
Many structures in science, engineering, and art can be viewed as curves in 3-space. The entanglement of these curves plays a crucial role in determining the functionality and physical properties of materials. Many concepts in knot theory provide theoretical tools to explore the complexity and entanglement of curves in 3-space. However, classical knot theory focuses on global topological properties and lacks the consideration of local structural information, which is critical in practical applications. In this work, two localized models based on the Jones polynomial were proposed, namely, the multi-scale Jones polynomial and the persistent Jones polynomial. The stability of these models, especially the insensitivity of the multi-scale and persistent Jones polynomial models to small perturbations in curve collections, was analyzed, thus ensuring their robustness for real-world applications.
Citation: Ruzhi Song, Fengling Li, Jie Wu, Fengchun Lei, Guo-Wei Wei. Multi-scale Jones polynomial and persistent Jones polynomial for knot data analysis[J]. AIMS Mathematics, 2025, 10(1): 1463-1487. doi: 10.3934/math.2025068
[1] | Wael W. Mohammed, Kalpasree Sharma, Partha Jyoti Hazarika, G. G. Hamedani, Mohamed S. Eliwa, Mahmoud El-Morshedy . Zero-inflated discrete Lindley distribution: Statistical and reliability properties, estimation techniques, and goodness-of-fit analysis. AIMS Mathematics, 2025, 10(5): 11382-11410. doi: 10.3934/math.2025518 |
[2] | Alanazi Talal Abdulrahman, Khudhayr A. Rashedi, Tariq S. Alshammari, Eslam Hussam, Amirah Saeed Alharthi, Ramlah H Albayyat . A new extension of the Rayleigh distribution: Methodology, classical, and Bayes estimation, with application to industrial data. AIMS Mathematics, 2025, 10(2): 3710-3733. doi: 10.3934/math.2025172 |
[3] | Khaled M. Alqahtani, Mahmoud El-Morshedy, Hend S. Shahen, Mohamed S. Eliwa . A discrete extension of the Burr-Hatke distribution: Generalized hypergeometric functions, different inference techniques, simulation ranking with modeling and analysis of sustainable count data. AIMS Mathematics, 2024, 9(4): 9394-9418. doi: 10.3934/math.2024458 |
[4] | Ehab M. Almetwally, Ahlam H. Tolba, Dina A. Ramadan . Bayesian and non-Bayesian estimations for a flexible reduced logarithmic-inverse Lomax distribution under progressive hybrid type-Ⅰ censored data with a head and neck cancer application. AIMS Mathematics, 2025, 10(4): 9171-9201. doi: 10.3934/math.2025422 |
[5] | Mohamed S. Eliwa, Essam A. Ahmed . Reliability analysis of constant partially accelerated life tests under progressive first failure type-II censored data from Lomax model: EM and MCMC algorithms. AIMS Mathematics, 2023, 8(1): 29-60. doi: 10.3934/math.2023002 |
[6] | Mohamed S. Algolam, Mohamed S. Eliwa, Mohamed El-Dawoody, Mahmoud El-Morshedy . A discrete extension of the Xgamma random variable: mathematical framework, estimation methods, simulation ranking, and applications to radiation biology and industrial engineering data. AIMS Mathematics, 2025, 10(3): 6069-6101. doi: 10.3934/math.2025277 |
[7] | Juxia Xiao, Ping Yu, Zhongzhan Zhang . Weighted composite asymmetric Huber estimation for partial functional linear models. AIMS Mathematics, 2022, 7(5): 7657-7684. doi: 10.3934/math.2022430 |
[8] | Mohamed Ahmed Mosilhy . Discrete Erlang-2 distribution and its application to leukemia and COVID-19. AIMS Mathematics, 2023, 8(5): 10266-10282. doi: 10.3934/math.2023520 |
[9] | Nora Nader, Dina A. Ramadan, Hanan Haj Ahmad, M. A. El-Damcese, B. S. El-Desouky . Optimizing analgesic pain relief time analysis through Bayesian and non-Bayesian approaches to new right truncated Fréchet-inverted Weibull distribution. AIMS Mathematics, 2023, 8(12): 31217-31245. doi: 10.3934/math.20231598 |
[10] | Mustafa M. Hasaballah, Yusra A. Tashkandy, Oluwafemi Samson Balogun, M. E. Bakr . Reliability analysis for two populations Nadarajah-Haghighi distribution under Joint progressive type-II censoring. AIMS Mathematics, 2024, 9(4): 10333-10352. doi: 10.3934/math.2024505 |
Many structures in science, engineering, and art can be viewed as curves in 3-space. The entanglement of these curves plays a crucial role in determining the functionality and physical properties of materials. Many concepts in knot theory provide theoretical tools to explore the complexity and entanglement of curves in 3-space. However, classical knot theory focuses on global topological properties and lacks the consideration of local structural information, which is critical in practical applications. In this work, two localized models based on the Jones polynomial were proposed, namely, the multi-scale Jones polynomial and the persistent Jones polynomial. The stability of these models, especially the insensitivity of the multi-scale and persistent Jones polynomial models to small perturbations in curve collections, was analyzed, thus ensuring their robustness for real-world applications.
The count data sets emerge in various fields like the yearly number of destructive earthquakes, number of patients of a specific disease in a hospital ward, failure of machines, number of patients due to coronavirus, number of monthly traffic accidents, hourly bacterial growth, and so on. Various discrete probability models have been utilized to model these kinds of data sets. Poisson and negative binomial distributions are frequently for modeling count observations. On the other hand, in the advanced scientific eon, the data generated from different fields is getting complex day by day, however, existing discrete models do not provide an efficient fit.
Discretization of continuous distribution can be applied by using different approaches (survival discretization-mixed-Poisson-infinite series). The most widely used technique is the survival discretization approach by [1]. One of the important virtues of this methodology is that the generated discrete model retains the same functional form of the survival function as that of its continuous counterpart. Due to this feature, many survival characteristics of the distribution remain unchanged. The discretization approach to any continuous model depends on the domain of the random variable
Although various distributions are available in literate to analyze count observations, there is still a need to introduce a more flexible and suitable distribution under different conditions. The fundamental purpose of this paper is to propose discrete Ramos-Louzada distribution, which is a one-parameter lifetime distribution introduced by [24]. The proposed one-parameter distribution herein has distinctive properties which makes it among the best choice for modeling over-dispersed and positively skewed data with leptokurtic-shaped. A continuous random variable
g(x;λ)=1λ2(λ−1)(λ2−2λ+x)e−xλ;x≥0,λ≥2, | (1) |
where λ is the shape parameter. The corresponding survival function (sf) to Eq (1) can be formulated as
G(x;λ)=λ2−λ+xλ(λ−1)e−xλ;x≥0,λ≥2. | (2) |
In this article, the discrete version of Ramos and Louzada distribution is proposed and studied in detail. The following are some interesting features of the proposed distribution: Its statistical and reliability characteristics can be expressed as closed forms. Its failure rate is showing an increasing pattern. The suggested distribution evaluated time and count data sets more effectively than competing distributions. As a result, we feel that the proposed model is the greatest option for attracting a wider range of applications and industries.
The rest of the study is organized as follows: In Section 2, we introduce a new distribution using survival discretization methodology. Different mathematical properties are derived in Section 3. Parameter estimation and simulation study are presented in Section 4. Four data sets are utilized to show the flexibility of the proposed model in Section 5. Finally, Section 6 provides some conclusions.
Let
Pr(X=x;η)=G(x;η)−G(x+1;η);x∈Z+, |
where
Pr(X=x;λ)=p(x;λ)=e−xλλ(λ−1)[(λ2−λ+x)(1−e−1λ)−e−1λ];x=0,1,2,…, | (3) |
where
p(x+1;λ)=e−1λ[(λ2−λ+x+1)(1−e−1λ)−e−1λ][(λ2−λ+x)(1−e−1λ)−e−1λ]p(x;λ). |
Figure 1 illustrates some pmf plots of the DRL models based on different values of the model parameter
Based on
F(x;λ)=1−λ2−λ+x+1λ(λ−1)e−x+1λ;x=0,1,2,…. | (4) |
The corresponding sf to Eq (4) can be expressed as
S(x;λ)=λ2−λ+xλ(λ−1)e−xλ;x=0,1,2,…. | (5) |
The hazard rate function (hrf) of the DRL model is given by
h(x;λ)=(λ2−λ+x)(1−e−1λ)−e−1λ(λ2−λ+x);x=0,1,2,…, | (6) |
where
The reversed hazard rate function (rhrf) and the second rate of failure are given as
˘r=e−xλ[(λ2−λ+x)(1−e−1λ)−e−1λ]λ(λ−1)−(λ2−λ+x+1)e−x+1λ;x=0,1,2,… | (7) |
and
r∗(x)=log[(λ2−λ+x)e1λλ2−λ+x+1];x=0,1,2,…, | (8) |
where
In this Section, the probability generating function (pgf) as well as its rth moment are investigated. Assume the random variable
WX(z)=∞∑x=0zxPr(X=x;λ)=1+(z−1)∞∑x=1zx−1S(x;λ)=1+e−1λ(z−1)[1(1−ze−1λ)+1λ(λ−1)(1−ze−1λ)2]. | (9) |
On replacing z by
MX(z)=1+e−1λ(ez−1)[1(1−eze−1λ)+1λ(λ−1)(1−eze−1λ)2]. | (10) |
The first four moments around the origin
μ'1=λ−λ2+(λ2−λ+1)e1λλ(λ−1)(e1λ−1)2, |
μ'2=(λ(λ−1)+1)e2λ+3e1λ−λ(λ−1)λ(λ−1)(e1λ−1)3, |
μ'3=[λ(λ−1)+1]e3λ+[3λ(λ−1)+10]e2λ−[3λ(λ−1)−7]e1λ−λ(λ−1)λ(λ−1)(e1λ−1)4 |
and
μ'4=[λ(λ−1)+1]e4λ+[10λ(λ−1)+25]e3λ+67e2λ+[10λ(λ−1)−3]e1λ−λ(λ−1)λ(λ−1)(e1λ−1)5. |
Based on the rth moments, the variance can be expressed as
σ2=[λ4−2λ3+2λ2−λ]e3λ−[2λ4−4λ3+2λ2−1]e2λ+[λ4−2λ3+λ]e1λλ2(λ−1)2(e1λ−1)4. | (11) |
The dispersion index (di) is defined by variance to mean ratio. The di indicates that the reported model is suitable for under-, equi- or over-dispersed data sets. Using the derived moments, the coefficients skewness and kurtosis can be listed in closed forms. Some numerical computations for mean, variance, di, skewness, and kurtosis based on DRL parameters are listed in Table 1.
