	Optimization (MLE)	$\complement$	562.4069	1590.6532	3.5278	11,607.4026
		B1	7.5775	1.6863	4.3500	11.1750
		B2	31.7254	4.6026	21.3000	39.2000
		B3	1.9993	0.8892	0.2448	4.1809
		B4	4.5639	1.8122	1.4330	10.8203
Gumbel	Derivative (Moments)	μ	16.2938	2.3630	11.4856	20.1282
		β	4.0155	0.7772	2.7366	6.2913
	Optimization (MLE)	μ	16.2134	2.3629	11.3835	20.1133
		β	4.2340	0.7465	2.7812	6.5603
Johnson’s SB	Derivative (Moments)	ε	7.5775	1.6863	4.3500	11.1750
		λ	31.7254	4.6026	21.3000	39.2000
		γ	0.9011	0.3319	0.1835	1.8047
		δ	1.2655	0.2212	0.7413	1.7499
	Optimization (MLE)	ε	7.5775	1.6863	4.3500	11.1750
		λ	31.7254	4.6026	21.3000	39.2000
		γ	0.9090	0.3415	0.2055	1.7963
		δ	1.2961	0.2090	0.8062	1.7604
Weibull-2P	Derivative (Moments)	b	20.4748	2.4730	15.1130	25.0459
		c	4.2068	0.9295	2.1778	6.1998
	Optimization (MLE)	b	20.5079	2.4735	15.1411	25.0836
		c	3.9932	0.8209	2.2678	6.4435
Weibull-3P	Derivative (Moments)	a	7.5775	1.6863	4.3500	11.1750
		b	12.4075	1.7067	8.6744	16.6523
		c	2.3337	0.4648	1.3990	3.6479
	Optimization (MLE)	a	7.5775	1.6863	4.3500	11.1750
		b	12.4075	1.7067	8.6744	16.6523
		c	2.3337	0.4648	1.3990	3.6479

Param: parameter; St Dev: standard deviation; Min: minimum; Max: maximum.

, respectively. For distributions fitted by derivative and optimization, the values for the estimated shape parameters varied, particularly the scaling factor of the generalized beta distribution. The PDFs with location parameters (Weibull-3P, beta, generalized beta, and Johnson’s SB) had the same estimated average values of 6.353 in Q. robur and 7.5775 in T. grandis, due to similar constraints on the parameter. Similarly, the average values of the upper limit parameters for PDFs with four parameters were 37.9093 and 31.7254 in Q. robur and T. grandis, respectively. No constraint was imposed on the A Charlier and Gumbel distributions. The mean values of the estimated A Charlier parameters (µ, ${\sigma }^2$, asym, and Kurt) were 21.0193, 52.7453, 0.3697, and 0.0548, respectively, for Q. robur and 18.615, 27.5001, 0.5765, and 0.2409, respectively, for T. grandis.

Based on the evaluation statistics, the Johnson SB distribution had the best fit with a rank sum of 13 (

Table 4. Rank position for each function and statistic for Q. robur and T. grandis stands. Bold text indicates distributions with the lowest rank sum.

Species	Fit	Function	Dn	W2	Bias	MAE	MSE	-ΛΛ	ΣRank
Q. robur	Moments (Derivative)	A Charlier	2	6	7	1	1	-	17
		beta	4	8	2	5	5	-	24
		Gamma-2P	8	7	6	6	6	-	33
		generalized beta	5	3	3	2	3	-	16
		Gumbel	7	5	5	8	8	-	33
		Johnson’s SB	6	1	1	3	2	-	13
		Weibull-2P	3	4	8	7	7	-	29
		Weibull-3P	1	2	4	4	4	-	15
	MLE (Optimization)	A Charlier	-	-	-	-	-	-	-
		beta	-	-	-	-	-	-	-
		Gamma-2P	6	4	4	4	4	3	25
		generalized beta	4	3	2	3	3	6	21
		Gumbel	3	6	6	6	6	1	28
		Johnson’s SB	5	2	1	2	1	5	16
		Weibull-2P	2	5	5	5	5	2	24
		Weibull-3P	1	1	3	1	1	4	11
T. grandis	Moments (Derivative)	A Charlier	3	6	7	3	4	-	23
		beta	5	8	2	6	6	-	27
		Gamma-2P	7	4	6	5	5	-	27
		generalized beta	4	5	3	2	2	-	16
		Gumbel	2	2	4	7	7	-	22
		Johnson’s SB	6	1	1	1	1	-	10
		Weibull-2P	8	7	8	8	8	-	39
		Weibull-3P	1	3	5	4	2	-	15
	MLE (Optimization)	A Charlier	-	-	-	-	-	-	-
		beta	-	-	-	-	-	-	-
		Gamma-2P	6	5	4	5	4	3	27
		generalized beta	3	3	2	3	2	5	18
		Gumbel	2	2	5	4	5	2	20
		Johnson’s SB	4	1	1	1	1	6	14
		Weibull-2P	5	6	6	6	6	1	30
		Weibull-3P	1	4	3	2	2	4	16

