The Zero-Adjusted Log-Symmetric Distributions: Point and Intervalar Estimation

RISCO-COSAVALENTE, DIEGO; CYSNEIROS, FRANCISCO JOSÉ A.

doi:10.1590/0001-3765202320200841

DIEGO RISCO-COSAVALENTE

manuscript, analytical results and application

Universidade Federal de Pernambuco, Departamento de Estatística, Av. Prof Luiz Freire, s/n, Cidade Universitária, 50670-901 Recife, PE, Brazil

https://orcid.org/0009-0000-0784-8985

FRANCISCO JOSÉ A. CYSNEIROS

proposed the research topic

checked by

Universidade Federal de Pernambuco, Departamento de Estatística, Av. Prof Luiz Freire, s/n, Cidade Universitária, 50670-901 Recife, PE, Brazil

https://orcid.org/0000-0001-6757-6969

Correspondence to: Francisco José A. Cysneiros E-mail: cysneiros@de.ufpe.br

Distribution	$d_{g}$	$f_{g}$
Normal	1	3
Student-t	$\frac{(v + 1)}{(v + 3}$	$\frac{3 (v + 1)}{(v + 3}$
Power-exponential	$\frac{2^{1 - v} Γ [(3 - v) / 2]}{{(1 + v)}^{2} Γ [(1 + v /) 2]}$	$\frac{(v + 3)}{(v + 1)}$
Logistic-I	0.09232751	0.2508615

