Type II Topp Leone Power Lomax Distribution with Applications
Abstract
:1. Introduction
1.1. From the Lomax Distribution to the Power Lomax Distribution
1.2. On the Extensions of the PL Distribution through the Use of General Families of Distributions
1.3. Novelty and Contributions
- To improve the characteristics and flexibility of the PL distribution by using the TIITL-G family (as motivated above). In particular, increasing, decreasing, J, and reverse J shapes are observed for the probability density and hazard rate functions, illustrating this claim.
- To introduce an extended version of the PL distribution whose quantile function has a closed-form.
- To study important statistical properties of the TIITLPL distribution, including the skewness, kurtosis, and various kinds of moments and order statistics.
- To explore the inferential features of the TIITLPL distribution through the use of the maximum likelihood method, providing a comprehensive methodology for the practitioner.
1.4. Paper Organization
2. The TIITLPL Distribution
2.1. Definition
2.2. Common Reliability Functions
3. Mathematical Properties
3.1. Quantile Function and Applications
3.1.1. Definition
3.1.2. Generated Values
3.1.3. Some Related Functions
3.1.4. Skewness and Kurtosis Based on the Quantile Function
3.2. Useful Expansions
3.3. Moments
3.3.1. Ordinary moments
3.3.2. Incomplete Moments
3.4. Order Statistics
- The limiting distribution of is the Weibull distribution with the following cdf: , , i.e., there exist and such that
- The limiting distribution of is the Fréchet distribution with the following cdf: , , i.e., there exist and such that
4. Inference
4.1. Maximum Likelihood Estimation Method
4.2. Simulation
- One thousand random samples of size , 100, 200, and 500 are generated from the TIITLPL distribution by the use of the quantile function (see Section 3.1.2).
- Eight different sets of values of true parameters , , , and in order, are taken asSet 1 , Set 2 , Set 3 ,Set 4 , Set 5 , Set 6 ,Set 7 , Set 8 .
- The mean of the obtained MLEs (Estimates) and the mean squared errors (MSEs) for the selected sets of parameters are calculated.
5. Applications
5.1. Aircraft Windshield Data
5.2. Cancer Patient Data
6. Final Remarks
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
cdf | cumulative distribution function |
probability distribution function | |
PL | Power Lomax |
TIITL-G | type II Topp-Leone-G |
TIITLPL | type II Topp-Leone power Lomax |
sf | survival function |
hrf | hazard rate function |
MLE | maximum likelihood estimate |
MSE | mean squared error |
AIC | Akaike information criterion |
CAIC | consistent Akaike information criterion |
HQIC | Hannan-Quinn information criterion |
Anderson-Darling | |
Cramér-von Mises | |
TTT | total time on test |
GL | gamma Lomax |
BL | beta Lomax |
TWL | Transmuted Weibull Lomax |
KwW | Kumaraswamy Weibull |
McW | McDonald Weibull |
BW | beta Weibull |
TLL | transmuted log-logistic |
LL | log-logistic |
TMOFr | transmuted Marshall-Olkin Fréchet |
SE | standard error |
TMW | transmuted modified Weibull |
TAW | transmuted additive Weibull |
GIG | generalized inverse gamma |
