A group acceptance sampling plan based on flexible new Kumaraswamy exponential distribution: An application to quality control reliability

Figures & data

Figure 1. Different graphical representations for pdf of the NKwE distribution.

Figure 2. Different graphical representation for hrf of the NKwE distribution.

Table 1. GASP displaying minimal $g$ and $c$ for $\alpha =0.5$ and $s=1.02$

		r = 5						r = 10
		a1 = 0.5			a1 = 1			a1 = 0.5			a1 = 1
η	r2	g	c	p(a)	g	c	p(a)	g	c	p(a)	g	c	p(a)
	2	-	-	-	-	-	-	-	-	-	-	-	-
0.25	4	4	2	0.9539	3	3	0.9733	2	3	0.9627	1	4	0.9555
	6	4	2	0.9849	1	2	0.9746	1	2	0.9657	1	3	0.9528
	8	4	2	0.9933	1	2	0.9883	1	2	0.9837	1	3	0.9812
	2	-	-	-	-	-	-	-	-	-	-	-	-
0.1	4	26	3	0.9817	4	3	0.9645	6	4	0.9839	1	4	0.9555
	6	7	2	0.9737	4	3	0.9917	3	3	0.9861	1	3	0.9528
	8	7	2	0.9883	2	2	0.9767	2	2	0.9677	1	3	0.9812
	2	-	-	-	-	-	-	-	-	-	-	-	-
0.05	4	34	3	0.9761	23	4	0.9893	7	4	0.9812	3	5	0.9719
	6	8	2	0.9700	6	3	0.9876	4	3	0.9815	1	3	0.9528
	8	8	2	0.9866	2	2	0.9767	2	2	0.9677	1	3	0.9812
	2	-	-	-	-	-	-	-	-	-	-	-	-
0.01	4	52	3	0.9637	35	4	0.9837	11	4	0.9706	4	5	0.9628
	6	13	2	0.9517	8	3	0.9835	5	3	0.9769	2	4	0.9818
	8	13	2	0.9784	4	2	0.9539	3	2	0.9520	2	3	0.9627

Note: A large sample length in needed cells contains hyphens (-).

Table 2. GASP displaying minimal $g$ and $c$ for $\alpha =0.8$ and $s=2.0$

	r = 5								r = 10
	a1 = 0.5					a1 = 1			a1 = 0.5			a1 = 1
η	r2		g	c	p(a)	g	c	p(a)	g	c	p(a)	g	c	p(a)
	2		18	3	0.9565	6	4	0.9508	4	4	0.9565	-	-	-
0.25	4		5	2	0.9924	1	2	0.9714	1	2	0.9851	1	4	0.9889
	6		2	1	0.9838	1	2	0.9946	1	1	0.9668	1	2	0.9532
	8		2	1	0.9932	1	1	0.9729	1	1	0.9856	1	2	0.9851
	2		274	4	0.9765	-	-	-	16	5	0.9748	-	-	-
0.1	4		7	2	0.9893	3	3	0.9926	2	2	0.9704	1	4	0.9889
	6		3	1	0.9758	2	2	0.9893	1	1	0.9668	1	2	0.9532
	8		3	1	0.9898	1	1	0.9729	2	1	0.9856	1	2	0.9851
	2		357	4	0.9695	-	-	-	20	5	0.9686	-	-	-
0.05	4		9	2	0.9863	4	3	0.9902	2	2	0.9704	1	4	0.9889
	6		3	1	0.9758	2	2	0.9893	2	2	0.9948	1	2	0.9532
	8		3	1	0.9898	1	1	0.9729	2	1	0.9714	1	2	0.9851
	2		584	4	0.9536	-	-	-	31	5	0.9517	-	-	-
0.01	4		14	2	0.9788	5	3	0.9877	3	2	0.9559	2	4	0.9780
	6		5	1	0.9599	3	2	0.9839	3	2	0.9922	1	2	0.9532
	8		5	1	0.9831	3	2	0.9954	2	1	0.9714	1	2	0.9851

Note: A large sample length in needed cells contains hyphens (-).

Table 3. $s=1.02$$\eta =0.25$ $r=5$ and $a_1=0.5$

$v$/$v_0$=$r_2$	4	6	8
$g$	4	4	4
$c$	2	2	2
$p(a)$	0.9539	0.9849	0.9933

Figure 3. Tables 1 and 2 [1] and [2] are used as the sources of some parametric values for the graphical representation of g and OC.

Table 4. The MLEs (SE), KS, and p-value for data set I

Parameter	MLEs (SE)	KS	PV
$\hat{\theta }$	84.7012 (49.9827)	0.1114	0.8499
$\hat{\alpha }$	0.2927 (0.0960)
$\hat{s}$	0.3132 (0.2748)

Figure 4. (a) Estimate the density, (b) estimate the cdf, (c) P-P, (d) Q-Q, (e) TTT, and (f) estimate the hrf.