𝜂	𝜙	n	Mean			Median			SD			MSE			√MSE			Bias
𝜂	𝜙	n	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋
0.1	1	10	2.183	1.007	0.100	1.985	0.827	0.100	1.012	0.717	0.094	1.058	0.513	0.009	1.029	0.716	0.094	0.183	0.007	0.000
		30	2.057	1.009	0.101	2.012	0.948	0.100	0.467	0.376	0.055	0.221	0.141	0.003	0.470	0.376	0.055	0.057	0.009	0.001
		50	2.037	1.006	0.100	2.010	0.967	0.100	0.368	0.286	0.043	0.137	0.082	0.002	0.370	0.286	0.043	0.037	0.006	0.000
	2	10	2.321	2.004	0.098	1.983	1.657	0.100	1.469	1.402	0.094	2.261	1.966	0.009	1.504	1.402	0.094	0.321	0.004	-0.002
		30	2.106	1.998	0.100	1.998	1.893	0.100	0.716	0.747	0.055	0.523	0.557	0.003	0.723	0.746	0.055	0.106	-0.002	0.000
		50	2.060	2.007	0.100	1.980	1.933	0.100	0.523	0.574	0.042	0.277	0.329	0.002	0.526	0.574	0.042	0.060	0.007	0.000
	3	10	2.561	3.057	0.101	1.996	2.477	0.100	2.123	2.331	0.095	4.819	5.438	0.009	2.195	2.332	0.095	0.561	0.057	0.001
		30	2.150	2.997	0.101	1.989	2.842	0.100	0.894	1.109	0.055	0.821	1.229	0.003	0.906	1.108	0.055	0.150	-0.003	0.001
		50	2.087	3.012	0.099	1.993	2.892	0.100	0.653	0.855	0.043	0.434	0.731	0.002	0.659	0.855	0.043	0.087	0.012	-0.001
0.2	1	10	2.183	0.998	0.196	1.988	0.812	0.200	1.042	0.758	0.124	1.119	0.575	0.015	1.058	0.758	0.124	0.183	-0.002	-0.004
		30	2.049	1.001	0.200	1.989	0.935	0.200	0.508	0.399	0.074	0.260	0.159	0.005	0.510	0.399	0.074	0.049	0.001	0.000
		50	2.029	1.002	0.200	1.991	0.959	0.200	0.381	0.306	0.056	0.146	0.094	0.003	0.382	0.306	0.056	0.029	0.002	-0.000
	2	10	2.447	2.014	0.197	2.016	1.589	0.200	1.901	1.609	0.123	3.812	2.588	0.015	1.952	1.609	0.123	0.447	0.014	-0.003
		30	2.130	1.999	0.199	2.009	1.865	0.200	0.755	0.801	0.072	0.587	0.641	0.005	0.766	0.801	0.072	0.130	-0.001	-0.001
		50	2.084	2.006	0.202	2.013	1.924	0.200	0.581	0.612	0.057	0.344	0.374	0.003	0.586	0.611	0.057	0.084	0.006	0.002
	3	10	2.741	3.003	0.197	1.998	2.431	0.200	2.817	2.422	0.123	8.480	5.863	0.015	2.912	2.421	0.123	0.741	0.003	-0.003
		30	2.179	3.011	0.200	1.972	2.798	0.200	0.990	1.218	0.072	1.011	1.483	0.005	1.006	1.218	0.072	0.179	0.011	0.000
		50	2.121	3.026	0.199	1.999	2.891	0.200	0.737	0.921	0.057	0.557	0.850	0.003	0.747	0.922	0.057	0.121	0.026	-0.001
0.3	1	10	2.226	1.006	0.281	1.998	0.788	0.300	1.127	0.849	0.127	1.322	0.721	0.016	1.150	0.849	0.128	0.226	0.006	-0.019
		30	2.081	0.993	0.297	2.012	0.919	0.300	0.570	0.425	0.082	0.332	0.181	0.007	0.576	0.425	0.082	0.081	-0.007	-0.003
		50	2.033	0.999	0.300	1.991	0.947	0.300	0.408	0.328	0.066	0.167	0.107	0.004	0.409	0.328	0.066	0.033	-0.001	0.000
	2	10	2.455	1.992	0.286	1.976	1.555	0.300	1.934	1.679	0.128	3.946	2.817	0.017	1.986	1.678	0.129	0.455	-0.008	-0.014
		30	2.137	2.012	0.299	1.995	1.864	0.300	0.833	0.862	0.082	0.712	0.743	0.007	0.844	0.862	0.082	0.137	0.012	-0.001
		50	2.076	2.004	0.299	1.996	1.906	0.300	0.608	0.664	0.064	0.376	0.440	0.004	0.613	0.663	0.064	0.076	0.004	-0.001
	3	10	2.824	3.046	0.284	2.002	2.383	0.300	3.072	2.487	0.128	10.111	6.188	0.017	3.180	2.488	0.129	0.824	0.046	-0.016
		30	2.203	3.002	0.299	1.971	2.797	0.300	1.091	1.297	0.082	1.230	1.681	0.007	1.109	1.296	0.082	0.203	0.002	-0.001
		50	2.131	3.007	0.300	2.004	2.883	0.300	0.774	0.988	0.063	0.616	0.977	0.004	0.785	0.988	0.063	0.131	0.007	-0.000
0.4	1	10	2.277	1.002	0.349	2.018	0.760	0.400	1.260	0.892	0.120	1.663	0.795	0.017	1.290	0.892	0.131	0.277	0.002	-0.051
		30	2.070	1.007	0.381	1.980	0.928	0.400	0.603	0.470	0.075	0.368	0.221	0.006	0.607	0.470	0.077	0.070	0.007	-0.019
		50	2.049	1.005	0.392	1.995	0.956	0.400	0.453	0.351	0.061	0.207	0.123	0.004	0.455	0.351	0.061	0.049	0.005	-0.008
	2	10	2.527	2.053	0.356	1.987	1.541	0.400	2.025	1.840	0.117	4.377	3.386	0.016	2.092	1.840	0.125	0.527	0.053	-0.044
		30	2.158	2.010	0.383	1.999	1.847	0.400	0.913	0.923	0.074	0.859	0.852	0.006	0.927	0.923	0.076	0.158	0.010	-0.017
		50	2.117	1.997	0.390	2.027	1.892	0.400	0.663	0.714	0.061	0.453	0.510	0.004	0.673	0.714	0.062	0.117	-0.003	-0.010
	3	10	2.894	3.032	0.352	1.998	2.268	0.400	3.363	3.026	0.119	12.107	9.155	0.017	3.480	3.026	0.129	0.894	0.032	-0.048
		30	2.259	2.990	0.382	1.996	2.735	0.400	1.221	1.384	0.076	1.557	1.914	0.006	1.248	1.384	0.078	0.259	-0.010	-0.018
		50	2.150	3.009	0.392	1.999	2.855	0.400	0.841	1.052	0.061	0.729	1.107	0.004	0.854	1.052	0.061	0.150	0.009	-0.008