BEBXII | beta exponential Burr XII |
BFr | beta Fréchet |
KwLL | Kumaraswamy log-logistic |
TCWG | transmuted complementary Weibull geometric |
KwEBXII | Kumaraswamy exponentiated Burr XII |
GTW | generalized transmuted-W |
ETGR | exponentiated transmuted generalized Rayleigh |
Nomenclature | |
cdf of the Lomax distribution | |
pdf of the Lomax distribution | |
cdf of the power Lomax distribution | |
pdf of the power Lomax distribution | |
cdf of the TIITL-G family | |
pdf of the TIITL-G family | |
cdf of the TIITLPL distribution | |
pdf of the TIITLPL distribution | |
sf of the TIITLPL distribution | |
hrf of the TIITLPL distribution | |
reverse hazard rate function of the TIITLPL distribution | |
cumulative hazard rate function of the TIITLPL distribution | |
quantile function of the TIITLPL distribution | |
M | Median of the TIITLPL distribution |
quantile density function of the TIITLPL distribution | |
hazard quantile function of the TIITLPL distribution | |
S | Galton skewness of the TIITLPL distribution |
K | Moors kurtosis of the TIITLPL distribution |
coefficient in the considered series expansion for | |
coefficient in the considered series expansion for | |
sth ordinary moment of the TIITLPL distribution | |
coefficient in the considered series expansion for | |
sth central moment of the TIITLPL distribution | |
sth cumulant of the TIITLPL distribution | |
sth incomplete moment taken on t of the TIITLPL distribution | |
mean deviation about the mean of the TIITLPL distribution | |
mean deviation about the median of the TIITLPL distribution | |
lower conditional moment taken on t of the TIITLPL distribution | |
upper conditional moment taken on t of the TIITLPL distribution | |
ith order statistic of the TIITLPL distribution | |
pdf of the ith order statistic of the TIITLPL distribution | |
coefficient in the considered series expansion for | |
sth moment of the ith order statistic of the TIITLPL distribution | |
likelihood function of the TIITLPL distribution | |
log-likelihood function of the TIITLPL distribution | |
, , and | MLEs of , , and , respectively |
n | size of the considered sample or the number of data |
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M | |
---|---|
540.1999 | |
54019.99 | |
1800.6660 | |
232.4650 | |
141.8530 | |
1206.2180 | |
26.97185 | |
7.6697 | |
2.8015 | |
9.6382 | |
0.0358 | |
0.0282 | |
0.0077 |
6.5310 | 329.8982 | 21948 | 1642994 | |
3.0469 | 83.6055 | 4496.53 | 304584 | |
0.2716 | 0.8662 | 18.6814 | 905.1954 | |
0.03667 | 0.0061 | 0.0038 | 0.0111 | |
0.0210 | 0.0017 | 0.0004 | 0.0003 | |
1.2956 | 6.6656 | 141.9529 | 6614.0360 | |
0.1528 | 0.03667 | 0.0126 | 0.0061 | |
0.3661 | 0.1528 | 0.0712 | 0.03667 | |
0.6575 | 0.4426 | 0.3044 | 0.2139 | |
0.1467 | 0.0980 | 0.2437 | 2.3221 | |
3.6090 | 51.7157 | 1679.7620 | 84843.84 | |
1.0420 | 1.1118 | 1.2121 | 1.3482 | |
0.8527 | 0.7423 | 0.6583 | 0.5939 |
Set 1 | Set 2 | Set 3 | Set 4 | |||||
---|---|---|---|---|---|---|---|---|
n | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs |
50 | 0.471 | 0.090 | 0.602 | 0.091 | 0.587 | 0.058 | 0.554 | 0.128 |
0.537 | 0.238 | 0.663 | 0.223 | 2.145 | 2.429 | 1.614 | 1.138 | |