) when the derivative method (Moments) was used to fit all the distributions to Q. robur stands. The Weibull-3P, generalized beta, and A Charlier followed this. The Gumbel and Gamma-2P distributions performed poorly (rank sum = 33) in describing the distribution of Q. robur. With the optimization method (MLE), the Weibull-3P distribution function had the lowest rank sum of 11 and, as such, ranked best. Johnson’s SB ranked second, while the Gumbel and Gamma-2P distributions performed poorly with a rank sum of 28 and 25, respectively.

In the case of T. grandis, Johnson’s SB had the best fit with both the derivative and the optimization methods with rank sums of 10 and 14, respectively. Weibull-3P and generalized beta performed relatively well compared to the others. Gumbel distribution was more suitable than beta, gamma, and Weibull-2P. Weibull-2P provided poor fits to the T. grandis stands, irrespective of the fitting method. Among all the evaluation statistics, the log-likelihood value (-ΛΛ) was more sensitive to the number of parameters on the distributions, being better in functions with fewer parameters. For detailed information on the fit statistics of the PDFs, see Appendix A

Table A1. Fits and statistics used in the comparison between Moments and Maximum Likelihood Estimation (MLE) methods for Quercus robur stands.

Function	Fit	Dn	W2	Bias	MAE	MSE	-ΛΛ
A Charlier	Moments	0.1088	0.0686	0.001370	0.018684	0.000610
	MLE	No fit	No fit	No fit	No fit	No fit	No fit
Beta	Moments	0.1125	0.0705	0.000670	0.019086	0.000632
	MLE	No fit	No fit	No fit	No fit	No fit	No fit
Gamma-2P	Moments	0.1284	0.0692	0.001240	0.019169	0.000648
	MLE	0.1275	0.0663	0.001353	0.019192	0.000649	−189.76
Generalized beta	Moments	0.1154	0.0578	0.000897	0.018852	0.000618
	MLE	0.1187	0.0584	0.000913	0.019092	0.000633	−175.34
Gumbel	Moments	0.1263	0.0678	0.001229	0.019739	0.000694
	MLE	0.1185	0.0667	0.001746	0.019390	0.000664	−190.59
Johnson’s SB	Moments	0.1228	0.0435	0.000641	0.018863	0.000617
	MLE	0.1239	0.0523	0.000568	0.019088	0.000627	−187.85
Weibull-2P	Moments	0.1121	0.0650	0.001566	0.019324	0.000659
	MLE	0.1119	0.0661	0.001555	0.019330	0.000660	−190.50
Weibull-3P	Moments	0.1043	0.0489	0.001012	0.018901	0.000623
	MLE	0.1073	0.0513	0.000923	0.018957	0.000627	−188.51

and

Table A2. Fits and statistics used in the comparison between Moments and Maximum Likelihood Estimation (MLE) methods for Tectona grandis stands.

