On Modified Burr XII-Inverse Exponential Distribution: Properties, Characterizations and Applications

Fiaz Ahmad Bhatti; Gholamhossein Hamedani; Haitham M. Yousof; Azeem Ali; Munir Ahmad

On Modified Burr XII-Inverse Exponential Distribution: Properties, Characterizations and Applications

Fiaz Ahmad Bhatti^1*, Gholamhossein Hamedani², Haitham M. Yousof³, Azeem Ali¹, Munir Ahmad¹

¹National College of Business Administration and Economics, Lahore, Pakistan

²Marquette University, Milwaukee, WI, USA

³Department of Statistics Mathematics and Insurance, Benha University, Al Qalyubia Governorate, Egypt

^*Corresponding author: Fiaz Ahmad Bhatti, National College of Business Administration and Economics, Lahore, Pakistan. Email: fiazahmad72@gmail.com

Received Date: 06 September, 2018; Accepted Date: 25 September, 2018; Published Date: 05 October, 2018

Citation: Bhatti FA, Hamedani G, Yousof HM, Ali A, Ahmad M (2018) On Modified Burr XII-Inverse Exponential Distribution: Properties, Characterizations and Applications. J Biostat Biom: JBSB-106. DOI: 10.29011/JBSB-106.100006

Abstract

In this paper, a flexible lifetime distribution with increasing, increasing and decreasing and modified bathtub hazard rate called Modified Burr XII-Inverse Exponential (MBXII-IE) is introduced. The density function of MBXII-IE has exponential, left-skewed, right-skewed and symmetrical shapes. Descriptive measures such as moments, moments of order statistics, incomplete moments, inequality measures, residual life function and reliability measures are theoretically established. The MBXII-IE distribution is characterized via different techniques. Parameters of MBXII-IE distribution are estimated using maximum likelihood method. The simulation study is performed to illustrate the performance of the Maximum Likelihood Estimates (MLEs) of the parameters of the MBXII-IE distribution. The potentiality of MBXII-IE distribution is demonstrated by its application to real data sets: fracture toughness, taxes revenue’s data and coal mining disaster data.

Keywords: Characterizations; Moments; Maximum Likelihood Estimation; Reliability

Figure 1: Plots of pdf of the MBXII-IE distribution for selected parameter values.

Figure 2: Plots of hrf of the MBXII-IE distribution for selected parameter values.

Figure 3: Fitted pdf, cdf, survival and pp plots of the MBXII-IE distribution for fracture toughness.

Figure 4: Fitted pdf, cdf, survival and pp plots of the MBXII-IE distribution for Tax Revenue.

Figure 5: Fitted pdf, cdf, survival and pp plots of the MBXII-IE distribution for coal mining disasters data.

1	α	β	γ	λ	MBXII-IE distribution
2	α	β	1	λ	BXII-IE distribution
3	α	1	1	λ	Lomax-IE distribution
4	1	β	1	λ	Log-logistic-IE distribution
5	α	β	γ→0	λ	Weibull-IE distribution

Table 1: Sub-models of the MBXII-IE Distribution.

Sample	Statistics
n=50	Means	0.508	0.5275	0.4009	1.5208	0.8235	0.9994	1.1443	1.3154
	Bias	0.008	0.0275	-0.0991	0.6208	0.0235	0.0994	0.1443	0.8154
	MSE	0.0691	0.6688	14.5017	3.0359	1.0307	8.3995	318.6126	3.4638
n=100	Means	0.5047	0.5165	0.4233	1.1072	0.7974	0.9218	0.6404	0.8864
	Bias	0.0047	0.0165	-0.0767	0.2072	-0.0026	0.0218	-0.3596	0.3864
	MSE	0.0335	0.0542	0.1008	0.5377	0.2057	0.8449	2.629	1.2674
n=200	Means	0.5009	0.5092	0.4617	0.9831	0.7882	0.926	0.7403	0.6479
	Bias	9e-04	0.0092	-0.0383	0.0831	-0.0118	0.026	-0.2597	0.1479
	MSE	0.0177	0.0229	0.0401	0.1467	0.1524	0.1817	0.2849	0.2877
n=300	Means	0.5016	0.5071	0.4804	0.9509	0.7864	0.9253	0.8063	0.5879
	Bias	0.0016	0.0071	-0.0196	0.0509	-0.0136	0.0253	-0.1937	0.0879
	MSE	0.0117	0.0132	0.0266	0.0875	0.1211	0.1281	0.1785	0.1394
n=500	Means	0.5001	0.5029	0.4854	0.932	0.7921	0.9131	0.8687	0.5514
	Bias	1e-04	0.0029	-0.0146	0.032	-0.0079	0.0131	-0.1313	0.0514
	MSE	0.007	0.0071	0.0153	0.0481	0.0841	0.077	0.096	0.064

Table 2: Means, Bias and MSEs of the MBXII-IE distribution (0.5, 0.5, 0.5, 0.9) and (0.8, 0.9, 1, 0.5).