|
|||||||||
Measure | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 | 4.5 | 5.0 | 5.5 | 8.0 |
Mean | 3.500 | 3.678 | 4.014 | 4.414 | 4.847 | 5.299 | 5.762 | 6.234 | 8.652 |
Variance | 8.079 | 11.80 | 15.71 | 20.03 | 24.82 | 30.10 | 35.87 | 42.13 | 80.90 |
di | 2.308 | 3.207 | 3.913 | 4.538 | 5.121 | 5.680 | 6.224 | 6.757 | 9.351 |
Skewness | 1.395 | 1.519 | 1.629 | 1.708 | 1.764 | 1.806 | 1.837 | 1.862 | 1.928 |
Kurtosis | 5.940 | 6.353 | 6.821 | 7.199 | 7.495 | 7.727 | 7.910 | 8.057 | 8.484 |
According to Table 1, it is noted that the DRL model can be used effectively to model overdispersion data as di is greater than one, which makes it a proper probability tool to discuss actuarial data. Moreover, the new discrete probabilistic model can be utilized to analyze positively skewed data with leptokurtic-shaped.
In this section, six estimation methods are used to estimate the unknown parameter of DRL distribution. The considered estimation methods are maximum likelihood estimation (mle), method of moments (mom), least-squares estimation (lse), Anderson-Darling estimation (ade), Cramer von-Misses estimation (cvme), and maximum product of spacing estimator (mpse).
Assume a random sample
L(λ|x)=−1λ∑ni=1xi+∑ni=1ln[(λ2−λ+xi)(1−e−1λ)−e−1λ]−nlnλ−nln(λ−1). | (12) |
Differentiating the Eq (12) with respect to the parameter
∂L(λ|x)∂λ=1λ2∑ni=1xi+∑ni=1[(2λ−1)(1−e−1λ)−(1−1λ+1λ2+xiλ2)e−1λ][(λ2−λ+xi)(1−e−1λ)−e−1λ]−nλ−nλ−1, | (13) |
the exact solution of Eq (13) is not easy, so we will maximize it by using optimization approaches, for example, the Newton-Raphson approach using R software.
Based on the mom definition, we must equate the sample mean to the corresponding population mean, and then solve the non-linear equation for the parameter
λ−λ2+(λ2−λ+1)e1λλ(λ−1)(e1λ−1)2=1n∑ni=1xi. | (14) |
To solve Eq (14), the uniroot function should be utilized.
To estimate the parameter minimizing the sum of squares of residuals, a standard approach like the lse should be used. For the estimation of the parameter of DRL distribution, the lse can be obtained by minimizing
lse(λ)=∑ni=1[1−(λ2−λ+xi:n)λ(λ−1)e−xi:n+1λ−in+2]2, | (15) |
with respect to the parameter
The ade of the parameter can be derived by minimizing the following equation
ade(λ)=−n−1n∑ni=1(2i−1)[log(1−(λ2−λ+xi:n)λ(λ−1)e−xi:n+1λ)+log((λ2−λ+xi:n)λ(λ−1)e−xi:n+1λ)]2, | (16) |
with respect to the parameter
The cvme is an estimation method. This method is derived as the difference between the empirical cdf and fitted cdf where
cvme(λ)=112n+∑ni=1[1−(λ2−λ+xi:n)λ(λ−1)e−xi:n+1λ−2i−12n]2. | (17) |
For
mpse(λ)=[∏h+1u=1Du(λ)]1h+1, | (18) |
with respect to the parameter
In this section, we discussed the results of the simulation study to compare the estimation performance of the proposed estimators based on the DRL model. The performance of considered estimators is evaluated via absolute biases and mean square errors. We simulate
Para. | n | Bias | mse | ||||||||||
|
mle | mom | ade | cvme | lse | mpse | mle | mom | ade | cvme | lse | mpse | |
2.0 | 10 | 0.099 | 0.072 | 0.878 | 0.606 | 0.956 | 0.801 | 0.762 | 1.496 | 2.897 | 2.059 | 3.034 | 1.797 |
20 | 0.034 | 0.035 | 0.486 | 0.259 | 0.444 | 0.435 | 0.307 | 1.103 | 1.408 | 0.749 | 1.212 | 0.626 | |
50 | 0.017 | 0.160 | 0.122 | 0.037 | 0.065 | 0.172 | 0.055 | 0.811 | 0.304 | 0.086 | 0.148 | 0.111 | |
100 | 0.013 | 0.226 | 0.015 | 0.003 | 0.005 | 0.086 | 0.017 | 0.697 | 0.033 | 0.006 | 0.009 | 0.026 | |
200 | 0.009 | 0.277 | 0.000 | 0.000 | 0.000 | 0.050 | 0.007 | 0.620 | 0.000 | 0.000 | 0.000 | 0.008 | |
500 | 0.002 | 0.338 | 0.000 | 0.000 | 0.000 | 0.028 | 0.003 | 0.569 | 0.000 | 0.000 | 0.000 | 0.002 | |
2.5 | 10 | 0.081 | 0.131 | 1.109 | 1.140 | 1.397 | 0.958 | 1.105 | 2.032 | 3.694 | 3.919 | 4.392 | 2.609 |
20 | 0.065 | 0.239 | 0.837 | 0.869 | 1.060 | 0.622 | 0.485 | 1.487 | 2.368 | 2.560 | 2.765 | 1.173 | |
50 | 0.058 | 0.293 | 0.595 | 0.695 | 0.798 | 0.320 | 0.186 | 1.063 | 1.436 | 1.723 | 1.785 | 0.391 | |
100 | 0.027 | 0.298 | 0.475 | 0.602 | 0.672 | 0.195 | 0.091 | 0.816 | 1.082 | 1.373 | 1.428 | 0.165 | |
200 | 0.013 | 0.255 | 0.374 | 0.525 | 0.589 | 0.098 | 0.045 | 0.602 | 0.864 | 1.139 | 1.185 | 0.065 | |
500 | 0.007 | 0.146 | 0.273 | 0.484 | 0.532 | 0.047 | 0.018 | 0.316 | 0.674 | 0.979 | 1.014 | 0.022 | |
3.0 | 10 | 0.019 | 0.224 | 1.085 | 1.103 | 1.304 | 0.996 | 1.574 | 2.613 | 3.976 | 4.160 | 4.504 | 3.186 |
20 | 0.005 | 0.268 | 0.877 | 0.924 | 1.044 | 0.639 | 0.751 | 1.793 | 2.525 | 2.701 | 2.865 | 1.512 | |
50 | 0.003 | 0.193 | 0.738 | 0.844 | 0.893 | 0.357 | 0.313 | 1.003 | 1.540 | 1.728 | 1.760 | 0.537 | |
100 | 0.004 | 0.130 | 0.754 | 0.868 | 0.872 | 0.222 | 0.163 | 0.567 | 1.160 | 1.352 | 1.341 | 0.242 | |
200 | 0.001 | 0.053 | 0.775 | 0.887 | 0.896 | 0.141 | 0.084 | 0.238 | 0.920 | 1.112 | 1.106 | 0.113 | |
500 | 0.001 | 0.006 | 0.811 | 0.929 | 0.940 | 0.070 | 0.035 | 0.066 | 0.769 | 0.965 | 0.980 | 0.042 | |
4.0 | 10 | 0.049 | 0.209 | 1.105 | 1.023 | 1.181 | 1.100 | 2.450 | 3.694 | 5.254 | 5.458 | 5.552 | 4.850 |
20 | 0.043 | 0.169 | 0.896 | 0.891 | 0.985 | 0.748 | 1.339 | 2.065 | 3.000 | 3.234 | 3.315 | 2.269 | |
50 | 0.031 | 0.049 | 0.835 | 0.847 | 0.899 | 0.400 | 0.585 | 0.763 | 1.608 | 1.739 | 1.789 | 0.815 | |
100 | 0.008 | 0.014 | 0.849 | 0.878 | 0.891 | 0.255 | 0.291 | 0.328 | 1.161 | 1.240 | 1.285 | 0.377 | |
200 | 0.007 | 0.016 | 0.838 | 0.877 | 0.882 | 0.156 | 0.147 | 0.159 | 0.911 | 1.005 | 1.010 | 0.174 | |
500 | 0.004 | 0.002 | 0.831 | 0.877 | 0.878 | 0.066 | 0.058 | 0.064 | 0.773 | 0.861 | 0.862 | 0.062 |
Para. | n | Bias | mse | ||||||||||
|
mle | mom | ade | cvme | lse | mpse | mle | mom | ade | cvme | lse | mpse | |
5.0 | 10 | 0.106 | 0.077 | 1.058 | 1.013 | 1.242 | 1.164 | 3.881 | 4.545 | 6.333 | 6.748 | 7.579 | 6.421 |
20 | 0.061 | 0.036 | 0.955 | 0.915 | 1.007 | 0.824 | 2.026 | 2.335 | 3.527 | 3.991 | 4.014 | 3.056 | |
50 | 0.046 | 0.015 | 0.864 | 0.869 | 0.904 | 0.447 | 0.821 | 0.839 | 1.793 | 1.906 | 1.970 | 1.070 | |
100 | 0.013 | 0.002 | 0.857 | 0.866 | 0.857 | 0.270 | 0.399 | 0.411 | 1.244 | 1.325 | 1.303 | 0.513 | |
200 | 0.013 | 0.006 | 0.833 | 0.843 | 0.849 | 0.161 | 0.198 | 0.214 | 0.946 | 1.006 | 1.009 | 0.238 | |
500 | 0.001 | 0.003 | 0.832 | 0.846 | 0.843 | 0.082 | 0.079 | 0.078 | 0.792 | 0.829 | 0.822 | 0.086 | |
6.0 | 10 | 0.167 | 0.115 | 1.192 | 1.073 | 1.169 | 1.260 | 5.339 | 5.663 | 8.785 | 8.977 | 9.398 | 8.321 |
20 | 0.049 | 0.027 | 0.946 | 0.952 | 0.975 | 0.890 | 2.794 | 2.926 | 4.236 | 4.861 | 4.686 | 3.860 | |
50 | 0.008 | 0.019 | 0.873 | 0.889 | 0.881 | 0.499 | 1.081 | 1.079 | 2.081 | 2.225 | 2.292 | 1.417 | |
100 | 0.007 | 0.016 | 0.859 | 0.843 | 0.870 | 0.294 | 0.523 | 0.517 | 1.392 | 1.421 | 1.522 | 0.631 | |
