A new class of distributions as a finite functional mixture using functional weights

BOUALI, DALAL LALA; CHESNEAU, CHRISTOPHE; SHARMA, VIKAS KUMAR; BAKOUCH, HASSAN S.

doi:10.1590/0001-3765202120181019

DALAL LALA BOUALI

Université de Caen Basse-Normandie, LMNO, UFR de Sciences, 14000 Caen, France

Laboratory of Applied Mathematics, Mohamed Khider University of Biskra, 07020 Biskra, Algeria

https://orcid.org/0000-0001-7496-9154

CHRISTOPHE CHESNEAU

Université de Caen Basse-Normandie, LMNO, UFR de Sciences, 14000 Caen, France

https://orcid.org/0000-0002-1522-9292

VIKAS KUMAR SHARMA

Department of Statistics, Institute of Science, Banaras Hindu University (BHU), 221005 Varanasi, India

https://orcid.org/0000-0003-0169-4094

HASSAN S. BAKOUCH

Department of Mathematics, Faculty of Science, Tanta University, 31512 Tanta, Egypt

https://orcid.org/0000-0002-3189-0670

Correspondence to: Christophe Chesneau E-mail: christophe.chesneau@gmail.com

AUTHOR CONTRIBUTIONS

The findings in this paper are part of the first author’s (Dalal Lala Bouali) doctoral thesis, which was supervised by the second author (Christophe Chesneau). The paper was co-written by all of the contributors, and they all contributed to the discussion and text analysis. Some properties of the new model were obtained by Dalal Lala Bouali and Christophe Chesneau. The simulation analysis and applications to real data sets were carried out by Hassan S. Bakouch and Vikas Kumar Sharma.

\(f(x)\)	\(w(x)\)	\(G(x)\)	\(h(x)\)
\(\theta e^{-\theta x}\)	\(e^{-\alpha x}\)	\(x^{\sigma}\)	\(\theta e^{-(\theta+\alpha) x}+ \theta \sigma \left[1-e^{-\alpha x^{\sigma}}\right] x^{\sigma-1} e^{-\theta x^{\sigma}}\)
\(\theta e^{-\theta x}\)	\(\left[\cos( \alpha x)\right]^2\)	\(x^{\sigma}\)	\(\theta \left[\cos( \alpha x)\right]^2 e^{-\theta x}+ \theta \sigma \left[\sin\left( \alpha x^{\sigma}\right)\right]^2x^{\sigma-1} e^{-\theta x^{\sigma}}\)
\(\theta e^{-\theta x}\)	\(\left[\cos( \alpha x)\right]^2\)	\(e^{\sigma x}-1\)	\(\theta \left[\cos( \alpha x)\right]^2 e^{-\theta x}+ \theta \sigma \left[\sin\left( \alpha x^{\sigma}\right)\right]^2 e^{\sigma x}e^{-\theta (e^{\sigma x}-1)}\)
\(2 \theta x e^{-\theta x^2}\)	\(1- e^{-\alpha x}\)	\(x+x^\sigma\)	\(2 \theta x e^{-\theta x^2} (1- e^{-\alpha x})+ \theta e^{-\alpha (x+x^\sigma)} (1+\sigma x) e^{-\theta (x+x^\sigma)}\)
\(\theta e^{-\theta x}\)	\(\displaystyle\frac{1}{1+\alpha x^2}\)	\(\displaystyle\frac{x}{\sigma}\)	\(\frac{\theta}{1+\alpha x^2} e^{-\theta x} + \frac{ \alpha \theta}{\sigma (\sigma^2 +\alpha x^2)} x^{2} e^{-\theta \frac{x}{\sigma}}\)
\(\theta \beta e^{-\theta x} \left(1-e^{-\theta x} \right)^{\beta-1}\)	\(\displaystyle\frac{1}{1+\alpha x^2}\)	\(\displaystyle\frac{x}{\sigma}\)	\(\theta \beta \left( \frac{1}{1+\alpha x^2}e^{-\theta x} \left(1-e^{-\theta x} \right)^{\beta-1}+ \frac{ \alpha }{\sigma} \frac{x^2}{\sigma^2 +\alpha x^2} e^{-\theta \frac{x}{\sigma}} \left(1-e^{-\theta \frac{x}{\sigma}} \right)^{\beta-1}\right)\)
\(\theta\beta (\theta x)^{\beta-1} e^{-\left(\theta x \right)^{\beta}}\)	\(\displaystyle\frac{1}{1+\alpha x^2}\)	\(\displaystyle\frac{x}{\sigma}\)	\(\theta \beta \left( \frac{1}{1+\alpha x^2}(\theta x)^{\beta-1} e^{-\left(\theta x \right)^{\beta}}+ \frac{ \alpha }{\sigma} \frac{x^2}{\sigma^2 +\alpha x^2} \left(\theta \frac{x}{\sigma}\right)^{\beta-1} e^{-\left(\theta \frac{x}{\sigma} \right)^{\beta}}\right)\)

