On some aspects of a bivariate alternative zero-inflated logarithmic series distribution

Figures & data

Table 1. Observed frequencies and computed values of expected frequencies of the BZILSD, the BAZILSD and the BPD by method of maximum likelihood for the first data set.

	0	1	2	3	4	∑
0	34	20	4	6	4	68
	34.99	12.03	7.15	4.66	3.38	65.21
	36.58	18.01	5.01	4.13	3.49	67.22
	34.51	16.22	3.81	0.59	0.07	55.20
1	17	7	0	0	0	24
	12.29	10.16	1.56	0.51	0.21	24.73
	16.53	6.01	1.23	0.12	0.03	23.92
	23.12	11.21	2.72	0.44	0.05	37.54
2	6	4	1	0	0	11
	7.56	2.17	1.01	0.25	0.01	11
	5.56	3.38	1.89	0.14	0.01	10.98
	7.75	3.87	0.97	0.16	0.02	12.77
3	0	4	2	2	0	4
	3.15	1.12	0.91	0.21	0.01	5.4
	3.11	1.70	0.56	0.09	0.01	5.53
	1.73	0.89	0.23	0.04	0.005	2.89
4	0	0	0	1	0	0
	1.40	0.71	0.11	0.09	0.01	2.32
	1.03	0.05	0.03	0.02	0.01	1.14
	0.29	0.15	0.04	0.007	0.0009	0.50
5	2	0	0	0	0	2
	0.15	0.08	0.07	0.03	0.01	0.34
	0.09	0.05	0.04	0.02	0.01	0.21
	0.04	0.02	0.005	0.001	0.04	0.11
∑	59	35	5	6	4	109
	62.54	26.27	10.81	5.75	3.63	109
	62.90	29.60	8.76	4.52	3.56	109
	67.44	32.36	7.78	1.24	0.18	109

Table 2. Observed frequencies and computed values of expected frequencies of the BZILSD, the BAZILSD and the BPD by method of maximum likelihood for the second data set.

	0	1	2	3	∑
0	27	9	3	2	41
	35.52	8.01	4.61	2.52	50.06
	28.01	7.98	3.93	1.23	41.15
	26.91	16.42	5.01	1.02	49.36
1	24	13	1	0	38
	16.01	11.21	0.91	0.11	28.24
	23.12	14.21	1.23	0.16	38.72
	16.69	10.45	3.27	0.68	31.09
2	8	2	1	0	11
	5.61	4.31	1.76	0.51	12.19
	7.62	3.02	0.76	0.28	11.68
	5.17	3.32	1.07	0.23	9.79
3	1	0	2	0	4
	1.04	1.12	0.91	0.21	5.4
	1.28	1.7	0.56	0.09	5.53
	1.07	0.7	0.23	0.76	2.76
∑	60	24	7	2	93
	58.42	24.04	7.39	3.15	93
	59.79	25.52	6.01	1.68	93
	49.84	30.89	9.58	2.69	93

Table 3. The computed Chi-square value and $P$ value while fitting the models – BZILSD, BAZILSD and BPD for the Data set 1 and Data set 2.

Data set/models	Chi-square value	Degrees of freedom	P-value
Data set 1
BZILSD	12.28	4	.02
BAZILSD	1.56	4	.82
BPD	17.96	3	<.0001
Data set 2
BZILSD	8.16	3	.04
BAZILSD	0.43	2	0.81
BPD	8.55	3	0.04

Table 4. The computed the values of $\log L(\underset{\_}{\hat{\mathrm{\Omega }}};x),\log L(\underset{\_}{{\hat{\mathrm{\Omega }}}^{\ast }};x)$ and the generalized likelihood ratio test statistic under $H_0$.

	$\log L(\underset{\_}{{\hat{\mathrm{\Omega }}}^{\ast }};x)$	$\log L(\underset{\_}{\hat{\mathrm{\Omega }}};x)$	Test statistic
Data set 1	−154.03	−146.16	15.74
Data set 2	−98.42	−95.49	5.86

Table 5. Bias and standard errors (within parenthesis) of the estimators of the parameters ${\theta }_1$,${\theta }_2$, ${\theta }_3$ and $\alpha$ of the BAZILSD for the simulated data sets.

Parameters set	Sample size	${\hat{\theta }}_1$	${\hat{\theta }}_2$	${\hat{\theta }}_3$	$\hat{\alpha }$
Set (i) ${\theta }_1=0.4361$, ${\theta }_2=0.2679$, ${\theta }_3=0.1905$, $\alpha =0.0110$	150	0.0889 (0.1446)	0.0744 (0.1533)	0.0613 (0.1308)	0.0503 (0.1153)
	300	0.0598 (0.1281)	0.0506 (0.1273)	0.0432 (0.1005)	0.0297 (0.0927)
	600	0.0288 (0.1049)	0.0118 (0.0949)	0.0232 (0.0917)	0.0074 (0.0693)
Set (ii) ${\theta }_1=0.0847$, ${\theta }_2=0.0439$, ${\theta }_3=0.0216$, $\alpha =0.0112$	150	0.0466 (0.1456)	0.0263 (0.1122)	0.0113 (0.1204)	0.0121 (0.0917)
	300	0.0284 (0.1136)	0.0176 (0.0927)	0.0099 (0.0866)	0.0071 (0.0721)
	600	0.0084 (0.1030)	0.0097 (0.0843)	0.0044 (0.0748)	0.0026 (0.0583)