200 | 0.007 | 0.006 | 0.845 | 0.839 | 0.846 | 0.181 | 0.261 | 0.269 | 1.030 | 1.061 | 1.065 | 0.308 | |
500 | 0.001 | 0.001 | 0.832 | 0.832 | 0.835 | 0.085 | 0.107 | 0.108 | 0.823 | 0.841 | 0.847 | 0.112 | |
8.0 | 10 | 0.141 | 0.089 | 1.294 | 1.077 | 1.320 | 1.622 | 8.569 | 8.656 | 12.77 | 12.61 | 13.63 | 13.41 |
20 | 0.074 | 0.009 | 1.050 | 0.991 | 1.111 | 1.110 | 4.265 | 4.178 | 6.399 | 6.736 | 7.355 | 6.250 | |
50 | 0.029 | 0.003 | 0.883 | 0.860 | 0.918 | 0.577 | 1.685 | 1.630 | 2.726 | 3.019 | 3.118 | 2.151 | |
100 | 0.015 | 0.014 | 0.867 | 0.861 | 0.859 | 0.372 | 0.834 | 0.814 | 1.787 | 1.862 | 1.859 | 1.029 | |
200 | 0.008 | 0.019 | 0.837 | 0.808 | 0.842 | 0.219 | 0.416 | 0.422 | 1.202 | 1.204 | 1.285 | 0.493 | |
500 | 0.006 | 0.001 | 0.826 | 0.814 | 0.820 | 0.097 | 0.169 | 0.175 | 0.877 | 0.879 | 0.902 | 0.176 | |
10.0 | 10 | 0.106 | 0.086 | 1.365 | 1.333 | 1.446 | 1.868 | 12.89 | 12.67 | 17.66 | 19.87 | 19.51 | 20.26 |
20 | 0.001 | 0.063 | 1.143 | 1.079 | 1.067 | 1.308 | 6.232 | 6.279 | 8.767 | 9.553 | 9.777 | 9.056 | |
50 | 0.025 | 0.036 | 0.899 | 0.892 | 0.933 | 0.686 | 2.483 | 2.373 | 3.774 | 4.076 | 4.248 | 3.246 | |
100 | 0.018 | 0.016 | 0.868 | 0.886 | 0.872 | 0.457 | 1.247 | 1.246 | 2.255 | 2.421 | 2.464 | 1.489 | |
200 | 0.001 | 0.009 | 0.852 | 0.827 | 0.829 | 0.267 | 0.636 | 0.599 | 1.467 | 1.501 | 1.501 | 0.712 | |
500 | 0.002 | 0.007 | 0.823 | 0.801 | 0.818 | 0.119 | 0.246 | 0.240 | 0.967 | 0.971 | 1.005 | 0.255 |
Based on the simulation criteria, it is observed that all estimation approaches work quite well in estimating the parameter λ of the DRL distribution.
In this section, the importance of the proposed distribution is discussed by using data sets from different areas. We shall compare the fits of the DRL distribution with different competitive distributions such as Poisson (Poi), discrete Pareto (DPr), discrete Rayleigh (DR), discrete inverse Rayleigh (DIR), discrete Burr-Hatke (DBH), discrete Bilal (DBi), discrete Lindley (DL), new discrete Lindley (NDL), and discrete Burr-XII (DBXII) distributions. The fitted probability distributions are compared using some criteria, namely, the negative log-likelihood (
The first data set represents the number of deaths due to coronavirus in Pakistan during the period March 18, 2020, to April 30, 2020, which were obtained from the public reports of the National Institute of Health (NIH), Islamabad, Pakistan (https://covid.gov.pk/stats/pakistan). The mean, variance, and di of data set I are 9.4773,102.39, and 10.804, respectively. The mle(s) along with standard error(s) "se(s)" and goodness-of-fit measures for this data are presented in Table 4.
Model | |
|
Goodness-of-fit measures | |||||
mle | se | mle | se | |
aic | ks | p-value | |
DRL | 8.8686 | 1.5033 | - | - | 145.22 | 292.43 | 0.156 | 0.2300 |
Poi | 9.4773 | 0.4641 | - | - | 283.94 | 569.89 | 0.391 | < 0.0001 |
DPr | 0.5021 | 0.0757 | - | - | 162.19 | 326.38 | 0.401 | < 0.0001 |
DR | 9.9883 | 0.7535 | - | - | 168.85 | 339.70 | 0.339 | < 0.0001 |
DIR | 7.4291 | 1.2625 | - | - | 166.31 | 334.61 | 0.382 | < 0.0001 |
DBH | 0.9950 | 0.0115 | - | - | 175.37 | 352.74 | 0.647 | < 0.0001 |
DBi | 11.838 | 1.2932 | - | - | 151.29 | 304.59 | 0.213 | 0.0370 |
DL | 0.8313 | 0.0165 | - | - | 149.17 | 300.33 | 0.184 | 0.1000 |
NDL | 0.1640 | 0.0161 | - | - | 148.44 | 298.89 | 0.237 | 0.0140 |
DBXII | 0.9536 | 0.0434 | 11.907 | 11.305 | 150.70 | 305.40 | 0.302 | 0.0007 |
The results in Table 4 show that the DRL distribution provides a better fit over other competing discrete models since it has the minimum aic, and ks values with the highest p-value. Figure 4 shows the probability-probability (pp) plots for all tested models which prove the empirical results listed in Table 4.
The second data set was reported in [25], which represents the exceedance of flood peaks in m3/s of the Wheaton River near Carcross in Yukon Territory, Canada based on the discretization concept. The mean, variance, and di of this data are 11.806,152.38, and 12.908, respectively. The mle(s), se(s), and goodness-of-fit measures for data set II are reported in Table 5.
Model | |
|
Goodness-of-fit measures | |||||
mle | se | mle | se | |
aic | ks | p-value | |
DRL | 11.214 | 1.4497 | - | - | 252.71 | 507.43 | 0.133 | 0.1600 |
Poi | 11.805 | 0.4049 | - | - | 564.38 | 1130.8 | 0.408 | < 0.0001 |
DPr | 0.4770 | 0.0563 | - | - | 276.82 | 555.64 | 0.311 | < 0.0001 |
DR | 12.280 | 0.7239 | - | - | 300.65 | 603.29 | 0.323 | < 0.0001 |
DIR | 4.7947 | 0.6303 | - | - | 331.46 | 664.92 | 0.497 | < 0.0001 |
DBH | 0.9966 | 0.0072 | - | - | 302.29 | 606.57 | 0.572 | < 0.0001 |
DBi | 14.621 | 1.2479 | - | - | 272.50 | 546.99 | 0.257 | 0.0002 |
DL | 0.8592 | 0.0109 | - | - | 264.30 | 530.59 | 0.232 | 0.0009 |
NDL | 0.1373 | 0.0107 | - | - | 262.09 | 526.17 | 0.271 | < 0.0001 |
DBXII | 0.8205 | 0.0591 | 2.6112 | 0.9287 | 270.50 | 544.99 | 0.228 | 0.0011 |
It is observed that the DRL model is the best among all competitive distributions. Figure 5 illustrates the pp plots for all tested distributions which prove the empirical results reported in Table 5.
The third data set was listed in [26] and represents the number of fires in Greece forest districts for the period from 1st July 1998 to 31 August 1998. The mean, variance, and di measures are 5.2, 32.382, and 6.2272, respectively. The mle(s), se(s), and goodness-of-fit measures for data set II are listed in Table 6.
Model | |
|
Goodness-of-fit measures | |||||
mle | se | mle | se | |
aic | ks | p-value | |
DRL | 4.3673 | 0.5510 | - | - | 301.11 | 604.21 | 0.1510 | 0.0140 |
Poi | 5.2000 | 0.2174 | - | - | 434.16 | 870.32 | 0.282 | < 0.0001 |
DPr | 0.6250 | 0.0597 | - | - | 339.05 | 680.10 | 0.352 | < 0.0001 |
DR | 5.6788 | 0.2714 | - | - | 352.72 | 707.45 | 0.261 | < 0.0001 |
DIR | 3.5198 | 0.3748 | - | - | 360.90 | 723.80 | 0.413 | < 0.0001 |
DBH | 0.9833 | 0.0136 | - | - | 352.42 | 706.85 | 0.532 | < 0.0001 |
DBi | 6.7993 | 0.4693 | - | - | 310.75 | 623.49 | 0.107 | < 0.0001 |
DL | 0.7337 | 0.0156 | - | - | 303.88 | 609.75 | 0.193 | 0.0100 |
NDL | 0.2567 | 0.0152 | - | - | 302.73 | 607.47 | 0.169 | 0.0037 |
DBXII | 0.7486 | 0.0459 | 2.4582 | 0.4938 | 325.00 | 654.01 | 0.287 | < 0.0001 |
It is found that the new discrete model is the best among all tested distributions. Figure 6 shows the pp plots for all competitive distributions which prove the empirical results listed in Table 6.
The fourth data set represents the time to death (in weeks) of AG-positive leukemia patients [27]. The mean, variance, and di values are 62.471, 2954.3, and 47.29, respectively. The estimates and goodness-of-fit measures for all competitive distributions are listed in Table 7.
