
The central focus of our investigation revolved around the convergence of agents' behavior toward a particular invariant distribution and determining the characteristics of the optimal strategies' distribution within the framework of a dynamical Multi-Local-Worlds complex adaptive system. This system was characterized by the co-evolution of agent behavior and local topological configuration. The study established a representation of an agent's behavior and local graphic topology configuration to elucidate the interaction dynamics within this dynamical context. As an illustrative example, we introduced three distinct agent types—smart agent, normal agent, and stupid agent—each associated with specific behaviors. The findings underscored that an agent's decision-making process was influenced by the evolution of random complex networks driven by preferential attachment, coupled with a volatility mechanism linked to its payment—a dynamic that propels the evolution of the complex adaptive system. Through simulation, we drew a conclusive observation that even when considering irrational behaviors characterized by limited information and memory constraints, the system's state converges to a specific attractor. This underscored the robustness and convergence properties inherent in the dynamical Multi-Local-Worlds complex adaptive system under scrutiny.
Citation: Hebing Zhang, Xiaojing Zheng. Multi-Local-Worlds economic and management complex adaptive system with agent behavior and local configuration[J]. Electronic Research Archive, 2024, 32(4): 2824-2847. doi: 10.3934/era.2024128
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The central focus of our investigation revolved around the convergence of agents' behavior toward a particular invariant distribution and determining the characteristics of the optimal strategies' distribution within the framework of a dynamical Multi-Local-Worlds complex adaptive system. This system was characterized by the co-evolution of agent behavior and local topological configuration. The study established a representation of an agent's behavior and local graphic topology configuration to elucidate the interaction dynamics within this dynamical context. As an illustrative example, we introduced three distinct agent types—smart agent, normal agent, and stupid agent—each associated with specific behaviors. The findings underscored that an agent's decision-making process was influenced by the evolution of random complex networks driven by preferential attachment, coupled with a volatility mechanism linked to its payment—a dynamic that propels the evolution of the complex adaptive system. Through simulation, we drew a conclusive observation that even when considering irrational behaviors characterized by limited information and memory constraints, the system's state converges to a specific attractor. This underscored the robustness and convergence properties inherent in the dynamical Multi-Local-Worlds complex adaptive system under scrutiny.
In numerous practical situations, the datasets are measured in the number of cycles, runs, and/or shocks the device sustains before its failure. For example, the number of voltage fluctuations, the lifetime of a discrete random variable (rv), and frequency of a device switched on/off, the life of a weapon is measured by the number of rounds fired before failure, and the number of completed cycles measures the life of the equipment. Further, the number of patients, number of deaths due to a disease/virus, and number of days a patient stays in a hospital ward. Various discrete probability models can be adopted to analyze such types of datasets.
The well-known traditional discrete probability models, including the negative binomial, geometric, and Poisson distributions have limitations to use due to their specific behavior such as the Poisson distribution that performs better with datasets having dispersion equal to average; the NB distribution is applicable for over-dispersed datasets. The real-life datasets may be over-dispersed or under-dispersed, so there is always a clear need for flexible discrete distributions to have a good resolution.
Several discretized forms of continuous distributions have been derived to model different count datasets in the last few decades. The most notable discretization approach in the literature is the survival discretizing approach which has gained much attention.
Let rv X follows a continuous distribution with survival function (sf) S(x). Using the survival discretization approach introduced by Kemp (2004), the probability mass function (pmf) of a discrete rv follows as
p(x)=P(X=x)=S(x)−S(x+1),x=0,1,2,3,… | (1) |
The survival discretization approach has been adopted to develop many discrete models. For example, the discrete normal [1], discrete Rayleigh [2], discrete half-normal [3], discrete Burr and discrete Pareto [4], discrete inverse-Weibull [5], new generalization of the geometric [6], discrete Lindley [7], generalized exponential type Ⅱ [8], discrete inverse-Rayleigh [9], two-parameter discrete Lindley [10], discrete log-logistic [11], discrete extended Weibull [12], exponentiated discrete-Lindley [13], discrete Burr-Hutke [14], discrete Marshall-Olkin Weibull [15], natural discrete-Lindley [16], discrete Bilal [17], discrete inverted Topp-Leone [18], uniform Poisson–Ailamujia [19], exponentiated discrete Lindley [20], discrete exponentiated Burr–Hatke [21], discrete Ramos-Louzada [22] and [23], and discrete type-Ⅱ half-logistic exponential [24].
The main goal of the present study is to introduce a new discrete distribution to model over-dispersed as well as under-dispersed datasets. The proposed distribution is called the discrete power-Ailamujia (DsPA) distribution. The mathematical properties of the DsPA distribution are derived and its parameters are estimated using the maximum likelihood method. Three real count datasets are fitted using the DsPA model and other competing discrete distributions. The DsPA distribution provides a better fit to the three datasets than some well-known discrete models according to the results of the simulation.
The paper is organized in the following sections. Section 2 is devoted to the derivation of the new DsPA distribution. Its mathematical properties are explored in Section 3. Section 4 is devoted to estimating the DsPA parameters and providing a comprehensive simulation study. The usefulness of the DsPA distribution is addressed in Section 5. Finally, we conclude the study in Section 6.
Jamal et al. [25] proposed a new continuous lifetime distribution called the power-Ailamujia distribution. Its probability density function and sf can be expressed as
f(x)=θ2βx2β−1e−θxβ,x≥0,θ,β>0 | (2) |
and
S(x)=(1+θxβ)e−θxβ,x≥0,θ,β>0, | (3) |
respectively.
Applying the survival discretization approach in (1), the rv X is said to have the DsPA distribution with parameters 0<λ<1 and β>0, if its sf takes the form
S(x;λ,β)=λ(x+1)β[1−(x+1)βlnλ],x∈N0, | (4) |
where λ=e−θ and N0={0,1,2,…,w} for 0<w<∞.
The corresponding cumulative distribution function (cdf) and pmf can be expressed as
F(x;λ,β)=1−λ(x+1)β[1−(x+1)βlnλ],x∈N0 | (5) |
and
Px(x;λ)=λxβ[1−xβlnλ]−λ(x+1)β[1−(x+1)βlnλ],x∈N0. | (6) |
Plots of the DsPA pmf, for various values of the parameters λ and β, are presented in Figure 1.
The hazard rate function (hrf) of the DsPA distribution can be expressed as
h(x;λ)=p(x)S(x)=λxβ[1−xβlnλ]λ(x+1)β[1−(x+1)βlnλ]−1,x∈N0 | (7) |
where h(x;λ)=P(x)S(x). Figure 2 shows the DsPA hrf plots for different values of λ and β.
The quantile function of the DsPA distribution reduces to
Q(u)=⌊1−[1−(x+1)βlnλ]λ(x+1)β⌋,0<u<1, |
where ⌊x⌋ denotes the integer part of x.