Statistic	Failure times data	Survival times data	Statistic	Failure times data	Survival times data
Minimum	0.03	1.0	Q1	0.91	4
Maximum	9.10	156.0	Q3	2.30	65
Mean	1.96	40.9	Skewness	1.98	1.16
Median	1.74	22.0	Kurtosis	2.16	2.88
St. Deviation	1.57	46.7	Range	9.07	155

Model	Estimate (SE)
FWE\((\theta,\alpha,\) )	\(\hat{\theta}=\) 0.227(0.078)	\(\hat{\alpha}=\) 0.123(0.111)	\(\hat{\sigma}=\) 0.294(0.115)
EG\((\alpha,\beta,\theta)\)	\(\hat{\alpha}=\) 1.260(2.196)	\(\hat{\beta}=\) 1.334(2.513)	\(\hat{\theta}=\) 0.759(0.479)
MW\((\alpha,\beta,\lambda)\)	\(\hat{\alpha}=\) -0.063(0.453)	\(\hat{\beta}=\) 0.431(0.475)	\(\hat{\lambda}=\) 1.283(0.295)
EW\((\alpha,\beta,\lambda)\)	\(\hat{\alpha}=\) 1.443(0.647)	\(\hat{\beta}=\) 0.580(0.302)	\(\hat{\lambda}=\) 1.101(0.264)
EGG\((\alpha,\beta,\lambda)\)	\(\hat{\alpha}=\) 1.477(0.805)	\(\hat{\beta}=\) 0.502(0.063)	\(\hat{\lambda}=\) 1.065(0.338)
WtW\((\alpha,\beta,\lambda)\)	\(\hat{\alpha}=\) 0.907(0.130)	\(\hat{\beta}=\) 0.336(2.768)	\(\hat{\lambda}=\) 0.973(0.960)

Model	\(-\widehat{\ln(\ell)}\)	AIC	BIC	KS	PV
FWE	121.099	248.197	255.190	0.088	0.567
EG	122.235	250.469	257.461	0.096	0.460
MW	122.512	251.024	258.016	0.108	0.311
EW	122.164	250.327	257.319	0.099	0.422
EGG	122.230	250.460	257.452	0.100	0.402
WtW	122.366	250.733	257.725	0.094	0.489

Model	Estimate (SE)
FWE\((\theta,\alpha,\) )	\(\hat{\theta}=\) 0.070(0.0264)	\(\hat{\alpha}=\) 0.0195(0.0213)	\(\hat{\sigma}=\) 3.277(1.202)
EG\((\alpha,\beta,\theta)\)	\(\hat{\alpha}=\) 100.034(9675.29)	\(\hat{\beta}=0.601 \times 10 ^{-2}(3.697\times 10^{-5})\)	\(\hat{\theta}=\) 0.526(0.229)
MW\((\alpha,\beta,\lambda)\)	\(\hat{\alpha}=\) -0.0027(0.0424)	\(\hat{\beta}=\) 0.0645(0.0396)	\(\hat{\lambda}=\) 0.795(0.293)
EW\((\alpha,\beta,\lambda)\)	\(\hat{\alpha}=\) 1.535(5.286)	\(\hat{\beta}=\) 0.160(1.033 )	\(\hat{\lambda}=\) 0.608(1.137)
EGG\((\alpha,\beta,\lambda)\)	\(\hat{\alpha}=\) 2.574(7.181)	\(\hat{\beta}=\) 0.037(0.025)	\(\hat{\lambda}=\) 0.444(0.695)
WtW\((\alpha,\beta,\lambda)\)	\(\hat{\alpha}=\) 0.512(0.0675)	\(\hat{\beta}=\) 8.021\(\times 10^{-5}(1.515)\)	\(\hat{\lambda}=\) 0.3583(0.291)

Model	\(-\widehat{\ln(\ell)}\)	AIC	BIC	KS	PV
FWE	151.125	308.250	312.739	0.090	0.952
EGD	153.136	312.271	316.761	0.136	0.576
MW	153.585	313.169	317.659	0.129	0.644
EWD	153.585	313.169	317.659	0.137	0.564
EGG	153.501	313.003	317.492	0.137	0.562
WtW	153.507	313.013	317.503	0.137	0.565

[1] Correspondence to: Christophe Chesneau E-mail: christophe.chesneau@gmail.com

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A new class of distributions as a finite functional mixture using functional weights

Abstract