Model | |
|
Goodness-of-fit measures | |||||
mle | se | mle | se | |
aic | ks | p-value | |
DRL | 61.943 | 15.273 | - | - | 87.425 | 176.85 | 0.152 | 0.8300 |
Poi | 62.470 | 1.9169 | - | - | 475.26 | 952.52 | 0.470 | 0.0011 |
DPr | 0.2838 | 0.0688 | - | - | 98.335 | 198.67 | 0.324 | 0.0560 |
DR | 58.076 | 7.0429 | - | - | 96.794 | 195.59 | 0.309 | 0.0770 |
DIR | 25.310 | 6.543 | - | - | 128.59 | 259.18 | 0.681 | < 0.0001 |
DBH | 0.9999 | 0.0029 | - | - | 119.81 | 241.62 | 0.716 | < 0.0001 |
DBi | 75.109 | 13.164 | - | - | 92.886 | 187.77 | 0.219 | 0.3900 |
DL | 0.9692 | 0.0052 | - | - | 91.858 | 185.72 | 0.215 | 0.4100 |
NDL | 0.0306 | 0.0052 | - | - | 91.458 | 184.92 | 0.218 | 0.3900 |
DBXII | 0.9975 | 0.0008 | 117.30 | 45.982 | 96.151 | 196.30 | 0.327 | 0.0530 |
It is noted that the DRL is the best for this data. Figure 7 shows the pp plots for all tested distributions which prove the empirical results mentioned in Table 7.
In this article, a new one-parameter discrete model has been proposed entitled a discrete Ramos-Louzada (DRL) distribution. The new model can be used effectively in modeling asymmetric data with overdispersion phenomena. Some of its statistical properties have been derived. It was found that all its properties can be expressed in closed forms, which makes the new model can be utilized in different analysis, especially, in time series and regression. Various estimation techniques including maximum likelihood, moments, least squares, Anderson's-Darling, Cramer von-Mises, and maximum product of spacing estimator, have been investigated to get the best estimator for the real data. The estimation performance of these estimation techniques has been assessed via a comprehensive simulation study. The flexibility of the proposed discrete model has been tested utilizing four distinctive real data sets in various fields. Finally, we hope that the DRL distribution attracts wider sets of applications in different fields.
The authors declare that they have no conflict of interest to report regarding the present study.
[1] | R. H. Crowell, R. H. Fox, Introduction to knot theory, New York: Springer, 1963. https://dx.doi.org/10.1007/978-1-4612-9935-6 |
[2] |
J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc., 30 (1928), 275–306. https://dx.doi.org/10.1090/S0002-9947-1928-1501429-1 doi: 10.1090/S0002-9947-1928-1501429-1
![]() |
[3] |
V. F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc., 12 (1985), 103–111. https://dx.doi.org/10.1090/s0273-0979-1985-15304-2 doi: 10.1090/s0273-0979-1985-15304-2
![]() |
[4] | C. Manolescu, An introduction to knot Floer homology, 2014, arXiv: 1401.7107. |
[5] |
M. Khovanov, A categorification of the Jones polynomial, Duke Math. J., 101 (2000), 359–426. https://dx.doi.org/10.1215/S0012-7094-00-10131-7 doi: 10.1215/S0012-7094-00-10131-7
![]() |
[6] | T. Ohtsuki, Quantum invariants: A study of knots, 3-manifolds, and their sets, Singapore: World Scientific, 2001. https://dx.doi.org/10.1142/4746 |
[7] | C. Z. Liang, K. Mislow, Knots in proteins, J. Am. Chem. Soc., 116 (1994), 11189–11190. https://dx.doi.org/10.1021/ja00103a057 |
[8] | D. W. Sumners, The role of knot theory in DNA research, Boca Raton: CRC Press, 1986. |
[9] |
T. Schlick, Q. Y. Zhu, A. Dey, S. Jain, S. T. Yan, A. Laederach, To knot or not to knot: multiple conformations of the SARS-CoV-2 frameshifting RNA element, J. Am. Chem. Soc., 143 (2021), 11404–11422. https://dx.doi.org/10.1021/jacs.1c03003 doi: 10.1021/jacs.1c03003
![]() |
[10] |
K. C. Millett, E. J. Rawdon, A. Stasiak, J. I. Sułkowska, Identifying knots in proteins, Biochem. Soc. Trans., 41 (2013), 533–537. https://dx.doi.org/10.1042/bst20120339 doi: 10.1042/bst20120339
![]() |
[11] |
J. Qin, S. T. Milner, Counting polymer knots to find the entanglement length, Soft Matter, 7 (2011), 10676–10693. https://dx.doi.org/10.1039/c1sm05972f doi: 10.1039/c1sm05972f
![]() |
[12] |
Y. Z. Liu, M. O'Keeffe, M. M. J. Treacy, O. M. Yaghi, The geometry of periodic knots, polycatenanes and weaving from a chemical perspective: a library for reticular chemistry, Chem. Soc. Rev., 47 (2018), 4642–4664. https://dx.doi.org/10.1039/c7cs00695k doi: 10.1039/c7cs00695k
![]() |
[13] |
R. L. Ricca, Topology bounds energy of knots and links, Proc. R. Soc. A., 464 (2008), 293–300. https://dx.doi.org/10.1098/rspa.2007.0174 doi: 10.1098/rspa.2007.0174
![]() |
[14] |
E. Panagiotou, K. C. Millett, P. J. Atzberger, Topological methods for polymeric materials: characterizing the relationship between polymer entanglement and viscoelasticity, Polymers, 11 (2019), 437. https://dx.doi.org/10.3390/polym11030437 doi: 10.3390/polym11030437
![]() |
[15] |
J. Arsuaga, M. Vazquez, P. McGuirk, S. Trigueros, D. W. Sumners, J. Roca, DNA knots reveal a chiral organization of DNA in phage capsids, Proc. Natl. Acad. Sci. U.S.A., 102 (2005), 9165–9169. https://dx.doi.org/10.1073/pnas.0409323102 doi: 10.1073/pnas.0409323102
![]() |
[16] |
J. I. Sulkowska, E. J. Rawdon, K. C. Millet, J. N. Onuchic, A. Stasiak, Conservation of complex knotting and slipknotting patterns in proteins, Proc. Natl. Acad. Sci. U.S.A., 109 (2012), E1715–E1723. https://dx.doi.org/10.1073/pnas.1205918109 doi: 10.1073/pnas.1205918109
![]() |
[17] |
P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, A. Ocneanu, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc., 12 (1985), 239–246. https://dx.doi.org/10.1090/s0273-0979-1985-15361-3 doi: 10.1090/s0273-0979-1985-15361-3
![]() |
[18] |
L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc., 318 (1990), 417–471. https://dx.doi.org/10.1090/S0002-9947-1990-0958895-7 doi: 10.1090/S0002-9947-1990-0958895-7
![]() |
[19] |
J. H. Przytycki, P. Traczyk, Conway algebras and skein equivalence of links, Proc. Amer. Math. Soc., 100 (1987), 744–748. https://dx.doi.org/10.1090/S0002-9939-1987-0894448-2 doi: 10.1090/S0002-9939-1987-0894448-2
![]() |
[20] |
L. Shen, H. S. Feng, F. L. Li, F. C. Lei, J. Wu, G.-W. Wei, Knot data analysis using multiscale Gauss link integral, Proc. Natl. Acad. Sci. U.S.A., 121 (2024), e2408431121. https://dx.doi.org/10.1073/pnas.2408431121 doi: 10.1073/pnas.2408431121
![]() |
[21] | E. Panagiotou, K. W. Plaxco, A topological study of protein folding kinetics, 2018, arXiv: 1812.08721. |
[22] |
Q. Baldwin, E. Panagiotou, The local topological free energy of proteins, J. Theor. Biol., 529 (2021), 110854. https://dx.doi.org/10.1016/j.jtbi.2021.110854 doi: 10.1016/j.jtbi.2021.110854
![]() |
[23] |
Q. Baldwin, B. Sumpter, E. Panagiotou, The local topological free energy of the SARS-CoV-2 Spike protein, Polymers, 14 (2022), 3014. https://dx.doi.org/10.3390/polym14153014 doi: 10.3390/polym14153014
![]() |
[24] |