Function	Fit	Dn	W2	Bias	MAE	MSE	-ΛΛ
A Charlier	Moments	0.1273	0.0642	0.001879	0.022755	0.000902
	MLE	No fit	No fit	No fit	No fit	No fit	No fit
Beta	Moments	0.1303	0.0809	0.001456	0.023183	0.000927
	MLE	No fit	No fit	No fit	No fit	No fit	No fit
Gamma-2P	Moments	0.1324	0.0552	0.001797	0.023169	0.000925
	MLE	0.1324	0.0552	0.001706	0.023121	0.000925	−142.093
Generalized beta	Moments	0.1285	0.0553	0.001477	0.022736	0.000889
	MLE	0.1270	0.0490	0.001119	0.022851	0.000899	−141.270
Gumbel	Moments	0.1239	0.0477	0.001537	0.023632	0.000963
	MLE	0.1239	0.0477	0.001885	0.023033	0.000926	−142.248
Johnson’s SB	Moments	0.1319	0.0364	0.001100	0.022523	0.000874
	MLE	0.1294	0.0366	0.000878	0.022609	0.000881	−140.772
Weibull-2P	Moments	0.1333	0.0804	0.002460	0.024083	0.001011
	MLE	0.1312	0.0799	0.002793	0.024128	0.001012	−144.362
Weibull-3P	Moments	0.1138	0.0487	0.001575	0.022839	0.000899
	MLE	0.1170	0.0525	0.001575	0.022839	0.000899	−141.563

.

Figure 1 shows a graphical evaluation of the observed and predicted distributions for three even-aged plots each of Q. robur and T. grandis stands. The plots were selected based on their high density among the sample plots. The predicted diameter distributions were relatively similar to the observed distributions in both species.

4. Discussion

An empirical comparison of eight probability density functions (PDFs) fitted by the derivative and optimization methods for diameter distribution estimation in tropical and Atlantic forests has been evaluated. Among the eight PDFs evaluated for the even-aged Q. robur and T. grandis stands, Johnson’s SB was more consistent in describing the diameter distributions compared to the others. Using either the derivative (Moments) or optimization (MLE) methods, Johnson’s SB still provided superior fits than most of the PDFs except Weibull-3P, where better fits were observed with optimization in Q. robur. Scolforo et al. [38] also found the methods of Moments and MLE as the preferred fitting methods for Johnson’s SB in describing the diameter distribution in Loblolly pine. Johnson’s SB and Weibull have been used consistently in quantitative forestry studies because of their flexibility to describe different shapes (e.g., [6,12,20,39,40,41]). In addition to the derivative method by Moments, the Weibull and Johnson’s SB functions can be fitted by other derivative methods, for example, Weibull by percentiles [42], Johnson’s SB by mode [43], conditional maximum likelihood (CML) [44], linear regression [39], percentiles [45], etc. However, in this study, we used the method of Moments because it is the only one that can be used as a derivative method in the eight compared functions.

Gorgoso-Varela et al. [6] made a comparison vis à vis each function between several derivative methods and optimization methods. Their results varied depending on the PDF and species. For example, in their study, the generalized beta fitted by derivative and optimization methods performed better than Johnson’s SB in Gmelina (Gmelina arborea Roxb.) but less so in Tasmanian blue gum (Eucalyptus globulus Labill) and Monterrey pine (Pinus radiata D. Don) stands. Gorgoso-Varela et al. [24] modeled the diameter distributions with six PDFs—beta, generalized beta, Gamma-2P, Johnson’s SB, and Weibull (two- and three-parameter)—in Pinus halepensis Mill. plantations using low-density Airborne Laser Scanning data in Aragón (northeast Spain). The results showed that the beta distribution was the most accurate, followed by the generalized beta distribution. Similarly, the generalized beta function fitted with the derivative (Moments) was more suitable than Johnson’s SB in describing the diameter distribution of Douglas fir stands [7]. In our study, using Moments and MLE for the generalized beta yielded good fits but were less superior than the Johnson’s SB and Weibull-3P functions. Few studies have applied the generalized beta function to forestry; rather, the beta function by Loetsch et al. [5] has been used consistently, particularly in Mediterranean, Atlantic, and Boreal forests [6,12,22]. We found the generalized beta to outperform the beta function in predicting the diameter distribution of Q. robur and T. grandis stands. Compared to the generalized beta, only the derivative method by Moments could be used to fit the beta function—making it more conservative than the former. Nevertheless, the beta function was more suitable than generalized beta for describing the diameter distribution of E. globulus and P. radiata stands in the same region [6].