Model					W	A	K-S p-value
MBXII-IE	0.0181133061 (0.022788744)	4.1601365928 (0.413218923)	0.0000000001 (0.002022918)	1.5161119726 (0.352629351)	0.1182952	0.7247703	0.0796 (0.4387)
BXII-IE	10.836337 (8.8614837)	2.939956 (0.5268154)	1	5.420295 (1.3394846)	0.2103398	1.295879	0.1186 (0.07024)
L-IE	88.15512 (30.158622)	1	1	20.65397 (1.627536)	0.3337133	2.026292	0.1147 (0.08714)
LL-IE	1	(4.867383) 0.38248098	1	2.975707 (0.06835492)	0.4064326	2.522655	0.332 (8.12e-12)

Table 3: MLEs, their standard errors (in parentheses) and Goodness-of-fit statistics for fracture toughness.

Model	AIC	CAIC	BIC	HQIC	-l
MBXII-IE	346.419	346.7698	357.5355	350.933	169.2095
BXII-IE	351.0785	351.2872	359.4159	354.4641	172.5393
L-IE	356.4294	356.5328	361.9876	358.6864	176.2147
LL-IE	370.1331	370.2365	375.6913	372.3901	183.0665

Table 4: Goodness-of-fit statistics for fracture toughness.

Model					W	A	K-S (p-value)
MBXII-IE	0.01574783 (0.017879592)	2.66428346 (0.356379272)	0.00150912 (0.002280467)	0.54190197 (0.168851792)	0.08121186	0.4526733	0.1009 (0.2603)
BXII-IE	3.405735 (1.5676440)	1.865183 (0.3395644)	1	2.831844 (0.6706193)	0.2254017	1.182635	0.2072 (0.0003741)
Lomax-IE	9.196698 (2.1114083)	1	1	6.214104 (0.5991982)	0.3149302	1.693847	0.1703 (0.006058)
LL-IE	1	2.770423 (0.23340747)	1	1.676333 (0.07349948)	0.3281241	1.739495	0.3634 (6.722e-12)

Table 5: MLEs, their standard errors (in parentheses) and Goodness-of-fit statistics for Tax Revenue.

Model	AIC	CAIC	BIC	HQIC	-l
MBXII-IE	289.7468	290.1678	300.1675	293.9642	140.8734
BXII-IE	300.7174	300.9674	308.5329	303.8805	147.3587
L-IE	304.8322	304.956	310.0426	306.941	150.4161
LL-IE	307.8828	308.0065	313.0932	309.9915	151.9414

Table 6: Goodness-of-fit statistics for Tax Revenue.

Model					W	A	K-S p-value
MBXII-IE	0.014003462 (0.010293280)	0.974040069 (0.114284919)	0.002542779 (0.001955081)	2.319100102 (1.400766759)	0.06743529	0.4651009	0.1142 (0.1167)
BXII-IE	0.5257374 (0.1056558)	0.9798724 (0.1313540)	20.8102228 (4.3210242)	1	0.804424	4.659193	0.5874 (<2.2e-16)
L-IE	0.5129513 (0.0608668)	1	1	20.4062733 (3.3094989)	0.8043515	4.653696	0.592 (<2.2e-16)
LL-IE	1	0.6939298 (0.05621336)	1	36.3055045 (4.43815735)	0.9376481	5.457812	0.521 (<2.2e-16)

Table 7: MLEs, their standard errors (in parentheses) and Goodness-of-fit statistics for coal mining disasters.

Model	AIC	CAIC	BIC	HQIC	-l
MBXII-IE	1409.089	1409.474	1419.854	1413.455	700.5445
BXII-IE	1493.264	1493.493	1501.338	1496.539	743.6322
L-IE	1491.287	1491.4	1496.67	1493.47	743.6435
LL-IE	1503.023	1503.136	1508.406 1	505.206	749.5115

Table 8: Goodness-of-fit statistics for coal mining disasters.

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On Modified Burr XII-Inverse Exponential Distribution: Properties, Characterizations and Applications

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