E. Panagiotou, L. H. Kauffman, Knot polynomials of open and closed curves, Proc. R. Soc. A., 476 (2020), 20200124. https://dx.doi.org/10.1098/rspa.2020.0124 doi: 10.1098/rspa.2020.0124
![]() |
[25] |
K. Barkataki, E. Panagiotou, The Jones polynomial of collections of open curves in 3-space, Proc. R. Soc. A., 478 (2022), 20220302. https://dx.doi.org/10.1098/rspa.2022.0302 doi: 10.1098/rspa.2022.0302
![]() |
[26] |
E. Panagiotou, L. H. Kauffman, Vassiliev measures of complexity of open and closed curves in 3-space, Proc. R. Soc. A., 477 (2021), 20210440. https://dx.doi.org/10.1098/rspa.2021.0440 doi: 10.1098/rspa.2021.0440
![]() |
[27] |
J. Wang, E. Panagiotou, The protein folding rate and the geometry and topology of the native state, Sci. Rep., 12 (2022), 6384. https://dx.doi.org/10.1038/s41598-022-09924-0 doi: 10.1038/s41598-022-09924-0
![]() |
[28] |
T. Herschberg, K. Pifer, E. Panagiotou, A computational package for measuring Topological Entanglement in Polymers, Proteins and Periodic systems (TEPPP), Comput. Phys. Commun., 286 (2023), 108639. https://dx.doi.org/10.1016/j.cpc.2022.108639 doi: 10.1016/j.cpc.2022.108639
![]() |
[29] |
Z. X. Cang, G.-W. Wei, Persistent cohomology for data with multicomponent heterogeneous information, SIAM J. Math. Data Sci., 2 (2020), 396–418. https://dx.doi.org/10.1137/19m1272226 doi: 10.1137/19m1272226
![]() |
[30] |
J. K. Park, R. Jernigan, Z. J. Wu, Coarse grained normal mode analysis vs. refined gaussian network model for protein residue-level structural fluctuations, Bull. Math. Biol., 75 (2013), 124–160. https://dx.doi.org/10.1007/s11538-012-9797-y doi: 10.1007/s11538-012-9797-y
![]() |
[31] | V. Turaev, Knotoids, Osaka J. Math., 49 (2012), 195–223. https://dx.doi.org/10.18910/10080 |
[32] | N. Gügümcü, L. H. Kauffman, New invariants of knotoids, Eur. J. Combin., 65 (2017), 186–229. https://dx.doi.org/10.1016/j.ejc.2017.06.004 |
[33] | N. Gügümcü, S. Lambropoulou, Knotoids, braidoids and applications, Symmetry, 9 (2017), 315. https://dx.doi.org/10.3390/sym9120315 |
[34] | N. Gügümcü, L. Kauffman, Parity in knotoids, 2019, arXiv: 1905.04089. |
[35] |
M. Manouras, S. Lambropoulou, L. H. Kauffman, Finite type invariants for knotoids, Eur. J. Combin., 98 (2021), 103402. https://dx.doi.org/10.1016/j.ejc.2021.103402 doi: 10.1016/j.ejc.2021.103402
![]() |
[36] | D. Cohen-Steiner, H. Edelsbrunner, J. Harer, Stability of persistence diagrams, In: Proceedings of the twenty-first annual symposium on computational geometry, New York: Association for Computing Machinery, 2005,263–271. https://dx.doi.org/10.1145/1064092.1064133 |
[37] |
Z. X. Cang, E. Munch, G.-W. Wei, Evolutionary homology on coupled dynamical systems with applications to protein flexibility analysis, J. Appl. and Comput. Topology, 4 (2020), 481–507. https://dx.doi.org/10.1007/s41468-020-00057-9 doi: 10.1007/s41468-020-00057-9
![]() |
[38] |
D. Bramer, G.-W. Wei, Atom-specific persistent homology and its application to protein flexibility analysis, Comput. Math. Biophys., 8 (2020), 1–35. https://dx.doi.org/10.1515/cmb-2020-0001 doi: 10.1515/cmb-2020-0001
![]() |
[39] |
K. Opron, K. L. Xia, G.-W. Wei, Fast and anisotropic flexibility-rigidity index for protein flexibility and fluctuation analysis, J. Chem. Phys., 140 (2014), 234105. https://dx.doi.org/10.1063/1.4882258 doi: 10.1063/1.4882258
![]() |
1. | Muhammad Ahsan-ul-Haq, On Poisson Moment Exponential Distribution with Applications, 2022, 2198-5804, 10.1007/s40745-022-00400-0 | |
2. | Mohamed S. Eliwa, Muhammad Ahsan-ul-Haq, Amani Almohaimeed, Afrah Al-Bossly, Mahmoud El-Morshedy, Barbara Martinucci, Discrete Extension of Poisson Distribution for Overdispersed Count Data: Theory and Applications, 2023, 2023, 2314-4785, 1, 10.1155/2023/2779120 | |
3. | Muhammad Ahsan-ul-Haq, Afrah Al-Bossly, Mahmoud El-Morshedy, Mohamed S. Eliwa, Maciej Lawrynczuk, Poisson XLindley Distribution for Count Data: Statistical and Reliability Properties with Estimation Techniques and Inference, 2022, 2022, 1687-5273, 1, 10.1155/2022/6503670 | |
4. | Abdulaziz S. Alghamdi, Muhammad Ahsan-ul-Haq, Ayesha Babar, Hassan M. Aljohani, Ahmed Z. Afify, The discrete power-Ailamujia distribution: properties, inference, and applications, 2022, 7, 2473-6988, 8344, 10.3934/math.2022465 | |
5. | Elebe Emmanuel Nwezza, Uchenna Ugwunnaya Uwadi, Christian Osagie, Modeling the number of component failures: A Poison-geometric distribution, 2022, 16, 24682276, e01206, 10.1016/j.sciaf.2022.e01206 | |
6. | Muhammed Irshad, Christophe Chesneau, Veena D’cruz, Radhakumari Maya, Discrete Pseudo Lindley Distribution: Properties, Estimation and Application on INAR(1) Process, 2021, 26, 2297-8747, 76, 10.3390/mca26040076 | |
7. | Muhammad Ahsan-ul-Haq, Javeria Zafar, A new one-parameter discrete probability distribution with its neutrosophic extension: mathematical properties and applications, 2023, 2364-415X, 10.1007/s41060-023-00382-z | |
8. | Ahmed Sedky Eldeeb, Muhammad Ahsan-ul-Haq, Ayesha Babar, A new discrete XLindley distribution: theory, actuarial measures, inference, and applications, 2024, 17, 2364-415X, 323, 10.1007/s41060-023-00395-8 | |
9. | Osama Abdulaziz Alamri, Classical and Bayesian estimation of discrete poisson Agu-Eghwerido distribution with applications, 2024, 109, 11100168, 768, 10.1016/j.aej.2024.09.063 | |
10. | Hassan M. Aljohani, Muhammad Ahsan-ul-Haq, Javeria Zafar, Ehab M. Almetwally, Abdulaziz S. Alghamdi, Eslam Hussam, Abdisalam Hassan Muse, Analysis of Covid-19 data using discrete Marshall–Olkinin Length Biased Exponential: Bayesian and frequentist approach, 2023, 13, 2045-2322, 10.1038/s41598-023-39183-6 | |
11. | Safar M. Alghamdi, Muhammad Ahsan-ul-Haq, Olayan Albalawi, Majdah Mohammed Badr, Eslam Hussam, H.E. Semary, M.A. Abdelkawy, Binomial Poisson Ailamujia model with statistical properties and application, 2024, 17, 16878507, 101096, 10.1016/j.jrras.2024.101096 | |
12. | Amani Alrumayh, Hazar A. Khogeer, A New Two-Parameter Discrete Distribution for Overdispersed and Asymmetric Data: Its Properties, Estimation, Regression Model, and Applications, 2023, 15, 2073-8994, 1289, 10.3390/sym15061289 | |
13. | John Kwadey Okutu, Nana K. Frempong, Simon K. Appiah, Atinuke O. Adebanji, Pritpal Singh, A New Generated Family of Distributions: Statistical Properties and Applications with Real-Life Data, 2023, 2023, 2577-7408, 1, 10.1155/2023/9325679 | |
14. | M. R. Irshad, Muhammed Ahammed, R. Maya, Christophe Chesneau, INAR(1) process with Poisson-transmuted record type exponential innovations, 2024, 19, 15741699, 145, 10.3233/MAS-231458 | |
15. | Khaled M. Alqahtani, Mahmoud El-Morshedy, Hend S. Shahen, Mohamed S. Eliwa, A discrete extension of the Burr-Hatke distribution: Generalized hypergeometric functions, different inference techniques, simulation ranking with modeling and analysis of sustainable count data, 2024, 9, 2473-6988, 9394, 10.3934/math.2024458 | |
