On Maxwell–Lomax distribution: properties and applications

Figures & data

Figure 1. Plots of pdf of M–L distribution.

Figure 2. Plots of hazard function of M–L distribution.

Table 1. Skewness and Kurtosis results of the M–L distribution setting $\mathrm{\lambda }$= 0.05 and θ = 0.05.

$Q(.)$	β = 1.5	β = 2.5	β = 3.5	β = 4.5	β = 5.5
¼	0.7275	0.4334	0.3086	0.2396	0.1958
½	1.0127	0.6016	0.4279	0.3320	0.2713
¾	1.3294	0.7874	0.5593	0.4337	0.3541
1/8	0.5509	0.3288	0.2343	0.1820	0.1488
2/8	0.7275	0.4334	0.3086	0.2396	0.1958
3/8	0.8735	0.5196	0.3698	0.2870	0.2345
5/8	1.1590	0.6876	0.4887	0.3791	0.3096
6/8	1.3294	0.7874	0.5593	0.4337	0.3541
7/8	1.5668	0.9259	0.6571	0.5092	0.4157
Skewness	0.0523	0.0494	0.0482	0.0475	0.0471
Kurtosis	0.1417	0.1342	0.1310	0.1292	0.1281

Table 2. Skewness and Kurtosis results of the M–L distribution setting $\mathrm{\lambda }$=5 and θ = 1.6.

$Q(.)$	β = 1.9	β = 2.9	β = 3.9	β = 4.9	β = 5.9
¼	1.0497	0.5671	0.3852	0.2909	0.2335
½	1.3254	0.6923	0.4631	0.3467	0.2767
¾	1.5971	0.8098	0.5345	0.3971	0.3153
1/8	0.8574	0.4756	0.3270	0.2486	0.2004
2/8	1.0497	0.5671	0.3852	0.2909	0.2335
3/8	1.1952	0.6340	0.4271	0.3210	0.2568
5/8	1.4546	0.7489	0.4976	0.3712	0.2955
6/8	1.5971	0.8098	0.5345	0.3971	0.3153
7/8	1.7837	0.8877	0.5809	0.4296	0.3401
Skewness	−0.0074	−0.0317	−0.0435	−0.0505	−0.0551
Kurtosis	−0.0158	−0.0808	−0.1127	−0.1317	−0.1443

Figure 3. Plots of skewness and kurtosis of M–L distribution for different parameter values.

Table 3. Simulation results of the M–L distribution setting $\lambda =0.05,$$\beta =1.5$ and $\theta =0.05.$

Parameter	n	Estimate ($\widehat{\Omega }$)	Bias	MSE
	10	0.3689	0.3189	0.7535
	20	0.2359	0.1858	0.5403
$\lambda$= 0.05	30	0.1580	0.1080	0.4329
	50	0.1265	0.0765	0.3925
	100	0.0649	0.0149	0.3182
	200	0.0514	0.0014	0.3020
	10	6.0617	4.5617	25.6208
	20	5.5131	4.0131	20.3680
$\beta$= 1.5	30	4.0720	2.5720	9.4371
	50	3.4164	1.9164	5.8389
	100	2.0230	0.5230	1.0465
	200	1.6140	0.1140	0.3769
	10	0.1429	0.0929	0.3515
	20	0.1187	0.0687	0.3234
$\theta$= 0.05	30	0.1033	0.0533	0.3061
	50	0.0936	0.0436	0.2955
	100	0.0637	0.0137	0.2638
	200	0.0526	0.0026	0.2526

Table 4. Simulation results of the M–L distribution setting $\lambda =5,$$\beta =1.9$ and $\theta =0.05.$

Parameter	n	Estimate ($\widehat{\Omega }$)	Bias	MSE
	10	10.2406	5.2406	32.9544
	20	9.5269	4.5269	25.2697
$\lambda$= 5	30	7.7501	2.7501	10.5631
	50	6.8224	1.8224	5.3935
	100	6.2875	1.2875	3.1951
	200	5.0802	0.0802	0.3366
	10	6.1413	4.2413	22.4799
	20	4.9038	3.0038	12.2766
$\beta$= 1.9	30	4.2645	2.3645	8.2053
	50	3.4131	1.5131	4.0525
	100	2.6180	0.7180	1.4835
	200	1.9083	0.0083	0.2583
	10	3.0858	1.4858	3.9434
	20	2.5877	0.9877	2.2132
$\theta$= 1.6	30	2.2882	0.6882	1.4118
	50	2.1716	0.5716	1.1483
	100	1.8564	0.2564	0.5721
	200	1.7072	0.1072	0.3686

Table

Sample size (n)	107
Mean	0.4689
Standard Deviation	0.1920
Minimum	0.0168
Maximum	0.8781
Skewness	−0.3353
Kurtosis	2.6861

Table

Sample size (n)	128
Mean	9.366
Standard Deviation	10.5083
Minimum	0.080
Maximum	79.050
Skewness	3.2866
Kurtosis	18.4831

Figure 4. TTT plots of dataset 1 (a) dataset 2 (b).

Figure 5. Fitted densities of dataset 1.

Figure 6. Fitted densities of dataset 2.

Table 5. MLEs and Goodness-of-fit-statistics for dataset 1.

	M-L		E-L		MO-L		R-L
	Estimate	SE	Estimate	SE	Estimate	SE	Estimate	SE
$\widehat{\lambda }$	0.1986	0.1493	3.7420	0.5713	0.3200	0.0400	1.7512	0.3871
$\widehat{\theta }$	0.0270	0.0194	0.0061	0.0006	9.7630	0.2963	0.0021	0.0003
$\widehat{\beta }$	21.2522	9.5270	63.9000	29.9100	3.4350	0.1377	0.0804	0.0105
$\ell$	19.1700		4.9673		2.7547		−244.4227
$\mathit{AIC}$	−32.3400		−3.9347		0.4906		494.8454
$\mathit{CAIC}$	−32.1070		−3.7016		0.7236		495.0784
$\mathit{BIC}$	−24.3216		4.0838		8.5091		502.8639
$\mathit{HQIC}$	−29.0895		−0.6841		3.7412		498.0960
$A^{*}$	1.5819		6.0008		6.8619		2.0899
$W^{*}$	0.2412		1.0318		1.1895		0.3148

Table 6. MLEs and goodness-of-fit-statistics for dataset 2.

	M-L			E-L				MO-L		R-L
	Estimate		SE	Estimate		SE		Estimate	SE	Estimate		SE
$\widehat{\lambda }$	3.2684		1.0278	1.2200		0.1497		0.0024	0.0012	70.6600		3.8390
$\widehat{\theta }$	24.4847		8.8895	0.4321		0.0195		172.6000	14.4320	0.0015		0.0009
$\widehat{\beta }$	0.3550		0.0370	10.0000		4.2040		1.7860	0.1378	0.5238		0.0349
$\ell$		−411.5780			−413.0452		−412.3371				−414.1033
AIC		829.1560			832.0904		830.6742				834.2066
CAIC		829.3495			832.2839		830.8677				834.4001
BIC		837.7121			840.6465		839.2303				842.7627
HQIC		832.6324			835.5668		834.1506				837.6830
$A^{*}$		0.4028			0.5418		1.1898				1.4101
$W^{*}$		0.0586			0.0775		0.1788				0.2279