The reverse hrf (rhrf) of the DsPA distribution is defined as
r∗(x)=p(x)F(x)=λxβ[1−xβlnλ]−λ(x+1)β[1−(x+1)βlnλ]1−λ(x+1)β[1−(x+1)βlnλ],x∈N0, | (8) |
where r∗(x)=P(x)F(x). Figure 3 shows the DsPA rhrf plots for several values of λ and β.
The second failure rate of the DsPA distribution is expressed by
r∗∗(x)=log{S(x)S(x+1)}=log{λ(x+1)β[1−(x+1)βlnλ]λ(x+2)β[1−(x+2)βlnλ]},x∈N0. | (9) |
The recurrence relation of probabilities from the DsPA distribution has the form
P(x+1)P(x)=λ(x+1)β[1−(x+1)βlnλ]−λ(x+2)β[1−(x+2)βlnλ]λxβ[1−xβlnλ]−λ(x+1)β[1−(x+1)βlnλ]. | (10) |
Hence,
P(x+1)=λ(x+1)β[1−(x+1)βlnλ]−λ(x+2)β[1−(x+2)βlnλ]λxβ[1−xβlnλ]−λ(x+1)β[1−(x+1)βlnλ]P(x). |
In this section, we studied some mathematical properties of the DsPA distribution. In this section, we studied some mathematical properties of the DsPA distribution.
The probability generating function (pgf) of the DsPA distribution is given as follows
Gx(z)=1+(z−1)∞∑x=1zx−1(1−xβlnλ)λxβ, | (11) |
where Gx(z)=∑∞x=0zxP(x). The moment generating function (mgf) can be obtained by replacing z with ez in Eq (11). Thus, the mgf of the DsPA distribution can be expressed as
Mx(z)=1+(ez−1)∞∑x=1(ez)x−1(1−xβlnλ)λxβ. | (12) |
Thus, the first four moments of the DsPA distribution are
E(X)=∞∑x=1(1−xβlnλ)λxβ, | (13) |
E(X2)=∞∑x=1(2x−1)(1−xβlnλ)λxβ, |
E(X3)=∞∑x=1(3x2−3x+1)(1−xβlnλ)λxβ |
and
E(X4)=∞∑x=1(4x3−6x2+4x−1)(1−xβlnλ)λxβ. |
Using the above moments, the variance (σ2), coefficient of skewness (CS), and coefficient of kurtosis (CK) can be presented in closed-form expressions. Further, another classical concept, called dispersion index (DI). The DI is defined as a variance to mean ratio. If the DI value is less than 1, then the model is suitable for under-dispersed datasets. Conversely, if the DI is greater than 1, then it is suitable for over-dispersed datasets. Numerical values of the mean, E(X), σ2, CS, CK and DI are reported in Tables 1‒5.
β | λ | ||||||||
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
0.5 | 0.7603 | 1.8909 | 3.6766 | 6.5446 | 10.966 | 16.848 | 22.277 | 22.458 | 12.187 |
1.0 | 0.3954 | 0.7529 | 1.1657 | 1.6848 | 2.3863 | 3.4156 | 5.1075 | 8.4629 | 18.452 |
1.5 | 0.3415 | 0.5827 | 0.8221 | 1.0907 | 1.4187 | 1.8533 | 2.4910 | 3.5895 | 6.2446 |
2.0 | 0.3313 | 0.5338 | 0.7085 | 0.8884 | 1.0967 | 1.3599 | 1.7259 | 2.3141 | 3.5954 |
2.5 | 0.3303 | 0.5230 | 0.6698 | 0.8012 | 0.9444 | 1.1259 | 1.3763 | 1.7634 | 2.5557 |
3.0 | 0.3303 | 0.5219 | 0.6619 | 0.7720 | 0.8721 | 0.9919 | 1.1725 | 1.4628 | 2.0209 |
β | λ | ||||||||
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
0.5 | 2.7920 | 12.243 | 38.665 | 103.06 | 229.88 | 417.28 | 621.34 | 779.36 | 615.37 |
1.0 | 0.3901 | 0.8140 | 1.4378 | 2.4502 | 4.2371 | 7.7429 | 15.802 | 40.249 | 177.94 |
1.5 | 0.2475 | 0.3692 | 0.4988 | 0.6724 | 0.9299 | 1.3485 | 2.1203 | 3.8858 | 10.421 |
2.0 | 0.2236 | 0.2727 | 0.3017 | 0.3478 | 0.4232 | 0.5408 | 0.7369 | 1.1271 | 2.2935 |
2.5 | 0.2213 | 0.2517 | 0.2384 | 0.2287 | 0.2486 | 0.3033 | 0.3864 | 0.5260 | 0.8911 |
3.0 | 0.2212 | 0.2496 | 0.2252 | 0.1869 | 0.1626 | 0.1789 | 0.2448 | 0.3256 | 0.4729 |
β | λ | ||||||||
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
0.5 | 4.6187 | 4.3533 | 3.9822 | 3.2831 | 2.4657 | 1.7555 | 1.2590 | 1.1461 | 2.0439 |
1.0 | 1.6228 | 1.4038 | 1.3664 | 1.3697 | 1.3821 | 1.3943 | 1.4037 | 1.4099 | 1.3511 |
1.5 | 0.9461 | 0.5856 | 0.5417 | 0.5744 | 0.6191 | 0.6586 | 0.6895 | 0.7124 | 0.7285 |
2.0 | 0.7459 | 0.1163 | -0.0092 | 0.1018 | 0.2223 | 0.2912 | 0.3312 | 0.3598 | 0.3836 |
2.5 | 0.7226 | -0.0655 | -0.4988 | -0.4984 | -0.1012 | 0.1683 | 0.1697 | 0.1591 | 0.1742 |
3.0 | 0.7218 | -0.0869 | -0.6648 | -1.0922 | -0.9663 | -0.0481 | 0.3705 | 0.0738 | 0.0476 |
β | λ | ||||||||
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
0.5 | 41.471 | 36.784 | 28.534 | 18.030 | 10.188 | 5.7950 | 3.7130 | 3.1485 | 5.9942 |
1.0 | 5.9572 | 5.6739 | 5.7391 | 5.8240 | 5.8909 | 5.9380 | 5.9690 | 5.9875 | 5.4737 |
1.5 | 2.5630 | 2.7503 | 3.1474 | 3.3764 | 3.4912 | 3.5508 | 3.5857 | 3.6093 | 3.6262 |
2.0 | 1.6369 | 1.5891 | 2.4907 | 3.0833 | 3.1595 | 3.0954 | 3.0599 | 3.0514 | 3.0542 |
2.5 | 1.5246 | 1.0744 | 1.8108 | 3.2354 | 3.9993 | 3.4076 | 2.8767 | 2.8838 | 2.8977 |
3.0 | 1.5211 | 1.0098 | 1.4967 | 2.7867 | 4.9278 | 5.5909 | 3.4605 | 2.4008 | 2.8802 |
β | λ | ||||||||
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
0.5 | 3.6720 | 6.4746 | 10.516 | 15.747 | 20.964 | 24.768 | 27.891 | 34.703 | 50.494 |
1.0 | 0.9866 | 1.0810 | 1.2335 | 1.4543 | 1.7756 | 2.2669 | 3.0939 | 4.7559 | 9.6437 |
1.5 | 0.7248 | 0.6337 | 0.6068 | 0.6165 | 0.6555 | 0.7276 | 0.8512 | 1.0826 | 1.6688 |
2.0 | 0.6749 | 0.5109 | 0.4258 | 0.3915 | 0.3859 | 0.3977 | 0.4269 | 0.4870 | 0.6379 |
2.5 | 0.6699 | 0.4813 | 0.3559 | 0.2854 | 0.2632 | 0.2694 | 0.2808 | 0.2983 | 0.3487 |
3.0 | 0.6697 | 0.4782 | 0.3402 | 0.2422 | 0.1865 | 0.1804 | 0.2088 | 0.2226 | 0.2340 |
From Tables 1‒5, we can conclude that the mean is an increasing function of λ and a decreasing function of β. It is clear that the skewness of the DsPA distribution can be positive or negative. The DI showing increasing behavior for larger values of the parameter λ and small values of β. Further, the DsPA distribution is suitable for over-dispersed and under-dispersed data sets.