16. | Mohamed Eliwa, Mahmoud El-Morshedy, Hend Shahen, Modelling dispersed count data under various shapes of failure rates: A discrete probability analogue of odd Lomax generator, 2023, 37, 0354-5180, 6177, 10.2298/FIL2318177E | |
17. | M. Ahsan-ul-Haq, M. R. Irshad, E. S. Muhammed Ahammed, R. Maya, New Discrete Bilal Distribution and Associated INAR(1) Process, 2023, 44, 1995-0802, 3647, 10.1134/S1995080223090020 | |
18. | Yingying Qi, Dan Ding, Yusra A. Tashkandy, M.E. Bakr, M.M. Abd El-Raouf, Anoop Kumar, A novel probabilistic model with properties: Its implementation to the vocal music and reliability products, 2024, 107, 11100168, 254, 10.1016/j.aej.2024.07.035 | |
19. | Khlood Al-Harbi, Aisha Fayomi, Hanan Baaqeel, Amany Alsuraihi, A Novel Discrete Linear-Exponential Distribution for Modeling Physical and Medical Data, 2024, 16, 2073-8994, 1123, 10.3390/sym16091123 | |
20. | Thanasate Akkanphudit, The Discrete Gompertz–Weibull–Fréchet Distribution: Properties and Applications, 2023, 44, 1995-0802, 3663, 10.1134/S1995080223090032 | |
21. | Mohamed S. Algolam, Mohamed S. Eliwa, Mohamed El-Dawoody, Mahmoud El-Morshedy, A discrete extension of the Xgamma random variable: mathematical framework, estimation methods, simulation ranking, and applications to radiation biology and industrial engineering data, 2025, 10, 2473-6988, 6069, 10.3934/math.2025277 | |
22. | Howaida Elsayed, Mohamed Hussein, A New Discrete Analogue of the Continuous Muth Distribution for Over-Dispersed Data: Properties, Estimation Techniques, and Application, 2025, 27, 1099-4300, 409, 10.3390/e27040409 | |
23. | Amir Mushtaq, Mohamed Kayid, Ghadah Alomani, A new discrete generalized class of distribution with application to radiation and COVID-19 data, 2025, 18, 16878507, 101485, 10.1016/j.jrras.2025.101485 | |
24. | Ali M. Mahnashi, Abdullah A. Zaagan, Poisson copoun distribution: An alternative discrete model for count data analysis, 2025, 128, 11100168, 571, 10.1016/j.aej.2025.05.085 | |
25. | Ayse Alici, Elif Seren Tanriverdi, Gulgun Yenisehirli, Baris Otlu, Investigation of Clonal Relationship Between Candida Parapsilosis and Candida Glabrata Strains Isolated from Blood Culture by Pulse Field Gel Electrophoresis, 2025, 63, 13020072, 62, 10.4274/haseki.galenos.2025.36844 |
|
|||||||||
Measure | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 | 4.5 | 5.0 | 5.5 | 8.0 |
Mean | 3.500 | 3.678 | 4.014 | 4.414 | 4.847 | 5.299 | 5.762 | 6.234 | 8.652 |
Variance | 8.079 | 11.80 | 15.71 | 20.03 | 24.82 | 30.10 | 35.87 | 42.13 | 80.90 |
di | 2.308 | 3.207 | 3.913 | 4.538 | 5.121 | 5.680 | 6.224 | 6.757 | 9.351 |
Skewness | 1.395 | 1.519 | 1.629 | 1.708 | 1.764 | 1.806 | 1.837 | 1.862 | 1.928 |
Kurtosis | 5.940 | 6.353 | 6.821 | 7.199 | 7.495 | 7.727 | 7.910 | 8.057 | 8.484 |
Para. | n | Bias | mse | ||||||||||
|
mle | mom | ade | cvme | lse | mpse | mle | mom | ade | cvme | lse | mpse | |
2.0 | 10 | 0.099 | 0.072 | 0.878 | 0.606 | 0.956 | 0.801 | 0.762 | 1.496 | 2.897 | 2.059 | 3.034 | 1.797 |
20 | 0.034 | 0.035 | 0.486 | 0.259 | 0.444 | 0.435 | 0.307 | 1.103 | 1.408 | 0.749 | 1.212 | 0.626 | |
50 | 0.017 | 0.160 | 0.122 | 0.037 | 0.065 | 0.172 | 0.055 | 0.811 | 0.304 | 0.086 | 0.148 | 0.111 | |
100 | 0.013 | 0.226 | 0.015 | 0.003 | 0.005 | 0.086 | 0.017 | 0.697 | 0.033 | 0.006 | 0.009 | 0.026 | |
200 | 0.009 | 0.277 | 0.000 | 0.000 | 0.000 | 0.050 | 0.007 | 0.620 | 0.000 | 0.000 | 0.000 | 0.008 | |
500 | 0.002 | 0.338 | 0.000 | 0.000 | 0.000 | 0.028 | 0.003 | 0.569 | 0.000 | 0.000 | 0.000 | 0.002 | |
2.5 | 10 | 0.081 | 0.131 | 1.109 | 1.140 | 1.397 | 0.958 | 1.105 | 2.032 | 3.694 | 3.919 | 4.392 | 2.609 |
20 | 0.065 | 0.239 | 0.837 | 0.869 | 1.060 | 0.622 | 0.485 | 1.487 | 2.368 | 2.560 | 2.765 | 1.173 | |
50 | 0.058 | 0.293 | 0.595 | 0.695 | 0.798 | 0.320 | 0.186 | 1.063 | 1.436 | 1.723 | 1.785 | 0.391 | |
100 | 0.027 | 0.298 | 0.475 | 0.602 | 0.672 | 0.195 | 0.091 | 0.816 | 1.082 | 1.373 | 1.428 | 0.165 | |
200 | 0.013 | 0.255 | 0.374 | 0.525 | 0.589 | 0.098 | 0.045 | 0.602 | 0.864 | 1.139 | 1.185 | 0.065 | |
500 | 0.007 | 0.146 | 0.273 | 0.484 | 0.532 | 0.047 | 0.018 | 0.316 | 0.674 | 0.979 | 1.014 | 0.022 | |
3.0 | 10 | 0.019 | 0.224 | 1.085 | 1.103 | 1.304 | 0.996 | 1.574 | 2.613 | 3.976 | 4.160 | 4.504 | 3.186 |
20 | 0.005 | 0.268 | 0.877 | 0.924 | 1.044 | 0.639 | 0.751 | 1.793 | 2.525 | 2.701 | 2.865 | 1.512 | |
50 | 0.003 | 0.193 | 0.738 | 0.844 | 0.893 | 0.357 | 0.313 | 1.003 | 1.540 | 1.728 | 1.760 | 0.537 | |
100 | 0.004 | 0.130 | 0.754 | 0.868 | 0.872 | 0.222 | 0.163 | 0.567 | 1.160 | 1.352 | 1.341 | 0.242 | |
200 | 0.001 | 0.053 | 0.775 | 0.887 | 0.896 | 0.141 | 0.084 | 0.238 | 0.920 | 1.112 | 1.106 | 0.113 | |
500 | 0.001 | 0.006 | 0.811 | 0.929 | 0.940 | 0.070 | 0.035 | 0.066 | 0.769 | 0.965 | 0.980 | 0.042 | |
4.0 | 10 | 0.049 | 0.209 | 1.105 | 1.023 | 1.181 | 1.100 | 2.450 | 3.694 | 5.254 | 5.458 | 5.552 | 4.850 |
20 | 0.043 | 0.169 | 0.896 | 0.891 | 0.985 | 0.748 | 1.339 | 2.065 | 3.000 | 3.234 | 3.315 | 2.269 | |
50 | 0.031 | 0.049 | 0.835 | 0.847 | 0.899 | 0.400 | 0.585 | 0.763 | 1.608 | 1.739 | 1.789 | 0.815 | |
100 | 0.008 | 0.014 | 0.849 | 0.878 | 0.891 | 0.255 | 0.291 | 0.328 | 1.161 | 1.240 | 1.285 | 0.377 | |
200 | 0.007 | 0.016 | 0.838 | 0.877 | 0.882 | 0.156 | 0.147 | 0.159 | 0.911 | 1.005 | 1.010 | 0.174 | |
500 | 0.004 | 0.002 | 0.831 | 0.877 | 0.878 | 0.066 | 0.058 | 0.064 | 0.773 | 0.861 | 0.862 | 0.062 |
Para. | n | Bias | mse | ||||||||||
|
mle | mom | ade | cvme | lse | mpse | mle | mom | ade | cvme | lse | mpse | |
5.0 | 10 | 0.106 | 0.077 | 1.058 | 1.013 | 1.242 | 1.164 | 3.881 | 4.545 | 6.333 | 6.748 | 7.579 | 6.421 |
20 | 0.061 | 0.036 | 0.955 | 0.915 | 1.007 | 0.824 | 2.026 | 2.335 | 3.527 | 3.991 | 4.014 | 3.056 | |
50 | 0.046 | 0.015 | 0.864 | 0.869 | 0.904 | 0.447 | 0.821 | 0.839 | 1.793 | 1.906 | 1.970 | 1.070 | |
100 | 0.013 | 0.002 | 0.857 | 0.866 | 0.857 | 0.270 | 0.399 | 0.411 | 1.244 | 1.325 | 1.303 | 0.513 | |
200 | 0.013 | 0.006 | 0.833 | 0.843 | 0.849 | 0.161 | 0.198 | 0.214 | 0.946 | 1.006 | 1.009 | 0.238 | |
500 | 0.001 | 0.003 | 0.832 | 0.846 | 0.843 | 0.082 | 0.079 | 0.078 | 0.792 | 0.829 | 0.822 | 0.086 | |
6.0 | 10 | 0.167 | 0.115 | 1.192 | 1.073 | 1.169 | 1.260 | 5.339 | 5.663 | 8.785 | 8.977 | 9.398 | 8.321 |
20 | 0.049 | 0.027 | 0.946 | 0.952 | 0.975 | 0.890 | 2.794 | 2.926 | 4.236 | 4.861 | 4.686 | 3.860 | |
50 | 0.008 | 0.019 | 0.873 | 0.889 | 0.881 | 0.499 | 1.081 | 1.079 | 2.081 | 2.225 | 2.292 | 1.417 | |
100 | 0.007 | 0.016 | 0.859 | 0.843 | 0.870 | 0.294 | 0.523 | 0.517 | 1.392 | 1.421 | 1.522 | 0.631 | |