𝜂	𝜙	n	Mean			Median			SD			MSE			√MSE			Bias
𝜂	𝜙	n	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋
0.1	1	10	2.052	0.820	0.099	2.008	0.792	0.100	0.450	0.316	0.094	0.205	0.133	0.009	0.453	0.364	0.094	0.052	-0.180	-0.001
		30	2.013	0.944	0.100	1.998	0.939	0.100	0.239	0.194	0.054	0.057	0.041	0.003	0.240	0.202	0.054	0.013	-0.056	-0.000
		50	2.007	0.964	0.100	1.999	0.961	0.100	0.183	0.147	0.042	0.034	0.023	0.002	0.183	0.152	0.042	0.007	-0.036	0.000
	2	10	2.095	1.658	0.101	1.992	1.608	0.100	0.644	0.641	0.095	0.424	0.527	0.009	0.651	0.726	0.095	0.095	-0.342	0.001
		30	2.022	1.889	0.100	1.997	1.872	0.100	0.337	0.384	0.054	0.114	0.160	0.003	0.337	0.400	0.054	0.022	-0.111	-0.000
		50	2.014	1.936	0.100	2.000	1.937	0.100	0.255	0.299	0.042	0.065	0.093	0.002	0.256	0.305	0.042	0.014	-0.064	-0.000
	3	10	2.142	2.482	0.102	1.997	2.425	0.100	0.828	0.963	0.097	0.705	1.196	0.009	0.840	1.094	0.097	0.142	-0.518	0.002
		30	2.034	2.825	0.100	1.994	2.802	0.100	0.415	0.567	0.054	0.173	0.353	0.003	0.416	0.594	0.054	0.034	-0.175	-0.000
		50	2.023	2.906	0.100	1.996	2.902	0.100	0.317	0.440	0.043	0.101	0.202	0.002	0.318	0.450	0.043	0.023	-0.094	-0.000
0.2	1	10	2.053	0.800	0.198	1.997	0.776	0.200	0.476	0.337	0.124	0.229	0.153	0.015	0.478	0.391	0.124	0.053	-0.200	-0.002
		30	2.019	0.938	0.199	2.001	0.931	0.200	0.251	0.206	0.073	0.063	0.046	0.005	0.252	0.215	0.073	0.019	-0.062	-0.001
		50	2.008	0.966	0.202	2.000	0.964	0.200	0.189	0.157	0.056	0.036	0.026	0.003	0.189	0.161	0.057	0.008	-0.034	0.002
	2	10	2.110	1.618	0.196	2.010	1.565	0.200	0.705	0.681	0.120	0.509	0.609	0.014	0.714	0.781	0.120	0.110	-0.382	-0.004
		30	2.027	1.864	0.199	1.999	1.852	0.200	0.357	0.406	0.073	0.128	0.184	0.005	0.358	0.428	0.073	0.027	-0.136	-0.001
		50	2.014	1.929	0.199	1.995	1.919	0.200	0.270	0.314	0.057	0.073	0.104	0.003	0.270	0.322	0.057	0.014	-0.071	-0.001
	3	10	2.162	2.424	0.197	2.019	2.353	0.200	0.873	0.989	0.123	0.788	1.309	0.015	0.888	1.144	0.123	0.162	-0.576	-0.003
		30	2.052	2.793	0.200	2.002	2.761	0.200	0.454	0.609	0.074	0.209	0.414	0.005	0.457	0.643	0.074	0.052	-0.207	0.000
		50	2.030	2.894	0.200	2.000	2.876	0.200	0.334	0.481	0.057	0.113	0.242	0.003	0.336	0.492	0.057	0.030	-0.106	0.000
0.3	1	10	2.058	0.776	0.285	1.995	0.743	0.300	0.518	0.354	0.127	0.272	0.175	0.016	0.521	0.419	0.127	0.058	-0.224	-0.015
		30	2.011	0.927	0.299	1.990	0.921	0.300	0.270	0.213	0.082	0.073	0.051	0.007	0.270	0.225	0.082	0.011	-0.073	-0.001
		50	2.009	0.951	0.301	1.997	0.946	0.300	0.206	0.169	0.064	0.042	0.031	0.004	0.206	0.176	0.064	0.009	-0.049	0.001
	2	10	2.106	1.560	0.288	1.989	1.495	0.300	0.749	0.722	0.129	0.573	0.715	0.017	0.757	0.845	0.129	0.106	-0.440	-0.012
		30	2.032	1.859	0.298	1.997	1.843	0.300	0.383	0.432	0.082	0.147	0.207	0.007	0.384	0.455	0.082	0.032	-0.141	-0.002
		50	2.022	1.914	0.300	2.005	1.905	0.300	0.292	0.342	0.064	0.086	0.124	0.004	0.293	0.352	0.064	0.022	-0.086	-0.000
	3	10	2.175	2.323	0.283	1.983	2.240	0.300	0.989	1.048	0.127	1.009	1.555	0.016	1.004	1.247	0.128	0.175	-0.677	-0.017
		30	2.063	2.768	0.301	2.011	2.741	0.300	0.488	0.647	0.081	0.242	0.473	0.006	0.492	0.688	0.081	0.063	-0.232	0.001
		50	2.034	2.873	0.298	2.006	2.852	0.300	0.359	0.507	0.064	0.130	0.273	0.004	0.361	0.523	0.064	0.034	-0.127	-0.002
0.4	1	10	2.068	0.763	0.351	1.998	0.726	0.400	0.553	0.368	0.119	0.310	0.191	0.017	0.557	0.438	0.129	0.068	-0.237	-0.049
		30	2.019	0.918	0.385	2.000	0.909	0.400	0.294	0.231	0.074	0.087	0.060	0.006	0.295	0.245	0.076	0.019	-0.082	-0.015
		50	2.013	0.946	0.392	1.998	0.941	0.400	0.223	0.179	0.061	0.050	0.035	0.004	0.224	0.187	0.062	0.013	-0.054	-0.008
	2	10	2.142	1.513	0.351	2.000	1.434	0.400	0.834	0.736	0.117	0.715	0.779	0.016	0.846	0.883	0.127	0.142	-0.487	-0.049
		30	2.036	1.826	0.384	1.998	1.808	0.400	0.423	0.461	0.075	0.180	0.243	0.006	0.424	0.493	0.076	0.036	-0.174	-0.016
		50	2.025	1.894	0.390	2.007	1.883	0.400	0.308	0.364	0.063	0.095	0.144	0.004	0.309	0.379	0.064	0.025	-0.106	-0.010
	3	10	2.228	2.278	0.353	1.997	2.168	0.400	1.094	1.103	0.119	1.249	1.738	0.016	1.118	1.318	0.127	0.228	-0.722	-0.047
		30	2.062	2.762	0.381	2.006	2.739	0.400	0.515	0.690	0.074	0.269	0.533	0.006	0.518	0.730	0.076	0.062	-0.238	-0.019
		50	2.030	2.851	0.392	1.990	2.826	0.400	0.387	0.547	0.061	0.151	0.321	0.004	0.388	0.567	0.061	0.030	-0.149	-0.008