The MRL function is a helpful reliability characteristic to model and analyze the burn-in and maintenance policies. Consider the rv X that has the cdf F(.). For a discrete rv, the MRL function is defined by
MRL=ε(i)=E(X−i|X≥i)=11−F(i−1,λ)w∑j=i+1[1−F(j−1,λ)],i∈N0, |
where N0={0,1,2,…,w} and 0<w<∞.
Then, the MRL of the DsPA model reduces to
MRL=11−F(i−1,λ,β)w∑j=i+1[1−F(j−1,λ,β)] |
=1[1−(i)βlnλ]λ(i)βw∑j=i+1[1−(j)βlnλ]λ(j)β |
=1[1−(i)βlnλ]λ(i)β[w∑j=i+1λ(j)β−lnλw∑j=i+1(j)βλ(j)β]. |
In this section, the parameters λ and β are estimated using the maximum likelihood (ML) method.
Suppose x1,…,xn be a random sample from the DsPA distribution with pmf (6). Then the log-likelihood function takes the form
L=1[1−(i)βlnλ]λ(i)βw∑j=i+1[1−(j)βlnλ]λ(j)β. | (14) |
Now, by differentiating (14) w.r.t λ and β, we can write
∂L∂λ=n∑i=1lnλ[(xi+1)2βλ(xi+1)β−xi2βλxiβ]λ{[1−xiβlnλ]λxiβ−[1−(xi+1)βlnλ]λ(xi+1)β}=0 | (15) |
and
∂L∂β=n∑i=1(lnλ)2[λ(xi+1)β(xi+1)2βln(xi+1)−λxiβxi2βlnxi]{[1−xiβlnλ]λxiβ−[1−(xi+1)βlnλ]λ(xi+1)β}=0. | (16) |
The ML estimates (MLEs) of λ and β follow from the above equation. Eqs (15) and (16) can be solved using iterative procedures such as Newton-Raphson. For this purpose, we use the maxLik function of R software [26].
In this section, we carried out a numerical simulation to access the performance of the ML estimation method. This assessment is done by generating N=10,000 samples using the qf of the DsPA model for different sample sizes n=10,20,50, and 100 and for several values of the parameters λ and β, where (λ,β)=(0.50,0.50),(0.50,2.0),(0.90,1.20),(0.90,2.0). The assessment is completed using absolute bias, mean relative errors (MREs), and mean square errors (MSEs) which are defined by
Bias(δ)=1NN∑i=1|ˆδi−δ|,MSE(δ)=1NN∑i=1(ˆδi−δ)2andMRE(δ)=1NN∑i=1ˆδiδi, |
where δ=(λ,β).
The simulation results for λ and β are reported in Tables 6‒8. The bias, MSE and MRE of the parameters λ and β are computed using the R program using the ML method. For all values of λ and β, the ML estimation approach illustrates the consistency property, that is, the MSEs and MREs decrease as n increases.
n | E(λ) | E(β) | Bias(λ) | Bias(β) | MSE(λ) | MSE(β) | MRE(λ) | MRE(β) |
10 | 0.4605 | 0.5804 | -0.0395 | 0.0804 | 0.0483 | 0.0333 | 0.0789 | 0.1609 |
20 | 0.4796 | 0.5370 | -0.0204 | 0.0370 | 0.0247 | 0.0111 | 0.0409 | 0.0740 |
50 | 0.4910 | 0.5143 | -0.0090 | 0.0143 | 0.0098 | 0.0033 | 0.0180 | 0.0286 |
100 | 0.4966 | 0.5065 | -0.0034 | 0.0065 | 0.0049 | 0.0015 | 0.0069 | 0.0130 |
200 | 0.4975 | 0.5037 | -0.0025 | 0.0037 | 0.0024 | 0.0007 | 0.0050 | 0.0074 |
n | E(λ) | E(β) | Bias(λ) | Bias(β) | MSE(λ) | MSE(β) | MRE(λ) | MRE(β) |
10 | 0.4654 | 2.3160 | -0.0346 | 0.3160 | 0.0490 | 0.5537 | 0.0692 | 0.1580 |
20 | 0.4809 | 2.1419 | -0.0191 | 0.1419 | 0.0243 | 0.1730 | 0.0382 | 0.0709 |
50 | 0.4905 | 2.0571 | -0.0095 | 0.0571 | 0.0097 | 0.0514 | 0.0191 | 0.0286 |
100 | 0.4970 | 2.0243 | -0.0030 | 0.0243 | 0.0050 | 0.0247 | 0.0060 | 0.0122 |
200 | 0.4984 | 2.0125 | -0.0016 | 0.0125 | 0.0024 | 0.0115 | 0.0032 | 0.0062 |
n | E(λ) | E(β) | Bias(λ) | Bias(β) | MSE(λ) | MSE(β) | MRE(λ) | MRE(β) |
10 | 0.8630 | 1.3915 | -0.0370 | 0.1915 | 0.0990 | 0.1973 | 0.0411 | 0.1596 |
20 | 0.8843 | 1.2834 | -0.0157 | 0.0834 | 0.0473 | 0.0630 | 0.0174 | 0.0695 |
50 | 0.8926 | 1.2310 | -0.0074 | 0.0310 | 0.0181 | 0.0190 | 0.0082 | 0.0258 |
100 | 0.8972 | 1.2154 | -0.0028 | 0.0154 | 0.0088 | 0.0086 | 0.0031 | 0.0129 |
200 | 0.8980 | 1.2076 | -0.0020 | 0.0076 | 0.0045 | 0.0043 | 0.0022 | 0.0063 |
From Tables 6‒8, we conclude that:
1. The estimates of λ and β close to their true values with the increase of n for all studied cases.
2. The MSEs for λ and β decrease with the increase of n for all studied cases.
3. The MREs for λ and β decrease with the increase of n for all studied cases.
n | E(λ) | E(β) | Bias(λ) | Bias(β) | MSE(λ) | MSE(β) | MRE(λ) | MRE(β) |
10 | 0.8690 | 2.3159 | -0.0310 | 0.3159 | 0.1005 | 0.5381 | 0.0344 | 0.1580 |
20 | 0.8785 | 2.1443 | -0.0215 | 0.1443 | 0.0454 | 0.1699 | 0.0239 | 0.0722 |
50 | 0.8941 | 2.0523 | -0.0059 | 0.0523 | 0.0183 | 0.0535 | 0.0065 | 0.0261 |
100 | 0.8957 | 2.0278 | -0.0043 | 0.0278 | 0.0089 | 0.0246 | 0.0048 | 0.0139 |
200 | 0.8980 | 2.0139 | -0.0020 | 0.0139 | 0.0044 | 0.0117 | 0.0022 | 0.0070 |
In this section, we illustrate the importance of the newly DsPA distribution by utilizing three real-life datasets. We shall compare the fits of the DsPA distribution with the following competing discrete distributions which are reported in Table 10.