200 | 0.007 | 0.006 | 0.845 | 0.839 | 0.846 | 0.181 | 0.261 | 0.269 | 1.030 | 1.061 | 1.065 | 0.308 | |
500 | 0.001 | 0.001 | 0.832 | 0.832 | 0.835 | 0.085 | 0.107 | 0.108 | 0.823 | 0.841 | 0.847 | 0.112 | |
8.0 | 10 | 0.141 | 0.089 | 1.294 | 1.077 | 1.320 | 1.622 | 8.569 | 8.656 | 12.77 | 12.61 | 13.63 | 13.41 |
20 | 0.074 | 0.009 | 1.050 | 0.991 | 1.111 | 1.110 | 4.265 | 4.178 | 6.399 | 6.736 | 7.355 | 6.250 | |
50 | 0.029 | 0.003 | 0.883 | 0.860 | 0.918 | 0.577 | 1.685 | 1.630 | 2.726 | 3.019 | 3.118 | 2.151 | |
100 | 0.015 | 0.014 | 0.867 | 0.861 | 0.859 | 0.372 | 0.834 | 0.814 | 1.787 | 1.862 | 1.859 | 1.029 | |
200 | 0.008 | 0.019 | 0.837 | 0.808 | 0.842 | 0.219 | 0.416 | 0.422 | 1.202 | 1.204 | 1.285 | 0.493 | |
500 | 0.006 | 0.001 | 0.826 | 0.814 | 0.820 | 0.097 | 0.169 | 0.175 | 0.877 | 0.879 | 0.902 | 0.176 | |
10.0 | 10 | 0.106 | 0.086 | 1.365 | 1.333 | 1.446 | 1.868 | 12.89 | 12.67 | 17.66 | 19.87 | 19.51 | 20.26 |
20 | 0.001 | 0.063 | 1.143 | 1.079 | 1.067 | 1.308 | 6.232 | 6.279 | 8.767 | 9.553 | 9.777 | 9.056 | |
50 | 0.025 | 0.036 | 0.899 | 0.892 | 0.933 | 0.686 | 2.483 | 2.373 | 3.774 | 4.076 | 4.248 | 3.246 | |
100 | 0.018 | 0.016 | 0.868 | 0.886 | 0.872 | 0.457 | 1.247 | 1.246 | 2.255 | 2.421 | 2.464 | 1.489 | |
200 | 0.001 | 0.009 | 0.852 | 0.827 | 0.829 | 0.267 | 0.636 | 0.599 | 1.467 | 1.501 | 1.501 | 0.712 | |
500 | 0.002 | 0.007 | 0.823 | 0.801 | 0.818 | 0.119 | 0.246 | 0.240 | 0.967 | 0.971 | 1.005 | 0.255 |
Model | |
|
Goodness-of-fit measures | |||||
mle | se | mle | se | |
aic | ks | p-value | |
DRL | 8.8686 | 1.5033 | - | - | 145.22 | 292.43 | 0.156 | 0.2300 |
Poi | 9.4773 | 0.4641 | - | - | 283.94 | 569.89 | 0.391 | < 0.0001 |
DPr | 0.5021 | 0.0757 | - | - | 162.19 | 326.38 | 0.401 | < 0.0001 |
DR | 9.9883 | 0.7535 | - | - | 168.85 | 339.70 | 0.339 | < 0.0001 |
DIR | 7.4291 | 1.2625 | - | - | 166.31 | 334.61 | 0.382 | < 0.0001 |
DBH | 0.9950 | 0.0115 | - | - | 175.37 | 352.74 | 0.647 | < 0.0001 |
DBi | 11.838 | 1.2932 | - | - | 151.29 | 304.59 | 0.213 | 0.0370 |
DL | 0.8313 | 0.0165 | - | - | 149.17 | 300.33 | 0.184 | 0.1000 |
NDL | 0.1640 | 0.0161 | - | - | 148.44 | 298.89 | 0.237 | 0.0140 |
DBXII | 0.9536 | 0.0434 | 11.907 | 11.305 | 150.70 | 305.40 | 0.302 | 0.0007 |
Model | |
|
Goodness-of-fit measures | |||||
mle | se | mle | se | |
aic | ks | p-value | |
DRL | 11.214 | 1.4497 | - | - | 252.71 | 507.43 | 0.133 | 0.1600 |
Poi | 11.805 | 0.4049 | - | - | 564.38 | 1130.8 | 0.408 | < 0.0001 |
DPr | 0.4770 | 0.0563 | - | - | 276.82 | 555.64 | 0.311 | < 0.0001 |
DR | 12.280 | 0.7239 | - | - | 300.65 | 603.29 | 0.323 | < 0.0001 |
DIR | 4.7947 | 0.6303 | - | - | 331.46 | 664.92 | 0.497 | < 0.0001 |
DBH | 0.9966 | 0.0072 | - | - | 302.29 | 606.57 | 0.572 | < 0.0001 |
DBi | 14.621 | 1.2479 | - | - | 272.50 | 546.99 | 0.257 | 0.0002 |
DL | 0.8592 | 0.0109 | - | - | 264.30 | 530.59 | 0.232 | 0.0009 |
NDL | 0.1373 | 0.0107 | - | - | 262.09 | 526.17 | 0.271 | < 0.0001 |
DBXII | 0.8205 | 0.0591 | 2.6112 | 0.9287 | 270.50 | 544.99 | 0.228 | 0.0011 |
Model | |
|
Goodness-of-fit measures | |||||
mle | se | mle | se | |
aic | ks | p-value | |
DRL | 4.3673 | 0.5510 | - | - | 301.11 | 604.21 | 0.1510 | 0.0140 |
Poi | 5.2000 | 0.2174 | - | - | 434.16 | 870.32 | 0.282 | < 0.0001 |
DPr | 0.6250 | 0.0597 | - | - | 339.05 | 680.10 | 0.352 | < 0.0001 |
DR | 5.6788 | 0.2714 | - | - | 352.72 | 707.45 | 0.261 | < 0.0001 |
DIR | 3.5198 | 0.3748 | - | - | 360.90 | 723.80 | 0.413 | < 0.0001 |
DBH | 0.9833 | 0.0136 | - | - | 352.42 | 706.85 | 0.532 | < 0.0001 |
DBi | 6.7993 | 0.4693 | - | - | 310.75 | 623.49 | 0.107 | < 0.0001 |
DL | 0.7337 | 0.0156 | - | - | 303.88 | 609.75 | 0.193 | 0.0100 |
NDL | 0.2567 | 0.0152 | - | - | 302.73 | 607.47 | 0.169 | 0.0037 |
DBXII | 0.7486 | 0.0459 | 2.4582 | 0.4938 | 325.00 | 654.01 | 0.287 | < 0.0001 |
Model | |
|
Goodness-of-fit measures | |||||
mle | se | mle | se | |
aic | ks | p-value | |
DRL | 61.943 | 15.273 | - | - | 87.425 | 176.85 | 0.152 | 0.8300 |
Poi | 62.470 | 1.9169 | - | - | 475.26 | 952.52 | 0.470 | 0.0011 |
DPr | 0.2838 | 0.0688 | - | - | 98.335 | 198.67 | 0.324 | 0.0560 |
DR | 58.076 | 7.0429 | - | - | 96.794 | 195.59 | 0.309 | 0.0770 |
DIR | 25.310 | 6.543 | - | - | 128.59 | 259.18 | 0.681 | < 0.0001 |
DBH | 0.9999 | 0.0029 | - | - | 119.81 | 241.62 | 0.716 | < 0.0001 |
DBi | 75.109 | 13.164 | - | - | 92.886 | 187.77 | 0.219 | 0.3900 |
DL | 0.9692 | 0.0052 | - | - | 91.858 | 185.72 | 0.215 | 0.4100 |
NDL | 0.0306 | 0.0052 | - | - | 91.458 | 184.92 | 0.218 | 0.3900 |
DBXII | 0.9975 | 0.0008 | 117.30 | 45.982 | 96.151 | 196.30 | 0.327 | 0.0530 |
|
|||||||||
Measure | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 | 4.5 | 5.0 | 5.5 | 8.0 |
Mean | 3.500 | 3.678 | 4.014 | 4.414 | 4.847 | 5.299 | 5.762 | 6.234 | 8.652 |
Variance | 8.079 | 11.80 | 15.71 | 20.03 | 24.82 | 30.10 | 35.87 | 42.13 | 80.90 |
di | 2.308 | 3.207 | 3.913 | 4.538 | 5.121 | 5.680 | 6.224 | 6.757 | 9.351 |
Skewness | 1.395 | 1.519 | 1.629 | 1.708 | 1.764 | 1.806 | 1.837 | 1.862 | 1.928 |
Kurtosis | 5.940 | 6.353 | 6.821 | 7.199 | 7.495 | 7.727 | 7.910 | 8.057 | 8.484 |
Para. | n | Bias | mse | ||||||||||
|
mle | mom | ade | cvme | lse | mpse | mle | mom | ade | cvme | lse | mpse | |
2.0 | 10 | 0.099 | 0.072 | 0.878 | 0.606 | 0.956 | 0.801 | 0.762 | 1.496 | 2.897 | 2.059 | 3.034 | 1.797 |
20 | 0.034 | 0.035 | 0.486 | 0.259 | 0.444 | 0.435 | 0.307 | 1.103 | 1.408 | 0.749 | 1.212 | 0.626 | |
50 | 0.017 | 0.160 | 0.122 | 0.037 | 0.065 | 0.172 | 0.055 | 0.811 | 0.304 | 0.086 | 0.148 | 0.111 | |
100 | 0.013 | 0.226 | 0.015 | 0.003 | 0.005 | 0.086 | 0.017 | 0.697 | 0.033 | 0.006 | 0.009 | 0.026 | |
200 | 0.009 | 0.277 | 0.000 | 0.000 | 0.000 | 0.050 | 0.007 | 0.620 | 0.000 | 0.000 | 0.000 | 0.008 | |
500 | 0.002 | 0.338 | 0.000 | 0.000 | 0.000 | 0.028 | 0.003 | 0.569 | 0.000 | 0.000 | 0.000 | 0.002 | |
2.5 | 10 | 0.081 | 0.131 | 1.109 | 1.140 | 1.397 | 0.958 | 1.105 | 2.032 | 3.694 | 3.919 | 4.392 | 2.609 |
20 | 0.065 | 0.239 | 0.837 | 0.869 | 1.060 | 0.622 | 0.485 | 1.487 | 2.368 | 2.560 | 2.765 | 1.173 | |
50 | 0.058 | 0.293 | 0.595 | 0.695 | 0.798 | 0.320 | 0.186 | 1.063 | 1.436 | 1.723 | 1.785 | 0.391 | |
100 | 0.027 | 0.298 | 0.475 | 0.602 | 0.672 | 0.195 | 0.091 | 0.816 | 1.082 | 1.373 | 1.428 | 0.165 | |
200 | 0.013 | 0.255 | 0.374 | 0.525 | 0.589 | 0.098 | 0.045 | 0.602 | 0.864 | 1.139 | 1.185 | 0.065 | |
500 | 0.007 | 0.146 | 0.273 | 0.484 | 0.532 | 0.047 | 0.018 | 0.316 | 0.674 | 0.979 | 1.014 | 0.022 | |
3.0 | 10 | 0.019 | 0.224 | 1.085 | 1.103 | 1.304 | 0.996 | 1.574 | 2.613 | 3.976 | 4.160 | 4.504 | 3.186 |
20 | 0.005 | 0.268 | 0.877 | 0.924 | 1.044 | 0.639 | 0.751 | 1.793 | 2.525 | 2.701 | 2.865 | 1.512 | |
50 | 0.003 | 0.193 | 0.738 | 0.844 | 0.893 | 0.357 | 0.313 | 1.003 | 1.540 | 1.728 | 1.760 | 0.537 | |
