Cosine Topp–Leone family of distributions: Properties and Regression

Figures & data

Figure 1. Plots of density (left) and hazard rate (right) functions of CTLW distribution.

Figure 2. Plots of the density function of CTLF distribution.

Figure 3. Plot of hazard rate function of CTLF distribution.

Figure 4. Plots of density function of CTLB distribution.

Figure 5. Plot of hazard rate function of CTLB distribution.

Figure 6. Plots of density functions of CTLCh distribution.

Figure 7. Plot of hazard rate function of CTLCh distribution.

Figure 8. Plots of density (left) and hazard rate (right) functions of CTLCa distribution.

Table 1. Simulation results of AB, RMSE, and CP for CTLW distribution

Parameter	n	I				II
		AB	RMSE	CP	AW	AB	RMSE	CP	AW
α	25	−0.6179	2.1914	0.9113	27.2602	1.3313	2.7307	0.8933	17.485
	75	−0.6248	2.0585	0.8633	17.3854	0.5858	1.7869	0.8833	7.0163
	100	−0.6034	2.0132	0.8580	15.4048	0.4195	1.4498	0.9007	5.3519
	150	−0.3882	1.8339	0.8787	13.4463	0.3057	1.0822	0.9040	3.8753
	200	−0.3429	1.6978	0.8953	12.1186	0.1963	0.8550	0.9113	2.9800
β	25	0.3213	0.7200	0.9953	5.6802	0.1470	0.7831	0.9567	3.9377
	75	0.2705	0.6174	0.9951	3.5067	0.1031	0.5274	0.9500	2.1598
	100	0.2661	0.6003	0.9962	3.0662	0.0844	0.4544	0.9533	1.8379
	150	0.1905	0.5060	0.9963	2.4681	0.0461	0.3800	0.9560	1.4508
	200	0.1542	0.4354	0.9967	2.1627	0.0448	0.3204	0.9600	1.2404
λ	25	−0.0360	0.3552	0.9900	3.1315	0.3904	0.6587	0.9933	4.1251
	75	−0.0174	0.2534	0.9933	1.7476	0.2067	0.4820	0.9767	2.0721
	100	−0.0018	0.2286	0.9913	1.5140	0.1755	0.4266	0.9760	1.7239
	150	0.0005	0.1998	0.9913	1.2122	0.1312	0.3514	0.9760	1.3252
	200	0.0004	0.1858	0.9900	1.0320	0.0968	0.2984	0.9660	1.1047

Table 2. Data on survival times of lung cancer patients

72	411	228	126	118	10	82	110	314
13	123	97	153	59	117	16	151	22
122	27	54	7	63	392	10	8	92
553	278	12	260	200	156	182	143	105
1	587	389	33	25	357	467	201	1
20	7	24	99	8	99	61	25	95
73	8	36	48	7	140	186	84	19
100	42	8	144	25	11	30	384	4
56	21	18	139	20	31	52	287	18
35	117	132	12	162	3	95	177	162
103	250	100	999	112	87	231	242	991
30	44	283	15	25	103	21	13	87
80	51	29	24	18	83	31	51	90
45	80	52	164	19	53	15	43	340
111	231	378	49

Table 3. Parameter estimates and standard errors (in parentheses) for dataset 1

	$\widehat{\alpha }$	$\hat{\beta }$	$\hat{\lambda }$
CTLW	1.0695 (0.5246)	0.6085 (0.1519)	0.0444 (0.04510)
TLW	1001.5 (0.0000289)	0.1073 (0.0063)	2.3723 (0.0588)
B3P	1.0744 (0.1092)	2.6782 (1.266)	212.8368 (131.5044)
GINW	2.3345 (0.2758)	0.6759 (0.0393)	5.6179 (0.1696)
Ca	47.605 (5.500)	57.6688 (7.0219)
GoIE	0.0402 (0.0261)	0.804 (0.0713)	2.4822 (1.5500)
IE	20.336 (1.7374)
L	0.0163 (0.001)

Table 4. Goodness-of-fit statistics and information Criteria for dataset 1

Model	ℓ	KS(p-value)	CM	AD	AIC	BIC	CAIC	HQIC
CTLW	−790.14	0.0570(0.765)	0.0481	0.2609	1586.28	1595.04	1586.46	17.921
TLW	−803.38	0.1074(0.085)	0.3429	2.1516	1612.76	1621.52	1612.94	17.954
B3P	−790.67	0.0590(0.752)	0.0564	0.3090	1587.34	1596.10	1587.516	17.922
GINW	−815.55	0.1202(0.038)	0.6168	4.0854	1637.10	1645.86	1637.283	17.852
Ca	−843.53	0.2837(5E–10)	2.5416	13.9980	1691.07	1696.91	1691.246	18.052
GoIE	−790.37	0.0571(0.761)	0.0492	0.3018	1586.74	1595.50	1586.921	17.922
IE	−845.84	0.2843(5E–10)	3.9372	20.1440	1693.68	1696.60	1693.862	18.057
L	−835.14	0.2262(2E–6)	2.2246	19.0450	1672.28	1675.20	1672.461	18.029

Figure 9. Estimated CDF plots of fitted distributions for Data 1.