Model | Abbreviation | pmf |
Discrete Bur-Ⅻ | DsBⅫ | P(x)=λln(1+xα)−λln(1+(1+x)α). |
Uniform Poisson–Ailamujia | UPA | P(x)=2λ(1+2λ)−x−1. |
Poisson | Poi | P(x)=e−λλxx!. |
Discrete-Pareto | DsPr | P(x)=e−λln(1+x)−e−λln(2+x). |
Discrete-Rayleigh | DsR | P(x)=e−x22λ2−e−(x+1)22λ2. |
Discrete inverse-Rayleigh | DsIR | P(x)=e−λ(1+x)2−e−λx2. |
Discrete Burr-Hutke | DsBH | P(x)=(1x+1−λx+2)λx. |
The fitted distributions are compared using the negative maximum log-likelihood (-Loglik.), Akaike information criterion (AIC), Bayesian information criterion (BIC), and the p-value of Kolmogorov–Smirnov test (KS p-value).
Dataset Ⅰ: The first dataset is about the failure times for a sample of 15 electronic components in an acceleration life test [27]. The data observations are: 1.0, 5.0, 6.0, 11.0, 12.0, 19.0, 20.0, 22.0, 23.0, 31.0, 37.0, 46.0, 54.0, 60.0 and 66.0.
Dataset Ⅱ: The second dataset is about the number of fires in Greece from July 1, 1998 to August 31, 1998. This dataset is studied [28]. The data observations are: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 15, 15, 15, 15, 16, 20 and 43.
Dataset Ⅲ: The third dataset consists of 48 final mathematics examination marks for slow-paced students in the Indian Institute of Technology at Kanpur. The data is analyzed by [29]. The observations are: 29, 25, 50, 15, 13, 27, 15, 18, 7, 7, 8, 19, 12, 18, 5, 21, 15, 86, 21, 15, 14, 39, 15, 14, 70, 44, 6, 23, 58, 19, 50, 23, 11, 6, 34, 18, 28, 34, 12, 37, 4, 60, 20, 23, 40, 65, 19 and 31.
The MLEs of the competing discrete models, standard errors (SEs), and goodness-of-fit measures are listed in Tables 11‒13 for the three datasets, respectively. For visual comparisons, the P-P (probability–probability) plots of fitted distributions are displayed in Figures 4, 6 and 8 for the analyzed datasets, respectively. Furthermore, the estimated cdf, sf, hrf of the DsPA distribution are depicted in Figures 5, 7 and 9, respectively.
Model | λ | α | Measures | |||||
MLE | SEs | MLE | SEs | -Loglik. | AIC | BIC | KS p-value | |
DsBⅫ | 0.9839 | 0.0355 | 20.868 | 46.483 | 75.69 | 155.38 | 156.80 | 0.0150 |
UPA | 0.0182 | 0.0047 | - | - | 65.00 | 132.00 | 132.71 | 0.6734 |
Poi | 27.535 | 1.3548 | - | - | 151.21 | 304.41 | 305.12 | 0.0180 |
DsPr | 0.3283 | 0.0848 | - | - | 77.40 | 156.80 | 157.51 | 0.0097 |
DsR | 24.384 | 3.1487 | - | - | 66.39 | 134.79 | 135.50 | 0.4300 |
DsIR | 42.021 | 11.243 | - | - | 83.99 | 169.97 | 170.68 | 0.0000 |
DsBH | 0.9992 | 0.0076 | - | - | 91.37 | 184.74 | 185.44 | 0.0000 |
DsPA | 0.8886 | 0.0674 | 0.8588 | 0.1738 | 64.49 | 131.58 | 132.97 | 0.9500 |
Model | λ | α | Measures | |||||
MLE | SE | MLE | SE | -Loglik. | AIC | BIC | KS p-value | |
DsBⅫ | 0.7612 | 0.0427 | 2.5026 | 0.4870 | 373.39 | 750.79 | 756.41 | 0.0000 |
UPA | 0.0926 | 0.0090 | - | - | 341.14 | 684.28 | 687.09 | 0.0028 |
Poi | 5.3988 | 0.2095 | - | - | 467.83 | 937.65 | 940.47 | 0.0000 |
DsPr | 0.6046 | 0.0546 | - | - | 389.64 | 781.27 | 784.08 | 0.0000 |
DsR | 5.6792 | 0.2567 | - | - | 385.25 | 772.49 | 775.31 | 0.0000 |
DsIR | 3.9959 | 0.3995 | - | - | 412.72 | 827.44 | 830.25 | 0.0000 |
DsBH | 0.9836 | 0.0127 | - | - | 407.16 | 816.31 | 819.12 | 0.0000 |
DsPA | 0.5812 | 0.0407 | 0.7709 | 0.0562 | 340.33 | 684.67 | 690.29 | 0.2484 |
Model | λ | α | Measures | |||||
MLE | SE | MLE | SE | -Loglik. | AIC | BIC | KS p-value | |
DsBⅫ | 0.9382 | 0.1926 | 5.1500 | 16.5597 | 247.48 | 498.97 | 502.71 | 0.0000 |
UPA | 0.0193 | 0.0028 | - | - | 205.11 | 412.22 | 414.09 | 0.0174 |
Poi | 25.8950 | 0.7345 | - | - | 396.59 | 795.18 | 797.05 | 0.0000 |
DsPr | 0.3225 | 0.0466 | - | - | 215.18 | 504.36 | 506.23 | 0.0000 |
DsR | 22.7562 | 1.6427 | - | - | 201.89 | 405.79 | 407.66 | 0.0460 |
DsIR | 177.56 | 26.02 | - | - | 205.13 | 412.27 | 414.14 | 0.0000 |
DsBH | 0.9990 | 0.0046 | - | - | 297.68 | 597.35 | 599.22 | 0.0000 |
DsPA | 0.9409 | 0.0231 | 1.0621 | 0.1113 | 197.44 | 398.88 | 402.62 | 0.8102 |
The findings in Tables 11‒13 illustrate that the DsPA distribution provides a superior fit over other competing discrete models, since it has the lowest values for all measures and the largest K-S p-value.