100 | 0.004 | 0.130 | 0.754 | 0.868 | 0.872 | 0.222 | 0.163 | 0.567 | 1.160 | 1.352 | 1.341 | 0.242 | |
200 | 0.001 | 0.053 | 0.775 | 0.887 | 0.896 | 0.141 | 0.084 | 0.238 | 0.920 | 1.112 | 1.106 | 0.113 | |
500 | 0.001 | 0.006 | 0.811 | 0.929 | 0.940 | 0.070 | 0.035 | 0.066 | 0.769 | 0.965 | 0.980 | 0.042 | |
4.0 | 10 | 0.049 | 0.209 | 1.105 | 1.023 | 1.181 | 1.100 | 2.450 | 3.694 | 5.254 | 5.458 | 5.552 | 4.850 |
20 | 0.043 | 0.169 | 0.896 | 0.891 | 0.985 | 0.748 | 1.339 | 2.065 | 3.000 | 3.234 | 3.315 | 2.269 | |
50 | 0.031 | 0.049 | 0.835 | 0.847 | 0.899 | 0.400 | 0.585 | 0.763 | 1.608 | 1.739 | 1.789 | 0.815 | |
100 | 0.008 | 0.014 | 0.849 | 0.878 | 0.891 | 0.255 | 0.291 | 0.328 | 1.161 | 1.240 | 1.285 | 0.377 | |
200 | 0.007 | 0.016 | 0.838 | 0.877 | 0.882 | 0.156 | 0.147 | 0.159 | 0.911 | 1.005 | 1.010 | 0.174 | |
500 | 0.004 | 0.002 | 0.831 | 0.877 | 0.878 | 0.066 | 0.058 | 0.064 | 0.773 | 0.861 | 0.862 | 0.062 |
Para. | n | Bias | mse | ||||||||||
|
mle | mom | ade | cvme | lse | mpse | mle | mom | ade | cvme | lse | mpse | |
5.0 | 10 | 0.106 | 0.077 | 1.058 | 1.013 | 1.242 | 1.164 | 3.881 | 4.545 | 6.333 | 6.748 | 7.579 | 6.421 |
20 | 0.061 | 0.036 | 0.955 | 0.915 | 1.007 | 0.824 | 2.026 | 2.335 | 3.527 | 3.991 | 4.014 | 3.056 | |
50 | 0.046 | 0.015 | 0.864 | 0.869 | 0.904 | 0.447 | 0.821 | 0.839 | 1.793 | 1.906 | 1.970 | 1.070 | |
100 | 0.013 | 0.002 | 0.857 | 0.866 | 0.857 | 0.270 | 0.399 | 0.411 | 1.244 | 1.325 | 1.303 | 0.513 | |
200 | 0.013 | 0.006 | 0.833 | 0.843 | 0.849 | 0.161 | 0.198 | 0.214 | 0.946 | 1.006 | 1.009 | 0.238 | |
500 | 0.001 | 0.003 | 0.832 | 0.846 | 0.843 | 0.082 | 0.079 | 0.078 | 0.792 | 0.829 | 0.822 | 0.086 | |
6.0 | 10 | 0.167 | 0.115 | 1.192 | 1.073 | 1.169 | 1.260 | 5.339 | 5.663 | 8.785 | 8.977 | 9.398 | 8.321 |
20 | 0.049 | 0.027 | 0.946 | 0.952 | 0.975 | 0.890 | 2.794 | 2.926 | 4.236 | 4.861 | 4.686 | 3.860 | |
50 | 0.008 | 0.019 | 0.873 | 0.889 | 0.881 | 0.499 | 1.081 | 1.079 | 2.081 | 2.225 | 2.292 | 1.417 | |
100 | 0.007 | 0.016 | 0.859 | 0.843 | 0.870 | 0.294 | 0.523 | 0.517 | 1.392 | 1.421 | 1.522 | 0.631 | |
200 | 0.007 | 0.006 | 0.845 | 0.839 | 0.846 | 0.181 | 0.261 | 0.269 | 1.030 | 1.061 | 1.065 | 0.308 | |
500 | 0.001 | 0.001 | 0.832 | 0.832 | 0.835 | 0.085 | 0.107 | 0.108 | 0.823 | 0.841 | 0.847 | 0.112 | |
8.0 | 10 | 0.141 | 0.089 | 1.294 | 1.077 | 1.320 | 1.622 | 8.569 | 8.656 | 12.77 | 12.61 | 13.63 | 13.41 |
20 | 0.074 | 0.009 | 1.050 | 0.991 | 1.111 | 1.110 | 4.265 | 4.178 | 6.399 | 6.736 | 7.355 | 6.250 | |
50 | 0.029 | 0.003 | 0.883 | 0.860 | 0.918 | 0.577 | 1.685 | 1.630 | 2.726 | 3.019 | 3.118 | 2.151 | |
100 | 0.015 | 0.014 | 0.867 | 0.861 | 0.859 | 0.372 | 0.834 | 0.814 | 1.787 | 1.862 | 1.859 | 1.029 | |
200 | 0.008 | 0.019 | 0.837 | 0.808 | 0.842 | 0.219 | 0.416 | 0.422 | 1.202 | 1.204 | 1.285 | 0.493 | |
500 | 0.006 | 0.001 | 0.826 | 0.814 | 0.820 | 0.097 | 0.169 | 0.175 | 0.877 | 0.879 | 0.902 | 0.176 | |
10.0 | 10 | 0.106 | 0.086 | 1.365 | 1.333 | 1.446 | 1.868 | 12.89 | 12.67 | 17.66 | 19.87 | 19.51 | 20.26 |
20 | 0.001 | 0.063 | 1.143 | 1.079 | 1.067 | 1.308 | 6.232 | 6.279 | 8.767 | 9.553 | 9.777 | 9.056 | |
50 | 0.025 | 0.036 | 0.899 | 0.892 | 0.933 | 0.686 | 2.483 | 2.373 | 3.774 | 4.076 | 4.248 | 3.246 | |
100 | 0.018 | 0.016 | 0.868 | 0.886 | 0.872 | 0.457 | 1.247 | 1.246 | 2.255 | 2.421 | 2.464 | 1.489 | |
200 | 0.001 | 0.009 | 0.852 | 0.827 | 0.829 | 0.267 | 0.636 | 0.599 | 1.467 | 1.501 | 1.501 | 0.712 | |
500 | 0.002 | 0.007 | 0.823 | 0.801 | 0.818 | 0.119 | 0.246 | 0.240 | 0.967 | 0.971 | 1.005 | 0.255 |
Model | |
|
Goodness-of-fit measures | |||||
mle | se | mle | se | |
aic | ks | p-value | |
DRL | 8.8686 | 1.5033 | - | - | 145.22 | 292.43 | 0.156 | 0.2300 |
Poi | 9.4773 | 0.4641 | - | - | 283.94 | 569.89 | 0.391 | < 0.0001 |
DPr | 0.5021 | 0.0757 | - | - | 162.19 | 326.38 | 0.401 | < 0.0001 |
DR | 9.9883 | 0.7535 | - | - | 168.85 | 339.70 | 0.339 | < 0.0001 |
DIR | 7.4291 | 1.2625 | - | - | 166.31 | 334.61 | 0.382 | < 0.0001 |
DBH | 0.9950 | 0.0115 | - | - | 175.37 | 352.74 | 0.647 | < 0.0001 |
DBi | 11.838 | 1.2932 | - | - | 151.29 | 304.59 | 0.213 | 0.0370 |
DL | 0.8313 | 0.0165 | - | - | 149.17 | 300.33 | 0.184 | 0.1000 |
NDL | 0.1640 | 0.0161 | - | - | 148.44 | 298.89 | 0.237 | 0.0140 |
DBXII | 0.9536 | 0.0434 | 11.907 | 11.305 | 150.70 | 305.40 | 0.302 | 0.0007 |
Model | |
|
Goodness-of-fit measures | |||||
mle | se | mle | se | |
aic | ks | p-value | |
DRL | 11.214 | 1.4497 | - | - | 252.71 | 507.43 | 0.133 | 0.1600 |
Poi | 11.805 | 0.4049 | - | - | 564.38 | 1130.8 | 0.408 | < 0.0001 |
DPr | 0.4770 | 0.0563 | - | - | 276.82 | 555.64 | 0.311 | < 0.0001 |
DR | 12.280 | 0.7239 | - | - | 300.65 | 603.29 | 0.323 | < 0.0001 |
DIR | 4.7947 | 0.6303 | - | - | 331.46 | 664.92 | 0.497 | < 0.0001 |
DBH | 0.9966 | 0.0072 | - | - | 302.29 | 606.57 | 0.572 | < 0.0001 |
DBi | 14.621 | 1.2479 | - | - | 272.50 | 546.99 | 0.257 | 0.0002 |
DL | 0.8592 | 0.0109 | - | - | 264.30 | 530.59 | 0.232 | 0.0009 |
NDL | 0.1373 | 0.0107 | - | - | 262.09 | 526.17 | 0.271 | < 0.0001 |
DBXII | 0.8205 | 0.0591 | 2.6112 | 0.9287 | 270.50 | 544.99 | 0.228 | 0.0011 |
Model | |
|
Goodness-of-fit measures | |||||
mle | se | mle | se | |
aic | ks | p-value | |
DRL | 4.3673 | 0.5510 | - | - | 301.11 | 604.21 | 0.1510 | 0.0140 |
Poi | 5.2000 | 0.2174 | - | - | 434.16 | 870.32 | 0.282 | < 0.0001 |
DPr | 0.6250 | 0.0597 | - | - | 339.05 | 680.10 | 0.352 | < 0.0001 |
DR | 5.6788 | 0.2714 | - | - | 352.72 | 707.45 | 0.261 | < 0.0001 |
DIR | 3.5198 | 0.3748 | - | - | 360.90 | 723.80 | 0.413 | < 0.0001 |
DBH | 0.9833 | 0.0136 | - | - | 352.42 | 706.85 | 0.532 | < 0.0001 |
DBi | 6.7993 | 0.4693 | - | - | 310.75 | 623.49 | 0.107 | < 0.0001 |
DL | 0.7337 | 0.0156 | - | - | 303.88 | 609.75 | 0.193 | 0.0100 |
NDL | 0.2567 | 0.0152 | - | - | 302.73 | 607.47 | 0.169 | 0.0037 |
DBXII | 0.7486 | 0.0459 | 2.4582 | 0.4938 | 325.00 | 654.01 | 0.287 | < 0.0001 |
Model | |
|
Goodness-of-fit measures | |||||
mle | se | mle | se | |
aic | ks | p-value | |
DRL | 61.943 | 15.273 | - | - | 87.425 | 176.85 | 0.152 | 0.8300 |
Poi | 62.470 | 1.9169 | - | - | 475.26 | 952.52 | 0.470 | 0.0011 |
DPr | 0.2838 | 0.0688 | - | - | 98.335 | 198.67 | 0.324 | 0.0560 |
DR | 58.076 | 7.0429 | - | - | 96.794 | 195.59 | 0.309 | 0.0770 |
DIR | 25.310 | 6.543 | - | - | 128.59 | 259.18 | 0.681 | < 0.0001 |
DBH | 0.9999 | 0.0029 | - | - | 119.81 | 241.62 | 0.716 | < 0.0001 |
DBi | 75.109 | 13.164 | - | - | 92.886 | 187.77 | 0.219 | 0.3900 |
DL | 0.9692 | 0.0052 | - | - | 91.858 | 185.72 | 0.215 | 0.4100 |
NDL | 0.0306 | 0.0052 | - | - | 91.458 | 184.92 | 0.218 | 0.3900 |
DBXII | 0.9975 | 0.0008 | 117.30 | 45.982 | 96.151 | 196.30 | 0.327 | 0.0530 |