0.735 | 0.223 | 1.758 | 0.587 | 0.920 | 0.250 | 2.122 | 1.636 | |
0.771 | 0.431 | 0.662 | 0.344 | 1.790 | 1.041 | 1.650 | 1.552 | |
100 | 0.504 | 0.084 | 0.644 | 0.053 | 0.574 | 0.033 | 0.609 | 0.083 |
0.428 | 0.085 | 0.669 | 0.128 | 1.747 | 0.446 | 1.739 | 0.510 | |
0.660 | 0.122 | 1.633 | 0.258 | 0.865 | 0.115 | 1.665 | 0.239 | |
0.656 | 0.245 | 0.474 | 0.088 | 1.479 | 0.284 | 1.310 | 1.088 | |
200 | 0.525 | 0.081 | 0.617 | 0.046 | 0.577 | 0.017 | 0.642 | 0.058 |
0.546 | 0.075 | 0.585 | 0.041 | 1.623 | 0.141 | 1.719 | 0.199 | |
0.558 | 0.045 | 1.561 | 0.124 | 0.832 | 0.025 | 1.548 | 0.085 | |
0.685 | 0.233 | 0.475 | 0.071 | 1.309 | 0.149 | 1.066 | 0.293 | |
500 | 0.504 | 0.061 | 0.647 | 0.035 | 0.592 | 0.013 | 0.660 | 0.051 |
0.477 | 0.051 | 0.612 | 0.027 | 1.620 | 0.058 | 1.729 | 0.137 | |
0.539 | 0.013 | 1.449 | 0.035 | 0.815 | 5.258 * | 1.515 | 0.023 | |
0.636 | 0.169 | 0.419 | 0.018 | 1.226 | 0.106 | 0.942 | 0.141 |
Set 5 | Set 6 | Set 7 | Set 8 | |||||
---|---|---|---|---|---|---|---|---|
n | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs |
50 | 0.651 | 0.147 | 0.773 | 0.210 | 0.699 | 0.206 | 1.268 | 0.48 |
0.613 | 0.303 | 0.486 | 0.228 | 1.464 | 4.786 | 1.143 | 3.391 | |
2.442 | 1.194 | 2.549 | 1.534 | 1.588 | 0.918 | 1.603 | 0.247 | |
0.746 | 0.758 | 1.058 | 1.221 | 1.752 | 2.663 | 1.432 | 1.103 | |
100 | 0.667 | 0.133 | 0.806 | 0.147 | 0.767 | 0.161 | 1.199 | 0.248 |
0.621 | 0.156 | 0.437 | 0.074 | 1.326 | 0.386 | 0.774 | 0.895 | |
2.296 | 0.714 | 2.310 | 0.440 | 1.247 | 0.115 | 1.592 | 0.123 | |
0.652 | 0.484 | 0.845 | 0.612 | 1.286 | 0.630 | 1.148 | 0.600 | |
200 | 0.735 | 0.109 | 0.828 | 0.109 | 0.750 | 0.102 | 1.197 | 0.168 |
0.647 | 0.078 | 0.456 | 0.039 | 1.247 | 0.177 | 0.703 | 0.093 | |
2.064 | 0.121 | 2.099 | 0.169 | 1.253 | 0.061 | 1.468 | 0.033 | |
0.478 | 0.222 | 0.819 | 0.594 | 1.115 | 0.582 | 1.098 | 0.222 | |
500 | 0.725 | 0.07 | 0.898 | 0.036 | 0.768 | 0.090 | 1.167 | 0.140 |
0.680 | 0.056 | 0.483 | 0.017 | 1.269 | 0.139 | 0.639 | 0.043 | |
1.880 | 0.073 | 2.043 | 0.048 | 1.195 | 0.013 | 1.484 | 0.016 | |
0.406 | 0.043 | 0.612 | 0.121 | 1.004 | 0.314 | 1.054 | 0.221 |
n | Mean | Median | Standard Deviation | Skewness | Kurtosis |
---|---|---|---|---|---|
84 | 2.56 | 2.35 | 1.12 | 0.10 | −0.71 |
Distributions | MLEs and SEs | ||||
---|---|---|---|---|---|
TIITLPL(, , , ) | 213.2225 (8.0985) | 3.6880 (0.7192) | 1.2282 (0.1149) | 186.8420 (7.6076) | - |
BL(, , , ) | 3.6036 (0.6187) | 32.6387 (13.7145) | 4.8307 (9.2382) | 119.0374 (42.9269) | - |
KwW(, , , ) | 34.6604 (17.5270) | 81.8464 (52.0142) | 14.4338 (27.0952) | 0.2040 (0.0423) | - |
McW(, , , , ) | 17.6864 (6.2220) | 33.6392 (19.9941) | 1.9406 (1.0111) | 0.3062 (0.0454) | 16.7211 (9.6221) |
BW(, , , ) | 34.1808 (14.8389) | 11.4968 (6.7300) | 1.3609 (1.0020) | 0.2982 (0.0603) | - |
TMOFr(, , , ) | 200.7472 (87.2751) | 1.9524 (0.1252) | 0.1022 (0.0173) | −0.8692 (0.1012) | - |
(, , ) | 2.0589 (0.2759) | 3.1025 (0.3408) | −0.4839 (0.3853) | - | - |
(, ) | 2.3911 (0.1369) | 3.2235 (0.2971) | - | - | - |
Distributions | AIC | CAIC | HQIC | A | W |
---|---|---|---|---|---|