In this study, a new one-parameter discrete model is proposed as a good alternative to some well-known discrete distributions. The newly introduced model is called the discrete-power-Ailamujia (DsPA) distribution. Some statistical properties of the DsPA distribution are derived. Its parameters are estimated by the maximum likelihood method. A simulation study is carried out to check the performance of the estimators. It is observed that the maximum likelihood method is efficient in estimating the DsPA parameters for large samples. Finally, three real-world datasets are analyzed to check the usefulness and applicability of the DsPA distribution. The goodness-of-fit measures and figures show that the DsPA distribution is a useful attractive alternative for competing discrete models.
This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, Saudi Arabia under Grant No. (G: 047-662-1442). The authors, therefore, acknowledge with thanks DSR for technical and financial support.
The authors declare no conflict of interest.
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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, On Poisson Moment Exponential Distribution with Applications, 2022, 2198-5804, 10.1007/s40745-022-00400-0 | |
4. | 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 | |
5. | 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 | |
6. | Refah Alotaibi, Hoda Rezk, Chanseok Park, Ahmed Elshahhat, The Discrete Exponentiated-Chen Model and Its Applications, 2023, 15, 2073-8994, 1278, 10.3390/sym15061278 | |
7. | 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 | |
8. | 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 | |
9. | Muhammad Ahsan-ul-Haq, Muhammad N. S. Hussain, Ayesha Babar, Laila A. Al-Essa, Mohamed S. Eliwa, Barbara Martinucci, Analysis, Estimation, and Practical Implementations of the Discrete Power Quasi‐Xgamma Distribution, 2024, 2024, 2314-4629, 10.1155/2024/1913285 | |
10. | 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 | |
11. | Seth Borbye, Suleman Nasiru, Kingsley Kuwubasamni Ajongba, Vladimir Mityushev, Poisson XRani Distribution: An Alternative Discrete Distribution for Overdispersed Count Data, 2024, 2024, 0161-1712, 10.1155/2024/5554949 | |
12. | 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 |
β | λ | ||||||||
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
0.5 | 0.7603 | 1.8909 | 3.6766 | 6.5446 | 10.966 | 16.848 | 22.277 | 22.458 | 12.187 |
1.0 | 0.3954 | 0.7529 | 1.1657 | 1.6848 | 2.3863 | 3.4156 | 5.1075 | 8.4629 | 18.452 |
1.5 | 0.3415 | 0.5827 | 0.8221 | 1.0907 | 1.4187 | 1.8533 | 2.4910 | 3.5895 | 6.2446 |
2.0 | 0.3313 | 0.5338 | 0.7085 | 0.8884 | 1.0967 | 1.3599 | 1.7259 | 2.3141 | 3.5954 |
2.5 | 0.3303 | 0.5230 | 0.6698 | 0.8012 | 0.9444 | 1.1259 | 1.3763 | 1.7634 | 2.5557 |
3.0 | 0.3303 | 0.5219 | 0.6619 | 0.7720 | 0.8721 | 0.9919 | 1.1725 | 1.4628 | 2.0209 |
β | λ | ||||||||
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
0.5 | 2.7920 | 12.243 | 38.665 | 103.06 | 229.88 | 417.28 | 621.34 | 779.36 | 615.37 |
1.0 | 0.3901 | 0.8140 | 1.4378 | 2.4502 | 4.2371 | 7.7429 | 15.802 | 40.249 | 177.94 |
1.5 | 0.2475 | 0.3692 | 0.4988 | 0.6724 | 0.9299 | 1.3485 | 2.1203 | 3.8858 | 10.421 |
2.0 | 0.2236 | 0.2727 | 0.3017 | 0.3478 | 0.4232 | 0.5408 | 0.7369 | 1.1271 | 2.2935 |
2.5 | 0.2213 | 0.2517 | 0.2384 | 0.2287 | 0.2486 | 0.3033 | 0.3864 | 0.5260 | 0.8911 |
3.0 | 0.2212 | 0.2496 | 0.2252 | 0.1869 | 0.1626 | 0.1789 | 0.2448 | 0.3256 | 0.4729 |
β | λ | ||||||||
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
0.5 | 4.6187 | 4.3533 | 3.9822 | 3.2831 | 2.4657 | 1.7555 | 1.2590 | 1.1461 | 2.0439 |
1.0 | 1.6228 | 1.4038 | 1.3664 | 1.3697 | 1.3821 | 1.3943 | 1.4037 | 1.4099 | 1.3511 |
1.5 | 0.9461 | 0.5856 | 0.5417 | 0.5744 | 0.6191 | 0.6586 | 0.6895 | 0.7124 | 0.7285 |
2.0 | 0.7459 | 0.1163 | -0.0092 | 0.1018 | 0.2223 | 0.2912 | 0.3312 | 0.3598 | 0.3836 |
2.5 | 0.7226 | -0.0655 | -0.4988 | -0.4984 | -0.1012 | 0.1683 | 0.1697 | 0.1591 | 0.1742 |
3.0 | 0.7218 | -0.0869 | -0.6648 | -1.0922 | -0.9663 | -0.0481 | 0.3705 | 0.0738 | 0.0476 |
β | λ | ||||||||
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