Table 5. GASP under NKwE model, $\hat{\alpha }$ = 0.2927 and $\hat{s}$ = 0.3132, displaying minimal $g$ and $c$

		r = 5						r = 10
		a1 = 0.5			a1 = 1			a1 = 0.5			a1 = 1
η	r2	g	c	p(a)	g	c	p(a)	g	c	p(a)	g	c	p(a)
	2	-	-	-	-	-	-	-	-	-	-	-	-
0.25	4	34	3	0.9674	6	3	0.9625	7	4	0.9731	3	5	0.9824
	6	7	2	0.9540	6	3	0.9873	3	3	0.9718	1	3	0.9520
	8	7	2	0.9734	2	2	0.9708	3	3	0.9859	1	3	0.9752
	2	-	-	-	-	-	-	-	-	-	-	-	-
0.1	4	658	4	0.9829	62	4	0.9818	11	4	0.9581	5	5	0.9709
	6	57	3	0.9814	10	3	0.9790	5	3	0.9535	3	4	0.9722
	8	11	2	0.9585	10	3	0.9903	5	3	0.9766	2	1	0.9511
	2	-	-	-	-	-	-	-	-	-	-	-	-
0.05	4	856	4	0.9779	81	4	0.9762	44	5	0.9813	6	5	0.9652
	6	74	3	0.9759	13	3	0.9728	6	3	0.9720	3	4	0.9722
	8	74	3	0.9887	13	3	0.9874	15	4	0.9835	2	3	0.9511
	2	-	-	-	-	-	-	-	-	-	-	-	-
0.01	4	-	-	-	124	4	0.9639	68	5	0.9712	-	-	-
	6	113	3	0.9635	20	3	0.9584	22	4	0.9760	5	4	0.9541
	8	113	3	0.9828	20	3	0.9807	9	3	0.9582	5	4	0.9807

Note: A large sample length in needed cells contains hyphens (-).

Table 6. The MLEs (SE), KS, and p-value for data set II

Parameter	MLEs (SE)	KS	PV
$\hat{\theta }$	0.0071 (0.0015)	0.1377	0.7754
$\hat{\alpha }$	1.1721 (0.2574)
$\hat{s}$	12.1036 (3.9690)

Figure 5. (a) Estimate the density, (b) estimate the cdf, (c) P-P, (d) Q-Q, (e) TTT, and (f) estimate the hrf.

Table 7. GASP under NKwE model, $\hat{\alpha }$ = 2.5047 and $\hat{s}$ = 12.7570, displaying minimal $\mathit{g}$ and $\mathit{c}$

		r = 5						r = 10
		a1 = 0.5			a1 = 1			a1 = 0.5			a1 = 1
η	r2	g	c	p(a)	g	c	p(a)	g	c	p(a)	g	c	p(a)
	2	3	3	0.9789	-	-	-	-	-	-	-	-	-
0.25	4	1	1	0.9805	1	3	0.9929	2	2	0.9815	1	4	0.9653
	6	1	1	0.9967	1	2	0.9942	1	1	0.9860	1	2	0.9501
	8	1	0	0.9529	1	1	0.9805	1	1	0.9961	1	2	0.9907
	2	5	3	0.9650	-	-	-	-	-	-	-	-	-
0.1	4	1	1	0.9805	1	3	0.9929	2	2	0.9815	1	4	0.9653
	6	1	1	0.9967	1	2	0.9942	1	1	0.9860	1	2	0.9501
	8	1	0	0.9529	1	1	0.9805	1	1	0.9961	1	2	0.9907
	2	6	3	0.9582	-	-	-	-	-	-	-	-	-
0.05	4	2	1	0.9613	1	3	0.9929	3	2	0.9723	-	-	-
	6	2	1	0.9934	1	2	0.9942	2	1	0.9723	1	2	0.9501
	8	1	0	0.9529	1	1	0.9805	2	1	0.9921	1	2	0.9907
	2	41	4	0.986	-	-	-	-	-	-	-	-	-
0.01	4	2	1	0.9613	1	3	0.9929	4	2	0.9632	-	-	-
	6	2	1	0.9934	1	2	0.9942	2	1	0.9723	1	2	0.9501
	8	1	0	0.9529	1	1	0.9805	2	1	0.9921	1	2	0.9907

Note: A large sample length in needed cells contains hyphens (-).

Figure 6. Plots of descriptive analysis for (a) data set I and (b) data set II.

Table 8. Descriptive analysis of data sets

Data set	n	z0	Q1	$\tilde{\mathbf{z}}$	$\bar{\mathbf{z}}$	σ	Q3	zn	SK	K
I	30	0.58	0.81	1.11	1.15	0.475	1.29	2.77	1.741	6.281
II	23	17.88	43.86	55.56	67.87	36.524	88.62	173.40	1.137	4.1808

Table 9. True median lifetimes, sample sizes, $g$, $c$, and OC values of the two data sets

	Data set I				Data set II
$v$/$v_0$=$r_2$	2	4	6	8	$v$/$v_0$=$r_2$	2	4	6	8
$n$	430	30	30	30	$n$	-	40	20	20
$g$	86	6	6	6	$g$	-	4	2	2
$c$	3	1	1	1	$c$	-	2	1	1
$p(a)$	0.9659	0.9713	0.9949	0.9986	$p(a)$	-	0.9632	0.9723	0.9921

Table 10. Sample sizes of GASP and OSP

		Data set I		Data set II
		a1= 0.5		a1= 0.5
η	r2	GASP	OSP	GASP	OSP
	2	33	36	15	17
0.25	4	11	14	5	7
	6	12	15	5	7
	8	6	9	6	8

Data availability statement

The data presented in this study are available on request from the corresponding author.