𝜂	𝜙	n	90%									95%									99%
			LC(%)			UC(%)			Coverage(%)			LC(%)			UC(%)			Coverage(%)			LC(%)			UC(%)			Coverage(%)
			𝜂	𝜙	𝜋	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋
0.1	1	10	8.05	2.94	6.86	8.43	11.53	0.00	83.53	85.53	93.14	5.40	1.18	1.24	5.24	6.85	0.00	89.35	91.97	98.76	2.50	0.20	1.24	2.24	2.70	0.00	95.26	97.10	98.76
		30	6.08	3.32	6.60	5.86	8.14	0.00	88.06	88.54	93.40	3.38	1.56	2.16	3.28	4.94	0.00	93.34	93.50	97.84	0.74	0.34	0.62	0.78	1.52	0.00	98.48	98.14	99.38
		50	6.04	3.70	6.06	5.30	7.10	3.26	88.66	89.20	90.68	2.98	1.76	2.44	3.20	4.08	0.00	93.82	94.16	97.56	0.66	0.28	0.94	0.80	1.04	0.00	98.54	98.68	99.06
	2	10	8.03	2.24	6.78	7.53	11.04	0.00	84.44	86.72	93.22	4.91	1.18	1.12	4.75	6.91	0.00	90.35	91.91	98.88	2.04	0.14	1.12	1.88	2.72	0.00	96.07	97.14	98.88
		30	5.81	3.40	7.70	5.75	7.29	0.00	88.45	89.31	92.30	3.06	1.50	2.52	2.90	3.86	0.00	94.03	94.63	97.48	0.76	0.12	0.62	0.66	1.16	0.00	98.58	98.72	99.38
		50	4.97	3.75	6.10	5.31	7.34	2.56	89.71	88.91	91.34	2.95	1.91	2.78	2.85	3.87	0.00	94.20	94.22	97.22	0.80	0.34	1.04	0.66	0.94	0.00	98.54	98.72	98.96
	3	10	8.09	2.81	7.44	7.85	11.46	0.00	84.05	85.73	92.56	5.09	1.44	1.48	5.11	7.55	0.00	89.80	91.00	98.52	1.62	0.20	1.48	2.18	2.87	0.00	96.19	96.93	98.52
		30	5.94	3.31	7.64	5.16	7.91	0.00	88.90	88.78	92.36	3.25	1.35	2.42	2.87	4.42	0.00	93.88	94.24	97.58	0.78	0.22	0.62	0.64	1.06	0.00	98.57	98.71	99.38
		50	5.79	3.55	5.92	5.85	7.28	3.14	88.36	89.16	90.94	3.23	1.45	2.58	3.29	3.87	0.00	93.48	94.67	97.42	0.89	0.32	0.92	0.81	0.69	0.00	98.31	98.99	99.08
0.2	1	10	7.66	2.84	3.74	8.80	12.93	0.00	83.53	84.23	96.26	5.18	1.16	3.74	5.44	8.14	0.00	89.38	90.70	96.26	2.22	0.16	0.72	2.32	3.14	0.00	95.46	96.70	99.28
		30	5.74	3.28	5.76	6.28	7.98	4.12	87.97	88.73	90.12	3.08	1.44	2.16	3.70	4.56	0.00	93.22	94.00	97.84	0.80	0.30	0.96	1.20	1.54	0.00	98.00	98.16	99.04
		50	5.86	3.80	6.42	5.90	7.76	4.94	88.23	88.43	88.64	3.10	1.58	3.46	3.04	4.10	1.76	93.86	94.32	94.78	0.90	0.34	0.90	0.82	1.06	0.00	98.28	98.60	99.10
	2	10	7.87	2.40	2.90	9.23	12.87	0.00	82.91	84.73	97.10	5.40	0.90	2.90	5.68	8.35	0.00	88.91	90.75	97.10	2.22	0.18	0.58	2.66	3.04	0.00	95.12	96.78	99.42
		30	6.27	3.37	6.18	5.53	8.69	4.30	88.20	87.94	89.52	3.63	1.62	2.52	3.02	4.77	0.00	93.35	93.61	97.48	1.00	0.30	0.90	0.78	1.50	0.00	98.22	98.20	99.10
		50	5.59	3.39	6.08	5.35	7.18	4.42	89.05	89.43	89.50	3.05	1.70	3.12	2.81	4.13	1.90	94.15	94.17	94.98	0.62	0.18	0.84	0.74	1.18	0.00	98.64	98.64	99.16
	3	10	7.87	2.60	3.12	8.89	12.64	0.00	83.23	84.76	96.88	4.97	1.12	3.12	5.99	8.35	0.00	89.04	90.53	96.88	2.04	0.18	0.70	2.40	2.98	0.00	95.55	96.84	99.30
		30	5.73	3.62	6.14	6.31	7.98	4.64	87.96	88.40	89.22	2.96	1.57	2.50	3.68	4.40	0.00	93.37	94.03	97.50	1.03	0.24	0.98	1.07	1.53	0.00	97.91	98.23	99.02