0.5 | 41.471 | 36.784 | 28.534 | 18.030 | 10.188 | 5.7950 | 3.7130 | 3.1485 | 5.9942 |
1.0 | 5.9572 | 5.6739 | 5.7391 | 5.8240 | 5.8909 | 5.9380 | 5.9690 | 5.9875 | 5.4737 |
1.5 | 2.5630 | 2.7503 | 3.1474 | 3.3764 | 3.4912 | 3.5508 | 3.5857 | 3.6093 | 3.6262 |
2.0 | 1.6369 | 1.5891 | 2.4907 | 3.0833 | 3.1595 | 3.0954 | 3.0599 | 3.0514 | 3.0542 |
2.5 | 1.5246 | 1.0744 | 1.8108 | 3.2354 | 3.9993 | 3.4076 | 2.8767 | 2.8838 | 2.8977 |
3.0 | 1.5211 | 1.0098 | 1.4967 | 2.7867 | 4.9278 | 5.5909 | 3.4605 | 2.4008 | 2.8802 |
β | λ | ||||||||
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
0.5 | 3.6720 | 6.4746 | 10.516 | 15.747 | 20.964 | 24.768 | 27.891 | 34.703 | 50.494 |
1.0 | 0.9866 | 1.0810 | 1.2335 | 1.4543 | 1.7756 | 2.2669 | 3.0939 | 4.7559 | 9.6437 |
1.5 | 0.7248 | 0.6337 | 0.6068 | 0.6165 | 0.6555 | 0.7276 | 0.8512 | 1.0826 | 1.6688 |
2.0 | 0.6749 | 0.5109 | 0.4258 | 0.3915 | 0.3859 | 0.3977 | 0.4269 | 0.4870 | 0.6379 |
2.5 | 0.6699 | 0.4813 | 0.3559 | 0.2854 | 0.2632 | 0.2694 | 0.2808 | 0.2983 | 0.3487 |
3.0 | 0.6697 | 0.4782 | 0.3402 | 0.2422 | 0.1865 | 0.1804 | 0.2088 | 0.2226 | 0.2340 |
n | E(λ) | E(β) | Bias(λ) | Bias(β) | MSE(λ) | MSE(β) | MRE(λ) | MRE(β) |
10 | 0.4605 | 0.5804 | -0.0395 | 0.0804 | 0.0483 | 0.0333 | 0.0789 | 0.1609 |
20 | 0.4796 | 0.5370 | -0.0204 | 0.0370 | 0.0247 | 0.0111 | 0.0409 | 0.0740 |
50 | 0.4910 | 0.5143 | -0.0090 | 0.0143 | 0.0098 | 0.0033 | 0.0180 | 0.0286 |
100 | 0.4966 | 0.5065 | -0.0034 | 0.0065 | 0.0049 | 0.0015 | 0.0069 | 0.0130 |
200 | 0.4975 | 0.5037 | -0.0025 | 0.0037 | 0.0024 | 0.0007 | 0.0050 | 0.0074 |
n | E(λ) | E(β) | Bias(λ) | Bias(β) | MSE(λ) | MSE(β) | MRE(λ) | MRE(β) |
10 | 0.4654 | 2.3160 | -0.0346 | 0.3160 | 0.0490 | 0.5537 | 0.0692 | 0.1580 |
20 | 0.4809 | 2.1419 | -0.0191 | 0.1419 | 0.0243 | 0.1730 | 0.0382 | 0.0709 |
50 | 0.4905 | 2.0571 | -0.0095 | 0.0571 | 0.0097 | 0.0514 | 0.0191 | 0.0286 |
100 | 0.4970 | 2.0243 | -0.0030 | 0.0243 | 0.0050 | 0.0247 | 0.0060 | 0.0122 |
200 | 0.4984 | 2.0125 | -0.0016 | 0.0125 | 0.0024 | 0.0115 | 0.0032 | 0.0062 |
n | E(λ) | E(β) | Bias(λ) | Bias(β) | MSE(λ) | MSE(β) | MRE(λ) | MRE(β) |
10 | 0.8630 | 1.3915 | -0.0370 | 0.1915 | 0.0990 | 0.1973 | 0.0411 | 0.1596 |
20 | 0.8843 | 1.2834 | -0.0157 | 0.0834 | 0.0473 | 0.0630 | 0.0174 | 0.0695 |
50 | 0.8926 | 1.2310 | -0.0074 | 0.0310 | 0.0181 | 0.0190 | 0.0082 | 0.0258 |
100 | 0.8972 | 1.2154 | -0.0028 | 0.0154 | 0.0088 | 0.0086 | 0.0031 | 0.0129 |
200 | 0.8980 | 1.2076 | -0.0020 | 0.0076 | 0.0045 | 0.0043 | 0.0022 | 0.0063 |
n | E(λ) | E(β) | Bias(λ) | Bias(β) | MSE(λ) | MSE(β) | MRE(λ) | MRE(β) |
10 | 0.8690 | 2.3159 | -0.0310 | 0.3159 | 0.1005 | 0.5381 | 0.0344 | 0.1580 |
20 | 0.8785 | 2.1443 | -0.0215 | 0.1443 | 0.0454 | 0.1699 | 0.0239 | 0.0722 |
50 | 0.8941 | 2.0523 | -0.0059 | 0.0523 | 0.0183 | 0.0535 | 0.0065 | 0.0261 |
100 | 0.8957 | 2.0278 | -0.0043 | 0.0278 | 0.0089 | 0.0246 | 0.0048 | 0.0139 |
200 | 0.8980 | 2.0139 | -0.0020 | 0.0139 | 0.0044 | 0.0117 | 0.0022 | 0.0070 |
Model | Abbreviation | pmf |
Discrete Bur-Ⅻ | DsBⅫ | P(x)=λln(1+xα)−λln(1+(1+x)α). |
Uniform Poisson–Ailamujia | UPA | P(x)=2λ(1+2λ)−x−1. |
Poisson | Poi | P(x)=e−λλxx!. |
Discrete-Pareto | DsPr | P(x)=e−λln(1+x)−e−λln(2+x). |
Discrete-Rayleigh | DsR | P(x)=e−x22λ2−e−(x+1)22λ2. |
Discrete inverse-Rayleigh | DsIR | P(x)=e−λ(1+x)2−e−λx2. |
Discrete Burr-Hutke | DsBH | P(x)=(1x+1−λx+2)λx. |
Model | λ | α | Measures | |||||
MLE | SEs | MLE | SEs | -Loglik. | AIC | BIC | KS p-value | |
DsBⅫ | 0.9839 | 0.0355 | 20.868 | 46.483 | 75.69 | 155.38 | 156.80 | 0.0150 |
UPA | 0.0182 | 0.0047 | - | - | 65.00 | 132.00 | 132.71 | 0.6734 |
Poi | 27.535 | 1.3548 | - | - | 151.21 | 304.41 | 305.12 | 0.0180 |
DsPr | 0.3283 | 0.0848 | - | - | 77.40 | 156.80 | 157.51 | 0.0097 |
DsR | 24.384 | 3.1487 | - | - | 66.39 | 134.79 | 135.50 | 0.4300 |
DsIR | 42.021 | 11.243 | - | - | 83.99 | 169.97 | 170.68 | 0.0000 |
DsBH | 0.9992 | 0.0076 | - | - | 91.37 | 184.74 | 185.44 | 0.0000 |
DsPA | 0.8886 | 0.0674 | 0.8588 | 0.1738 | 64.49 | 131.58 | 132.97 | 0.9500 |
Model | λ | α | Measures | |||||
MLE | SE | MLE | SE | -Loglik. | AIC | BIC | KS p-value | |
DsBⅫ | 0.7612 | 0.0427 | 2.5026 | 0.4870 | 373.39 | 750.79 | 756.41 | 0.0000 |
