A new class of lifetime models and the evaluation of the confidence intervals by double percentile bootstrap

MARINHO, PEDRO R.D.; BOURGUIGNON, MARCELO; SILVA, RODRIGO B.; CORDEIRO, GAUSS M.

doi:10.1590/0001-3765201920180480

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Mathematical Sciences • An. Acad. Bras. Ciênc. 91 (01) • 2019 • https://doi.org/10.1590/0001-3765201920180480 copy

A new class of lifetime models and the evaluation of the confidence intervals by double percentile bootstrap

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract: In this paper, we introduce a new three-parameter distribution by compounding the Nadarajah-Haghighi and geometric distributions, which can be interpreted as a truncated Marshall-Olkin extended Weibull. The compounding procedure is based on the work by Marshall and Olkin 1997. We prove that the new distribution can be obtained as a compound model with mixing exponential distribution. It can have decreasing, increasing, upside-down bathtub, bathtub-shaped, constant and decreasing-increasing-decreasing failure rate functions depending on the values of the parameters. Some mathematical properties of the new distribution are studied including moments and quantile function. The maximum likelihood estimation procedure is discussed and a particle swarm optimization algorithm is provided for estimating the model parameters. The flexibility of the new model is illustrated with an application to a real data set.

Key words
Exponential distribution; Failure rate function; Geometric distribution; Maximum likelihood estimation; Nadarajah-Haghighi distribution.

n	Simple Bootstrap (In hours)	Double Bootstrap (In hours)
20	105.3212	150.4235
60	175.3455	191.9864
100	213.7565	265.5455
500	400.3253	450.4535

		Simple Bootstrap		Double Bootstrap
n	Parameters	Coverage (%)	Range	Coverage (%)	Range
n = 20	$α$	65.0327	2.1562	92.3441	3.2432
	$λ$	62.1665	2.1605	93.4551	2.3423
	$p$	67.3002	0.9663	94.2138	1.3563
n = 60	$α$	71.1454	0.7446	93.5233	3.2124
	$λ$	77.7465	2.1454	92.3453	2.3567
	p	66.7213	1.2423	94.4354	2.8433
n = 100	$α$	72.2198	1.4354	92.4345	3.3315
	$λ$	69.2300	2.1322	94.2130	1.0035
	p	69.4357	0.5493	93.4365	2.1252
n = 500	$α$	83.1233	2.1215	94.7834	3.2123
	$λ$	71.9011	1.2321	94.3254	3.2122
	p	72.2965	1.1190	93.9342	1.2134

Statistics	Real data sets
Statistics	Time between failures (hours)
Mean	1.5779
Median	0.6140
Mode	0.5000
Variance	3.7275
Skewness	1.3643
Kurtosis	0.5445
Maximum	6.5600
Minimum	0.0620
n	23

Distributions	Estimates of the parameters			Equation
NHG( $α, λ, p$ )	0.4195	5.7294	$-$ 0.7929	$F (x) = \frac{1 - e^{1 - (1 + λ x)^{α}}}{1 - p e^{1 - (1 + λ x)^{α}}}$
	(0.1153)	(2.7918)	(1.2611)
ENH( $α, λ, β$ )	0.2856	42.0728	3.1569	$F (x) = {[1 - e^{1 - (1 + λ x)^{α}}]}^{β}$
	(0.4808)	(19.0491)	(1.2583)
NH( $α, λ$ )	0.4934	2.5010		$F (x) = 1 - e^{1 - (1 + λ x)^{α}}$
	(0.3493)	(1.4762)

Distributions	Interval estimates
NHG( $α, λ, p$ )	(0.3578; 0.8177)	(0.8884; 13.8570)	( $-$ 3.7684; $-$ 0.0001)
ENH( $α, λ, β$ )	(0.2621; 0.7341)	(0.8400; 49.9999)	(0.9062; 5.8346)
NH( $α, λ$ )	(0.3804; 1.4111)	(0.3665; 5.9659)

	Statistics
Distribution	AIC	CAIC	BIC	HQIC	KS	A^*	W^*
NHG( $α, λ, p$ )	72.8293	74.0925	76.2358	73.6860	0.1684	0.2547	0.0323
ENH( $α, λ, β$ )	289.7261	290.9892	293.1326	290.5828	0.6630	0.3020	0.0424
NH( $α, λ$ )	239.9434	240.5434	242.2144	240.5146	0.7119	0.3546	0.0513

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[1] Correspondence to: Marcelo Bourguignon E-mail: m.p.bourguignon@gmail.com ORCid: 0000-0002-1182-5193

2.160	0.150	4.082	0.746	0.358	0.199	0.402	0.101	0.605	0.954
1.359	0.273	0.491	3.465	0.070	6.560	1.060	0.062	4.992	0.614
5.320	0.347	1.921