		50	6.06	3.76	5.96	5.50	6.36	5.30	88.45	89.88	88.74	3.32	1.81	3.08	2.98	3.22	2.06	93.70	94.97	94.86	1.03	0.28	0.70	0.89	0.79	0.00	98.09	98.93	99.30
0.3	1	10	9.06	2.66	4.02	9.24	13.06	0.00	81.70	84.28	95.98	5.82	1.20	0.86	5.84	8.68	0.00	88.34	90.12	99.14	2.58	0.20	0.06	2.62	3.58	0.00	94.80	96.22	99.94
		30	6.56	3.20	3.56	5.84	8.88	3.22	87.60	87.92	93.22	3.24	1.38	1.34	3.34	4.92	0.94	93.42	93.70	97.72	0.96	0.38	0.56	1.16	1.34	0.00	97.88	98.28	99.44
		50	4.84	3.72	4.92	5.26	7.54	4.46	89.90	88.74	90.62	2.70	1.84	2.44	3.20	4.20	2.10	94.10	93.96	95.46	0.58	0.28	0.56	0.60	1.10	0.28	98.82	98.62	99.16
	2	10	8.81	2.46	4.82	9.82	13.14	0.00	81.37	84.40	95.18	6.15	1.26	1.00	6.73	8.49	0.00	87.12	90.25	99.00	2.74	0.20	0.14	3.12	3.50	0.00	94.13	96.29	99.86
		30	5.67	3.14	4.58	6.55	9.13	2.78	87.78	87.72	92.64	2.94	1.50	1.84	3.73	5.19	0.98	93.33	93.31	97.18	0.74	0.38	0.86	1.14	1.50	0.00	98.12	98.12	99.14
		50	5.73	3.71	4.46	5.71	7.24	4.18	88.55	89.05	91.36	2.93	1.64	2.24	3.17	3.99	1.80	93.91	94.37	95.96	0.72	0.30	0.56	0.76	1.14	0.18	98.52	98.56	99.26
	3	10	8.60	2.69	4.76	8.84	13.51	0.00	82.56	83.80	95.24	5.47	1.32	1.12	6.09	8.94	0.00	88.43	89.74	98.88	2.71	0.22	0.16	3.01	3.61	0.00	94.29	96.17	99.84
		30	6.35	3.29	3.64	5.38	8.31	3.18	88.27	88.39	93.18	3.74	1.75	1.36	2.87	4.80	1.08	93.39	93.45	97.56	1.21	0.20	0.46	0.84	1.49	0.00	97.95	98.31	99.54
		50	5.55	3.84	4.24	5.59	6.70	4.00	88.86	89.46	91.76	2.86	1.87	2.00	3.02	3.80	1.72	94.13	94.33	96.28	0.64	0.26	0.48	0.82	1.07	0.14	98.53	98.67	99.38
0.4	1	10	10.53	2.21	5.21	10.14	15.46	4.35	79.33	82.34	90.44	7.28	1.14	1.08	6.90	9.94	0.00	85.83	88.91	98.92	3.63	0.24	0.00	3.23	4.47	0.00	93.14	95.29	100.00
		30	6.38	3.36	4.92	6.20	9.30	4.04	87.42	87.34	91.04	3.38	1.66	2.40	3.72	5.44	1.80	92.90	92.90	95.80	1.06	0.18	0.16	1.14	1.66	0.14	97.80	98.16	99.70
		50	5.98	3.70	5.74	5.54	7.94	5.18	88.47	88.35	89.08	3.22	1.66	2.96	3.18	4.36	2.40	93.60	93.98	94.64	0.84	0.32	0.28	0.70	1.28	0.10	98.46	98.40	99.62
	2	10	9.53	2.51	4.91	9.91	14.08	4.05	80.55	83.41	91.04	6.52	1.14	1.20	6.42	9.43	0.00	87.06	89.43	98.80	3.19	0.18	0.00	3.21	3.93	0.00	93.60	95.89	100.00
		30	6.77	3.06	4.62	6.23	8.43	3.74	87.00	88.50	91.64	3.91	1.52	2.26	3.27	5.09	1.46	92.83	93.39	96.28	1.28	0.20	0.30	0.80	1.56	0.20	97.92	98.24	99.50
		50	5.77	3.37	5.38	5.67	7.96	6.12	88.55	88.67	88.50	3.13	1.62	3.16	3.03	4.43	3.14	93.85	93.95	93.70	0.70	0.28	0.36	0.86	1.38	0.32	98.44	98.34	99.32
	3	10	9.95	2.23	5.93	9.55	14.08	3.85	80.50	83.70	90.22	6.58	1.10	0.92	6.76	9.73	0.00	86.66	89.17	99.08	3.15	0.24	0.00	3.55	4.35	0.00	93.30	95.41	100.00
		30	6.47	2.53	4.84	6.47	8.59	4.28	87.07	88.88	90.88	3.53	1.00	2.18	3.80	5.10	1.94	92.67	93.90	95.88	1.10	0.10	0.28	0.94	1.69	0.18	97.95	98.21	99.54
		50	5.88	3.03	5.24	5.00	7.81	5.40	89.11	89.15	89.36	3.13	1.49	2.62	2.75	4.14	3.02	94.12	94.38	94.36	0.74	0.20	0.28	0.80	1.27	0.16	98.45	98.53	99.56