UPA | 0.0926 | 0.0090 | - | - | 341.14 | 684.28 | 687.09 | 0.0028 |
Poi | 5.3988 | 0.2095 | - | - | 467.83 | 937.65 | 940.47 | 0.0000 |
DsPr | 0.6046 | 0.0546 | - | - | 389.64 | 781.27 | 784.08 | 0.0000 |
DsR | 5.6792 | 0.2567 | - | - | 385.25 | 772.49 | 775.31 | 0.0000 |
DsIR | 3.9959 | 0.3995 | - | - | 412.72 | 827.44 | 830.25 | 0.0000 |
DsBH | 0.9836 | 0.0127 | - | - | 407.16 | 816.31 | 819.12 | 0.0000 |
DsPA | 0.5812 | 0.0407 | 0.7709 | 0.0562 | 340.33 | 684.67 | 690.29 | 0.2484 |
Model | λ | α | Measures | |||||
MLE | SE | MLE | SE | -Loglik. | AIC | BIC | KS p-value | |
DsBⅫ | 0.9382 | 0.1926 | 5.1500 | 16.5597 | 247.48 | 498.97 | 502.71 | 0.0000 |
UPA | 0.0193 | 0.0028 | - | - | 205.11 | 412.22 | 414.09 | 0.0174 |
Poi | 25.8950 | 0.7345 | - | - | 396.59 | 795.18 | 797.05 | 0.0000 |
DsPr | 0.3225 | 0.0466 | - | - | 215.18 | 504.36 | 506.23 | 0.0000 |
DsR | 22.7562 | 1.6427 | - | - | 201.89 | 405.79 | 407.66 | 0.0460 |
DsIR | 177.56 | 26.02 | - | - | 205.13 | 412.27 | 414.14 | 0.0000 |
DsBH | 0.9990 | 0.0046 | - | - | 297.68 | 597.35 | 599.22 | 0.0000 |
DsPA | 0.9409 | 0.0231 | 1.0621 | 0.1113 | 197.44 | 398.88 | 402.62 | 0.8102 |
β | λ | ||||||||
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
0.5 | 0.7603 | 1.8909 | 3.6766 | 6.5446 | 10.966 | 16.848 | 22.277 | 22.458 | 12.187 |
1.0 | 0.3954 | 0.7529 | 1.1657 | 1.6848 | 2.3863 | 3.4156 | 5.1075 | 8.4629 | 18.452 |
1.5 | 0.3415 | 0.5827 | 0.8221 | 1.0907 | 1.4187 | 1.8533 | 2.4910 | 3.5895 | 6.2446 |
2.0 | 0.3313 | 0.5338 | 0.7085 | 0.8884 | 1.0967 | 1.3599 | 1.7259 | 2.3141 | 3.5954 |
2.5 | 0.3303 | 0.5230 | 0.6698 | 0.8012 | 0.9444 | 1.1259 | 1.3763 | 1.7634 | 2.5557 |
3.0 | 0.3303 | 0.5219 | 0.6619 | 0.7720 | 0.8721 | 0.9919 | 1.1725 | 1.4628 | 2.0209 |
β | λ | ||||||||
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
0.5 | 2.7920 | 12.243 | 38.665 | 103.06 | 229.88 | 417.28 | 621.34 | 779.36 | 615.37 |
1.0 | 0.3901 | 0.8140 | 1.4378 | 2.4502 | 4.2371 | 7.7429 | 15.802 | 40.249 | 177.94 |
1.5 | 0.2475 | 0.3692 | 0.4988 | 0.6724 | 0.9299 | 1.3485 | 2.1203 | 3.8858 | 10.421 |
2.0 | 0.2236 | 0.2727 | 0.3017 | 0.3478 | 0.4232 | 0.5408 | 0.7369 | 1.1271 | 2.2935 |
2.5 | 0.2213 | 0.2517 | 0.2384 | 0.2287 | 0.2486 | 0.3033 | 0.3864 | 0.5260 | 0.8911 |
3.0 | 0.2212 | 0.2496 | 0.2252 | 0.1869 | 0.1626 | 0.1789 | 0.2448 | 0.3256 | 0.4729 |
β | λ | ||||||||
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
0.5 | 4.6187 | 4.3533 | 3.9822 | 3.2831 | 2.4657 | 1.7555 | 1.2590 | 1.1461 | 2.0439 |
1.0 | 1.6228 | 1.4038 | 1.3664 | 1.3697 | 1.3821 | 1.3943 | 1.4037 | 1.4099 | 1.3511 |
1.5 | 0.9461 | 0.5856 | 0.5417 | 0.5744 | 0.6191 | 0.6586 | 0.6895 | 0.7124 | 0.7285 |
2.0 | 0.7459 | 0.1163 | -0.0092 | 0.1018 | 0.2223 | 0.2912 | 0.3312 | 0.3598 | 0.3836 |
2.5 | 0.7226 | -0.0655 | -0.4988 | -0.4984 | -0.1012 | 0.1683 | 0.1697 | 0.1591 | 0.1742 |
3.0 | 0.7218 | -0.0869 | -0.6648 | -1.0922 | -0.9663 | -0.0481 | 0.3705 | 0.0738 | 0.0476 |
β | λ | ||||||||
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
0.5 | 41.471 | 36.784 | 28.534 | 18.030 | 10.188 | 5.7950 | 3.7130 | 3.1485 | 5.9942 |
1.0 | 5.9572 | 5.6739 | 5.7391 | 5.8240 | 5.8909 | 5.9380 | 5.9690 | 5.9875 | 5.4737 |
1.5 | 2.5630 | 2.7503 | 3.1474 | 3.3764 | 3.4912 | 3.5508 | 3.5857 | 3.6093 | 3.6262 |
2.0 | 1.6369 | 1.5891 | 2.4907 | 3.0833 | 3.1595 | 3.0954 | 3.0599 | 3.0514 | 3.0542 |
2.5 | 1.5246 | 1.0744 | 1.8108 | 3.2354 | 3.9993 | 3.4076 | 2.8767 | 2.8838 | 2.8977 |
3.0 | 1.5211 | 1.0098 | 1.4967 | 2.7867 | 4.9278 | 5.5909 | 3.4605 | 2.4008 | 2.8802 |
β | λ | ||||||||
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
0.5 | 3.6720 | 6.4746 | 10.516 | 15.747 | 20.964 | 24.768 | 27.891 | 34.703 | 50.494 |
1.0 | 0.9866 | 1.0810 | 1.2335 | 1.4543 | 1.7756 | 2.2669 | 3.0939 | 4.7559 | 9.6437 |
1.5 | 0.7248 | 0.6337 | 0.6068 | 0.6165 | 0.6555 | 0.7276 | 0.8512 | 1.0826 | 1.6688 |
2.0 | 0.6749 | 0.5109 | 0.4258 | 0.3915 | 0.3859 | 0.3977 | 0.4269 | 0.4870 | 0.6379 |
2.5 | 0.6699 | 0.4813 | 0.3559 | 0.2854 | 0.2632 | 0.2694 | 0.2808 | 0.2983 | 0.3487 |
3.0 | 0.6697 | 0.4782 | 0.3402 | 0.2422 | 0.1865 | 0.1804 | 0.2088 | 0.2226 | 0.2340 |
n | E(λ) | E(β) | Bias(λ) | Bias(β) | MSE(λ) | MSE(β) | MRE(λ) | MRE(β) |