Table 5. Data on survival time of brain cancer patients

14	22	25	40	85	121
152	180	207	210	213	225
244	258	273	336	361	371
487	488	575	581	696	734
799	819	887	943	1084	1098

Table 6. Parameter estimates and standard errors (in parentheses) for dataset 2

	$\widehat{\alpha }$	$\hat{\beta }$	$\hat{\lambda }$	$\hat{\mu }$
CTLW	0.8682 (0.0531)	0.8139 (0.0927)	0.0049 (0.0027)
TLW	1000.8 (6.7E–05)	0.1179 (0.0148)	1.9079 (0.1469)
GINW	7.2315 (1.3099)	0.7513 (0.0957)	9.2599 (1.3616)
CTLCa	538.079 (494.541)		46.225 (22.672)	−236.762 (63.383)
Ca	180.719 (47.984)	275.04 (52.908)
AW	1.071 (0.1621)	0.0028 (0.003)	1.000 (2.6E–19)
IE	111.477 (20.353)
L	0.0048 (0.0006)

Table 7. Goodness-of-fit statistics and information Criteria for dataset 2

Model	ℓ	KS(p-value)	CM	AD	AIC	BIC	CAIC	HQIC
CTLW	−211.50	0.110(0.842)	0.065	0.493	428.996	433.200	429.919	14.557
TLW	−217.15	0.191(0.198)	0.236	1.478	440.292	444.496	441.215	14.609
GINW	−219.84	0.208(0.129)	0.322	1.911	445.687	449.890	446.610	14.634
CTLCa	−180.18	0.099(0.901)	0.053	0.453	366.361	370.564	367.284	14.239
Ca	−221.64	0.327(0.002)	0.908	4.619	447.279	450.081	448.202	14.650
AW	−210.92	0.125(0.695)	0.076	0.469	427.831	432.035	428.754	14.552
IE	−222.84	0.329(0.002)	1.489	15.334	447.672	449.073	448.595	15.043
L	−213.27	0.123(0.705)	0.128	1.404	428.538	429.939	429.461	14.696

Figure 10. Estimated CDF plot of fitted distribution for Data 2.

Figure 11. PDF Plots of the LCTLW distribution.

Table 8. The estimates of the regression parameters and goodness-of-fit statistics for the dataset

Model	Parameters	Estimates	SE	p-values	Goodness-of-fit
LCTLW	${\beta }_0$	90000.00	0.29	<0.0001
	${\beta }_1$	−1389.40	22.43	<0.0001	AIC = 2842.053
	${\beta }_2$	8997.60	0.06	<0.0001	BIC = 2856.653
	$\mathit{\alpha }$	0.40	0.04	-	CAIC = 2842.234
	$\mathit{\sigma }$	11111.00	0.37	-
LGIW	${\beta }_0$	1010000.00	1.78E–07	<0.0001
	${\beta }_1$	−23386.00	9.89E–06	<0.0001	AIC = 3818.738
	${\beta }_2$	920.00	8.59E–08	<0.0001	BIC = 3833.338
	$\mathit{\alpha }$	1.44	1.23E–01	-	CAIC = 3818.919
	$\mathit{\sigma }$	1000000.00	5.19E–08	-
LEW	${\beta }_0$	89997.00	4.98E–07	<0.0001
	${\beta }_1$	−1520.50	2.91E–05	<0.0001	AIC = 3777.121
	${\beta }_2$	4997.90	2.49E–07	<0.0001	BIC = 3791.721
	$\mathit{\alpha }$	2.26	1.93E–01	-	CAIC = 3777.302
	$\mathit{\sigma }$	1110000.00	1.63E–09	-

Figure 12. Cox-Snell residual plot of LCTLW regression model.

Table

Abbreviations/Symbols	Nomenclature
AB	Average bias
AD	Anderson - Darling test
AIC	Akaike information criteria
AW	Average width
BIC	Bayesian information criteria
CAIC	Consistent akaike information criteria
CDF	Cumulative distribution function
CM	Cramér-von misses
CP	Coverage probability
CTLB	Cosine Topp–Leone Burr XII distribution
CTLCa	Cosine Topp–Leone Cauchy distribution
CTLCh	Cosine Topp–Leone Chen distribution
CTLF	Cosine Topp–Leone Frechet distribution
CTL-G	Cosine Topp–Leone Family of distributions
CTLW	Cosine Topp–Leone Weibull distribution
F(x)	CDF of a distribution
f(x)	PDF of a distribution
G(x)	CDF of a baseline distribution
g(x)	PDF of a baseline distribution
h(x)	Hazard rate function of a distribution
HQIC	Hannan–Quinn information criterion
KS	Kolmogorov–Smirnov
LCTLW	log-cosine Topp–Leone Weibull
PDF	Probability density function
$\mathbb{R}$	Real number
α	Shape parameter
β	Shape parameter
σ	Shape parameter
λ	Scale parameter
μ	Location parameter