𝜂	𝜙	n	90%									95%									99%
			LC(%)			UC(%)			Coverage(%)			LC(%)			UC(%)			Coverage(%)			LC(%)			UC(%)			Coverage(%)
			𝜂	𝜙	𝜋	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋	𝜂	𝜙	𝜋
0.1	1	10	10.18	0.56	7.00	9.84	23.32	0.00	79.98	76.12	93.00	7.26	0.08	1.08	7.00	17.70	0.00	85.74	82.22	98.92	3.48	0.00	1.08	3.58	9.64	0.00	92.94	90.36	98.92
		30	6.70	1.36	6.90	6.38	13.76	0.00	86.92	84.88	93.10	3.98	0.48	2.36	3.98	8.88	0.00	92.04	90.64	97.64	1.44	0.04	0.56	1.44	3.54	0.00	97.12	96.42	99.44
		50	6.18	1.98	5.56	6.22	11.10	2.96	87.60	86.92	91.48	3.36	0.76	2.22	3.50	7.02	0.00	93.14	92.22	97.78	1.02	0.16	0.84	1.00	2.08	0.00	97.98	97.76	99.16
	2	10	9.54	0.66	6.90	9.66	22.96	0.00	80.80	76.38	93.10	6.40	0.10	1.36	6.60	17.04	0.00	87.00	82.86	98.64	3.42	0.00	1.36	3.00	8.90	0.00	93.58	91.10	98.64
		30	6.42	2.16	6.68	7.14	13.44	0.00	86.44	84.40	93.32	3.26	0.72	2.18	4.16	8.74	0.00	92.58	90.54	97.82	0.92	0.06	0.68	1.30	3.50	0.00	97.78	96.44	99.32
		50	6.24	2.42	5.50	5.50	10.64	2.82	88.26	86.94	91.68	3.56	0.90	2.34	3.14	6.62	0.00	93.30	92.48	97.66	0.96	0.00	0.78	0.96	2.24	0.00	98.08	97.76	99.22
	3	10	9.13	0.40	7.15	10.19	23.03	0.00	80.67	76.57	92.85	6.67	0.00	1.46	7.01	17.20	0.00	86.32	82.80	98.54	3.44	0.00	1.46	3.34	8.95	0.00	93.21	91.05	98.54
		30	6.62	1.44	7.36	6.40	13.46	0.00	86.98	85.10	92.64	3.52	0.64	2.46	3.68	8.46	0.00	92.80	90.90	97.54	1.08	0.00	0.76	1.08	3.24	0.00	97.84	96.76	99.24
		50	5.96	1.88	6.04	6.02	9.78	3.36	88.02	88.34	90.60	3.20	0.72	2.40	3.16	6.40	0.00	93.64	92.88	97.60	0.78	0.08	0.84	0.90	2.20	0.00	98.32	97.72	99.16
0.2	1	10	10.28	0.50	3.62	10.26	24.90	0.00	79.46	74.60	96.38	7.46	0.10	3.62	7.40	19.46	0.00	85.14	80.44	96.38	4.24	0.00	0.78	3.98	10.98	0.00	91.78	89.02	99.22
		30	6.74	1.72	6.06	6.26	13.68	4.48	87.00	84.60	89.46	4.08	0.58	2.64	3.44	9.14	0.00	92.48	90.28	97.36	1.48	0.02	1.00	0.96	3.78	0.00	97.56	96.20	99.00
		50	6.20	2.22	6.10	7.02	10.22	4.76	86.78	87.56	89.14	3.22	0.88	2.76	4.28	6.60	1.84	92.50	92.52	95.40	0.88	0.04	0.56	1.20	2.56	0.00	97.92	97.40	99.44
	2	10	10.34	0.60	3.06	9.90	25.48	0.00	79.76	73.92	96.94	7.24	0.12	3.06	6.90	18.88	0.00	85.86	81.00	96.94	4.02	0.00	0.64	3.46	10.60	0.00	92.52	89.40	99.36
		30	7.54	1.42	5.74	6.18	13.78	4.80	86.28	84.80	89.46	4.32	0.44	2.32	3.82	9.14	0.00	91.86	90.42	97.68	1.36	0.02	0.92	1.44	4.16	0.00	97.20	95.82	99.08
		50	5.78	2.02	6.00	5.38	11.16	4.52	88.84	86.82	89.48	3.50	0.80	2.78	3.10	7.22	1.92	93.40	91.98	95.30	1.10	0.06	0.50	0.68	2.66	0.00	98.22	97.28	99.50
	3	10	9.92	0.40	2.94	9.58	24.16	0.00	80.50	75.44	97.06	6.98	0.02	2.94	6.72	18.02	0.00	86.30	81.96	97.06	3.54	0.00	0.66	3.50	10.04	0.00	92.96	89.96	99.34
		30	6.86	1.30	6.42	6.50	13.80	4.32	86.64	84.90	89.26	4.02	0.52	2.56	3.90	8.98	0.00	92.08	90.50	97.44	1.38	0.02	0.94	1.44	3.44	0.00	97.18	96.54	99.06