10 | 0.4605 | 0.5804 | -0.0395 | 0.0804 | 0.0483 | 0.0333 | 0.0789 | 0.1609 |
20 | 0.4796 | 0.5370 | -0.0204 | 0.0370 | 0.0247 | 0.0111 | 0.0409 | 0.0740 |
50 | 0.4910 | 0.5143 | -0.0090 | 0.0143 | 0.0098 | 0.0033 | 0.0180 | 0.0286 |
100 | 0.4966 | 0.5065 | -0.0034 | 0.0065 | 0.0049 | 0.0015 | 0.0069 | 0.0130 |
200 | 0.4975 | 0.5037 | -0.0025 | 0.0037 | 0.0024 | 0.0007 | 0.0050 | 0.0074 |
n | E(λ) | E(β) | Bias(λ) | Bias(β) | MSE(λ) | MSE(β) | MRE(λ) | MRE(β) |
10 | 0.4654 | 2.3160 | -0.0346 | 0.3160 | 0.0490 | 0.5537 | 0.0692 | 0.1580 |
20 | 0.4809 | 2.1419 | -0.0191 | 0.1419 | 0.0243 | 0.1730 | 0.0382 | 0.0709 |
50 | 0.4905 | 2.0571 | -0.0095 | 0.0571 | 0.0097 | 0.0514 | 0.0191 | 0.0286 |
100 | 0.4970 | 2.0243 | -0.0030 | 0.0243 | 0.0050 | 0.0247 | 0.0060 | 0.0122 |
200 | 0.4984 | 2.0125 | -0.0016 | 0.0125 | 0.0024 | 0.0115 | 0.0032 | 0.0062 |
n | E(λ) | E(β) | Bias(λ) | Bias(β) | MSE(λ) | MSE(β) | MRE(λ) | MRE(β) |
10 | 0.8630 | 1.3915 | -0.0370 | 0.1915 | 0.0990 | 0.1973 | 0.0411 | 0.1596 |
20 | 0.8843 | 1.2834 | -0.0157 | 0.0834 | 0.0473 | 0.0630 | 0.0174 | 0.0695 |
50 | 0.8926 | 1.2310 | -0.0074 | 0.0310 | 0.0181 | 0.0190 | 0.0082 | 0.0258 |
100 | 0.8972 | 1.2154 | -0.0028 | 0.0154 | 0.0088 | 0.0086 | 0.0031 | 0.0129 |
200 | 0.8980 | 1.2076 | -0.0020 | 0.0076 | 0.0045 | 0.0043 | 0.0022 | 0.0063 |
n | E(λ) | E(β) | Bias(λ) | Bias(β) | MSE(λ) | MSE(β) | MRE(λ) | MRE(β) |
10 | 0.8690 | 2.3159 | -0.0310 | 0.3159 | 0.1005 | 0.5381 | 0.0344 | 0.1580 |
20 | 0.8785 | 2.1443 | -0.0215 | 0.1443 | 0.0454 | 0.1699 | 0.0239 | 0.0722 |
50 | 0.8941 | 2.0523 | -0.0059 | 0.0523 | 0.0183 | 0.0535 | 0.0065 | 0.0261 |
100 | 0.8957 | 2.0278 | -0.0043 | 0.0278 | 0.0089 | 0.0246 | 0.0048 | 0.0139 |
200 | 0.8980 | 2.0139 | -0.0020 | 0.0139 | 0.0044 | 0.0117 | 0.0022 | 0.0070 |
Model | Abbreviation | pmf |
Discrete Bur-Ⅻ | DsBⅫ | P(x)=λln(1+xα)−λln(1+(1+x)α). |
Uniform Poisson–Ailamujia | UPA | P(x)=2λ(1+2λ)−x−1. |
Poisson | Poi | P(x)=e−λλxx!. |
Discrete-Pareto | DsPr | P(x)=e−λln(1+x)−e−λln(2+x). |
Discrete-Rayleigh | DsR | P(x)=e−x22λ2−e−(x+1)22λ2. |
Discrete inverse-Rayleigh | DsIR | P(x)=e−λ(1+x)2−e−λx2. |
Discrete Burr-Hutke | DsBH | P(x)=(1x+1−λx+2)λx. |
Model | λ | α | Measures | |||||
MLE | SEs | MLE | SEs | -Loglik. | AIC | BIC | KS p-value | |
DsBⅫ | 0.9839 | 0.0355 | 20.868 | 46.483 | 75.69 | 155.38 | 156.80 | 0.0150 |
UPA | 0.0182 | 0.0047 | - | - | 65.00 | 132.00 | 132.71 | 0.6734 |
Poi | 27.535 | 1.3548 | - | - | 151.21 | 304.41 | 305.12 | 0.0180 |
DsPr | 0.3283 | 0.0848 | - | - | 77.40 | 156.80 | 157.51 | 0.0097 |
DsR | 24.384 | 3.1487 | - | - | 66.39 | 134.79 | 135.50 | 0.4300 |
DsIR | 42.021 | 11.243 | - | - | 83.99 | 169.97 | 170.68 | 0.0000 |
DsBH | 0.9992 | 0.0076 | - | - | 91.37 | 184.74 | 185.44 | 0.0000 |
DsPA | 0.8886 | 0.0674 | 0.8588 | 0.1738 | 64.49 | 131.58 | 132.97 | 0.9500 |
Model | λ | α | Measures | |||||
MLE | SE | MLE | SE | -Loglik. | AIC | BIC | KS p-value | |
DsBⅫ | 0.7612 | 0.0427 | 2.5026 | 0.4870 | 373.39 | 750.79 | 756.41 | 0.0000 |
UPA | 0.0926 | 0.0090 | - | - | 341.14 | 684.28 | 687.09 | 0.0028 |
Poi | 5.3988 | 0.2095 | - | - | 467.83 | 937.65 | 940.47 | 0.0000 |
DsPr | 0.6046 | 0.0546 | - | - | 389.64 | 781.27 | 784.08 | 0.0000 |
DsR | 5.6792 | 0.2567 | - | - | 385.25 | 772.49 | 775.31 | 0.0000 |
DsIR | 3.9959 | 0.3995 | - | - | 412.72 | 827.44 | 830.25 | 0.0000 |
DsBH | 0.9836 | 0.0127 | - | - | 407.16 | 816.31 | 819.12 | 0.0000 |
DsPA | 0.5812 | 0.0407 | 0.7709 | 0.0562 | 340.33 | 684.67 | 690.29 | 0.2484 |
Model | λ | α | Measures | |||||
MLE | SE | MLE | SE | -Loglik. | AIC | BIC | KS p-value | |
DsBⅫ | 0.9382 | 0.1926 | 5.1500 | 16.5597 | 247.48 | 498.97 | 502.71 | 0.0000 |
UPA | 0.0193 | 0.0028 | - | - | 205.11 | 412.22 | 414.09 | 0.0174 |
Poi | 25.8950 | 0.7345 | - | - | 396.59 | 795.18 | 797.05 | 0.0000 |
DsPr | 0.3225 | 0.0466 | - | - | 215.18 | 504.36 | 506.23 | 0.0000 |
DsR | 22.7562 | 1.6427 | - | - | 201.89 | 405.79 | 407.66 | 0.0460 |
DsIR | 177.56 | 26.02 | - | - | 205.13 | 412.27 | 414.14 | 0.0000 |
DsBH | 0.9990 | 0.0046 | - | - | 297.68 | 597.35 | 599.22 | 0.0000 |
DsPA | 0.9409 | 0.0231 | 1.0621 | 0.1113 | 197.44 | 398.88 | 402.62 | 0.8102 |