A Charlier provided relatively better fits than the beta, Gamma-2P, Weibull-2P, and Gumbel functions. This was expected because the A Charlier function is more suitable for even-aged stands, and it is the only function capable of representing bimodal distributions among those evaluated. One limitation of the A Charlier function is that it cannot represent uneven-aged stands that are characterized by a reversed J-shape [15]. In addition, the complex expression of the function may limit its application in practical forestry.

The performance of the two-parameter PDFs (Gamma-2P, Gumbel, and Weibull-2P) was poor and inconsistent in the two species. For example, the Weibull-2P function had the best fits in Q. robur stands and the worst fits in T. grandis stands among the two-parameter functions. On the other hand, the Gumbel function, which had a similar performance to Gamma-2P in Q. robur, was superior in T. grandis stands. Interestingly, the two-parameter functions had low log-likelihood values (deviance statistic) compared to the higher-level parameter PDFs (Table 3). Since this statistic is sensitive to the number of parameters, we recommend that it should not be used as the only index for comparing distributions; it should be used for comparing distributions with the same number of parameters.

Generally, the Weibull distribution is very flexible; however, imposing a constraint on the location parameter (a = 0) may reduce accuracy and thereby increase its inconsistency in some stands. The Gamma-2P function is very inflexible, and as such, may describe negative skewness and model symmetry diameter distribution poorly [45]. It performed poorly in predicting the diameter distribution in a Nauclea plantation (Nauclea dedirrichii [De Wild.] Merr.) [9] in Nigeria. The Gumbel function is sensitive to extreme values, thus making it suitable for modeling extreme value distributions. Different studies have used the Gumbel function to model the extreme value diameter and/or height distribution with varying degrees of success (e.g., [3,26]).

We did not perform a response calibration for distribution functions. To our knowledge, response calibration is applicable to mixed-effect models. However, if there is a new measured plot/stand, the parameters of the distributions (e.g., Weibull, Johnson’s, etc.) could be estimated either directly from the diameter values (MLE) or from the stand attributes, such as mean, variance, etc. (the derivative method). In most forest inventory, diameter is measured on all trees, so estimating the parameters of the distribution with the specific equations, especially using the derivative approach in the manuscript, would be a straightforward process.

Implications for Forestry

This study has a significant application in quantifying volume production based on diameter classes for both even-aged and uneven-aged forest stands. The primary focus was on even-aged stands, commonly found in planted forests, characterized by univariate diameter distributions. Utilizing Johnson’s SB, Weibull (3P), beta, and generalized beta distributions, fitted derivative methods, or optimization methods as applicable, effectively characterized the diameter distribution of even-aged stands, providing reliable estimates of yield by diameter classes. However, it is important to note that management practices and silvicultural interventions, such as selective harvest, may transform a unimodal even-aged stand into a bi-modal structure. In such cases, the A Charlier function emerged as the preferred probability density function for describing the forest stand’s structure, outperforming other evaluated distributions. This study identified two plots displaying a bi-modal structure, and in both instances, the A Charlier function exhibited superior performance compared to alternative distributions. Additionally, finite mixture models could be employed for stands with bi-modal structures, although they are often more challenging to implement than A Charlier fitted with Moments.

5. Conclusions

In this study, the suitability of eight probability density functions fitted using derivative and optimization methods for diameter distribution estimation in Q. robur and T. grandis stands was compared. Johnson’s SB function proved to be more suitable for describing the diameter distribution of the stands. The three-parameter Weibull and generalized beta functions exhibited relatively good performance. Estimating the frequency of trees in diameter classes with numerical methods for Johnson’s SB and generalized beta was straightforward. For those prioritizing the simplicity of the function expression, the three-parameter Weibull could be recommended for stands. The performance of the functions remained consistent regardless of the fitting methods, whether derivative or optimization. On the other hand, the fits of the two-parameter functions were generally poor and inconsistent in the Q. robur and T. grandis stands.