		50	6.14	2.18	5.76	5.60	11.14	4.94	88.26	86.68	89.30	3.08	0.86	3.16	3.02	7.02	1.92	93.90	92.12	94.92	0.88	0.08	0.62	1.00	2.68	0.00	98.12	97.24	99.38
0.3	1	10	10.74	0.38	4.96	11.88	27.80	0.00	77.38	71.82	95.04	7.84	0.04	0.96	9.54	21.36	0.00	82.62	78.60	99.04	4.16	0.00	0.12	5.54	12.80	0.00	90.30	87.20	99.88
		30	7.12	1.30	4.18	7.02	13.92	3.36	85.86	84.78	92.46	4.34	0.46	1.52	4.38	9.24	0.90	91.28	90.30	97.58	1.72	0.04	0.64	1.66	4.04	0.00	96.62	95.92	99.36
		50	6.24	1.70	4.76	6.12	12.36	4.28	87.64	85.94	90.96	3.30	0.58	2.60	3.24	7.26	2.08	93.46	92.16	95.32	1.00	0.06	0.56	1.00	2.44	0.34	98.00	97.50	99.10
	2	10	11.06	0.32	4.70	11.36	27.20	0.00	77.58	72.48	95.30	8.44	0.02	1.12	8.48	21.20	0.00	83.08	78.78	98.88	4.98	0.00	0.08	4.78	12.90	0.00	90.24	87.10	99.92
		30	7.24	1.26	3.68	6.40	14.36	2.94	86.36	84.38	93.38	4.34	0.40	1.62	3.88	9.58	0.94	91.78	90.02	97.44	1.70	0.06	0.60	1.32	4.08	0.00	96.98	95.86	99.40
		50	5.86	1.84	5.22	6.02	11.46	4.08	88.12	86.70	90.70	3.56	0.60	2.80	3.38	7.22	1.92	93.06	92.18	95.28	1.00	0.06	0.64	0.84	2.72	0.26	98.16	97.22	99.10
	3	10	10.47	0.18	5.26	12.07	27.32	0.00	77.47	72.50	94.74	7.54	0.04	1.32	8.87	21.31	0.00	83.59	78.65	98.68	4.24	0.00	0.08	5.10	12.75	0.00	90.65	87.25	99.92
		30	7.14	1.50	3.98	6.74	14.84	2.90	86.12	83.66	93.12	4.66	0.52	1.62	4.26	9.98	0.84	91.08	89.50	97.54	1.38	0.02	0.52	1.74	4.70	0.00	96.88	95.28	99.48
		50	5.58	1.92	4.80	5.68	11.44	3.86	88.74	86.64	91.34	3.02	0.80	2.52	3.10	6.78	1.62	93.88	92.42	95.86	0.82	0.08	0.66	0.98	2.34	0.30	98.20	97.58	99.04
0.4	1	10	12.76	0.38	5.29	13.26	30.68	4.01	73.98	68.94	90.71	9.67	0.02	0.88	10.37	24.63	0.00	79.95	75.35	99.12	5.87	0.00	0.00	6.27	15.36	0.00	87.86	84.64	100.00
		30	6.96	1.12	5.08	7.14	15.20	3.82	85.90	83.68	91.10	4.14	0.24	2.14	4.34	10.46	1.34	91.52	89.30	96.52	1.44	0.02	0.36	1.52	4.74	0.10	97.04	95.24	99.54
		50	6.02	1.78	5.98	6.52	12.14	5.20	87.46	86.08	88.82	3.06	0.60	3.18	3.84	7.84	2.74	93.10	91.56	94.08	1.12	0.00	0.46	1.14	2.86	0.18	97.74	97.14	99.36
	2	10	13.02	0.32	5.13	12.66	30.89	3.79	74.32	68.79	91.09	10.06	0.04	1.04	9.31	24.94	0.00	80.63	75.02	98.96	6.05	0.00	0.00	5.75	15.99	0.00	88.20	84.01	100.00
		30	7.16	1.22	4.68	6.94	14.66	4.24	85.90	84.12	91.08	4.24	0.30	2.12	4.46	10.04	1.64	91.30	89.66	96.24	1.22	0.00	0.18	1.70	4.52	0.18	97.08	95.48	99.64
		50	5.82	1.86	5.24	6.28	12.82	6.04	87.90	85.32	88.72	3.04	0.76	2.88	3.64	7.82	2.92	93.32	91.42	94.20	0.92	0.04	0.32	1.04	2.90	0.22	98.04	97.06	99.46
	3	10	12.23	0.22	5.01	12.45	31.07	4.01	75.33	68.71	90.99	9.48	0.00	1.00	9.32	24.75	0.00	81.20	75.25	99.00	6.31	0.00	0.00	6.03	16.32	0.00	87.65	83.68	100.00
		30	7.12	1.44	4.64	7.44	15.94	4.38	85.44	82.62	90.98	4.28	0.36	2.28	4.86	10.80	1.68	90.86	88.84	96.04	1.48	0.00	0.32	1.86	5.00	0.12	96.66	95.00	99.56
		50	6.44	1.80	5.16	6.06	12.68	5.38	87.50	85.52	89.46	3.44	0.64	3.08	3.52	7.94	3.08	93.04	91.42	93.84	1.02	0.14	0.46	1.32	3.08	0.18	97.66	96.78	99.36