269.0398 | 269.5461 | 272.9485 | 0.6245 | 0.0621 | |
285.4354 | 285.9355 | 289.3650 | 1.4080 | 0.1684 | |
281.4345 | 281.9411 | 291.1585 | 1.5062 | 0.1851 | |
283.8993 | 284.6692 | 296.0532 | 1.5917 | 0.1992 | |
BW | 305.0283 | 305.5343 | 314.7519 | 3.2277 | 0.4655 |
309.4725 | 309.9785 | 319.1953 | 2.4042 | 0.3209 | |
284.7719 | 285.0719 | 287.7034 | 1.4922 | 0.1842 | |
283.1625 | 283.3106 | 285.1168 | 1.5203 | 0.1866 |
n | Mean | Median | Standard Deviation | Skewness | Kurtosis |
---|---|---|---|---|---|
128 | 9.37 | 6.39 | 10.51 | 3.25 | 15.20 |
Distribution | MLEs and SEs | ||||
---|---|---|---|---|---|
(, , , ) | 3.7264 (0.1187) | 1.8637 (0.2335) | 0.8039 (0.1253) | 13.3937 (1.2256) | - |
(, , , ) | 4.6580 (13.1634) | 0.2984 (0.1675) | 7.8661 (4.4933) | 113.0181 (23.364) | - |
(, , , ) | 106.0695 (124.8000) | 1.7124 (0.0992) | 0.2173 (0.6104) | 0.0090 (0.0070) | - |
(, , , , ) | 2.7805 (44.5102) | 67.636 (104.728) | 0.3380 (0.3857) | 3.0833 (49.3534) | 0.8398 (1.7235) |
(, , , , ) | 22.1869 (21.9561) | 20.2778 (17.2968) | 0.2243 (0.1446) | 1.7804 (1.0763) | 1.3067 (1.0794) |
(, , , , ) | 2.3272 (0.3698) | 0.0002 (0.0002) | 17.9315 (7.3857) | 0.5430 (0.0420) | 0.0010 (0.0003) |
(, , , ) | 12.5268 (24.4699) | 33.342 (36.348) | 27.7533 (71.5078) | 0.1690 (0.1040) | - |
(, , , ) | 7.3765 (5.3893) | 0.0473 (0.0042) | 0.1182 (0.2600) | 0.0491 (0.0362) | - |
(, , , ) | 0.0002 (0.0114) | 0.1208 (0.0240) | 0.8955 (0.6260) | 0.4075 (0.4070) | - |
(, , , , ) | 0.00003 (0.0061) | 1.0065 (0.0353) | 0.1139 (0.0322) | 0.9722 (0.125) | −0.1630 (0.2803) |
(, , , , ) | 2.7965 (1.117) | 0.0128 (7.214) | 0.2991 (0.1512) | 0.6542 (0.1216) | 0.0020 (1.7691) |
Distributions | AIC | CAIC | HQIC | A | W |
---|---|---|---|---|---|
828.1981 | 828.5239 | 832.8331 | 0.2043 | 0.0340 | |
829.5312 | 829.8570 | 834.1672 | 0.3173 | 0.0494 | |
829.9953 | 830.3201 | 834.6335 | 0.3064 | 0.0435 | |
831.6514 | 832.1432 | 837.4453 | 0.3244 | 0.0485 | |
841.2684 | 841.7644 | 855.5283 | 0.9515 | 0.1345 | |
839.8243 | 840.3163 | 854.0853 | 2.6188 | 0.4100 | |
842.9651 | 843.2924 | 854.3735 | 1.1218 | 0.1689 | |
866.3500 | 866.6755 | 877.7588 | 2.3617 | 0.3980 | |
836.4555 | 836.7759 | 847.8586 | 0.1259 | 0.7609 | |
838.47833 | 838.9777 | 852.7390 | 0.1130 | 0.7030 | |
831.3475 | 831.8395 | 837.1411 | 0.3058 | 0.0469 |
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Al-Marzouki, S.; Jamal, F.; Chesneau, C.; Elgarhy, M. Type II Topp Leone Power Lomax Distribution with Applications. Mathematics 2020, 8, 4. https://doi.org/10.3390/math8010004
Al-Marzouki S, Jamal F, Chesneau C, Elgarhy M. Type II Topp Leone Power Lomax Distribution with Applications. Mathematics. 2020; 8(1):4. https://doi.org/10.3390/math8010004
Chicago/Turabian StyleAl-Marzouki, Sanaa, Farrukh Jamal, Christophe Chesneau, and Mohammed Elgarhy. 2020. "Type II Topp Leone Power Lomax Distribution with Applications" Mathematics 8, no. 1: 4. https://doi.org/10.3390/math8010004
APA StyleAl-Marzouki, S., Jamal, F., Chesneau, C., & Elgarhy, M. (2020). Type II Topp Leone Power Lomax Distribution with Applications. Mathematics, 8(1), 4. https://doi.org/10.3390/math8010004