Term	n	Min	Q1	Median	$\overline{y}$	Q3
$w$	55	0	0	3879.96	6060.98	7761
$w > 0$	37	221.93	3879.96	6759.58	9009.571	9259.03
Term	Max	SD	IQR	CV	CS	CK
$w$	55908.96	9615.13	7761	1.59	3.40	16.55
$w > 0$	55908.9	10553.33	5379.07	1.17	3.10	13.21

	ZALSt	ZALPE	ZALN	ZAGA	ZARBS	ZAIG
𝜂	5989.105	6023.433	5841.316	-	-	-
	(806.091)	(896.832)	(929.127)	-	-	-
𝜇	-	-	-	9009.571	8767.315	9009.571
	-	-	-	(1294.390)	(1694.320)	(2097.152)
𝜙	0.475	0.508	0.938	-	-	-
	(0.146)	(0.135)	(0.218)	-	-	-
𝜎	-	-	-	0.879	1.387	0.015
	-	-	-	(0.092)	(0.325)	(0.002)
−2log(𝐿̂)	809.13	811.23	813.94	815.97	822.20	825.15
AIC	815.13	817.23	819.94	821.97	828.20	831.15
BIC	821.15	823.25	825.97	827.99	834.22	837.18
KS	0.151	0.132	0.151	0.150	0.230	0.249
p-value	0.369	0.541	0.369	0.380	0.040	0.021

[1] Correspondence to: Francisco José A. Cysneiros E-mail: cysneiros@de.ufpe.br

Brasil

Brasil

The Zero-Adjusted Log-Symmetric Distributions: Point and Intervalar Estimation

Abstract