The generalized sigmoidal quantile function

Figures & data

Figure 1. Plot of $g{(x)}_{0.1}.$

Figure 2. Plot of $g{(x)}_{1000}.$

Figure 3. Example GSQF estimator for $X\sim N(0,1)$ for n = 10 simulated values for various values of τ.

Table 1. Monte Carlo bias and standard deviation (SD) estimates across competing quantile estimators for $X\sim \mathit{Laplace}(0,1).$

		n = 10		n = 25		n = 50
u	Estimator	bias	SD	bias	SD	bias	SD
0.50	${\widehat{Q}}_{\widehat{\tau }}$	−0.01	0.39	0.00	0.22	0.01	0.16
	${\widehat{Q}}_S$	0.12	0.41	0.00	0.23	0.03	0.16
	${\widehat{Q}}_L$	−0.01	0.39	0.00	0.23	0.01	0.16
	${\widehat{Q}}_H$	−0.01	0.39	0.00	0.22	0.01	0.16
	${\widehat{Q}}_{HD}$	−0.01	0.39	0.00	0.21	0.01	0.15
	${\widehat{Q}}_K$	−0.01	0.39	0.00	0.22	0.01	0.16
0.75	${\widehat{Q}}_{\widehat{\tau }}$	0.24	0.51	0.13	0.32	0.08	0.23
	${\widehat{Q}}_S$	0.04	0.55	−0.01	0.34	0.02	0.25
	${\widehat{Q}}_L$	0.16	0.56	0.07	0.35	0.04	0.25
	${\widehat{Q}}_H$	0.17	0.56	0.07	0.35	0.04	0.25
	${\widehat{Q}}_{HD}$	0.18	0.52	0.09	0.33	0.05	0.24
	${\widehat{Q}}_K$	0.07	0.52	0.04	0.33	0.03	0.24
0.90	${\widehat{Q}}_{\widehat{\tau }}$	0.33	0.91	0.13	0.53	0.06	0.39
	${\widehat{Q}}_S$	0.64	1.31	0.01	0.58	0.11	0.45
	${\widehat{Q}}_L$	0.54	1.22	0.20	0.61	0.09	0.44
	${\widehat{Q}}_H$	0.64	1.31	0.20	0.61	0.09	0.43
	${\widehat{Q}}_{HD}$	0.16	0.95	0.18	0.56	0.09	0.40
	${\widehat{Q}}_K$	0.13	0.91	0.03	0.57	0.01	0.41
0.99	${\widehat{Q}}_{\widehat{\tau }}$	−0.57	1.48	−0.41	1.24	−0.03	1.23
	${\widehat{Q}}_S$	−1.66	1.27	−0.80	1.25	−0.09	1.26
	${\widehat{Q}}_L$	−1.66	1.27	−0.80	1.25	−0.09	1.26
	${\widehat{Q}}_H$	−1.66	1.27	−0.80	1.25	−0.09	1.26
	${\widehat{Q}}_{HD}$	−1.69	1.25	−0.87	1.19	−0.28	1.14
	${\widehat{Q}}_K$	−1.66	1.27	−0.80	1.25	−0.09	1.36
	${\widehat{Q}}_{HB}$	−0.38	1.77	−0.31	1.35	−0.07	1.16
0.999	${\widehat{Q}}_{\widehat{\tau }}$	−1.91	1.74	−2.04	1.37	−1.79	1.31
	${\widehat{Q}}_S$	−3.99	1.24	−3.12	1.25	−2.24	1.27
	${\widehat{Q}}_L$	−3.99	1.24	−3.12	1.25	−2.24	1.27
	${\widehat{Q}}_H$	−3.99	1.24	−3.12	1.25	−2.24	1.27
	${\widehat{Q}}_{HD}$	−3.99	1.24	−3.12	1.25	−2.24	1.26
	${\widehat{Q}}_K$	−3.99	1.24	−3.12	1.25	−2.24	1.27
	${\widehat{Q}}_{HB}$	−1.72	2.01	−1.73	1.65	−1.45	1.53

Table 2. Monte Carlo bias and standard deviation (SD) estimates across competing quantile estimators for $X\sim N(0,1).$

		n = 10		n = 25		n = 50
u	Estimator	bias	SD	bias	SD	bias	SD
0.50	${\widehat{Q}}_{\widehat{\tau }}$	−0.01	0.34	0.01	0.21	0.00	0.16
	${\widehat{Q}}_S$	0.12	0.39	0.01	0.24	0.02	0.17
	${\widehat{Q}}_L$	−0.01	0.37	0.01	0.24	0.00	0.17
	${\widehat{Q}}_H$	−0.01	0.37	0.01	0.23	0.00	0.17
	${\widehat{Q}}_{HD}$	−0.01	0.34	0.01	0.21	0.00	0.16
	${\widehat{Q}}_K$	0-.01	0.36	0.00	0.22	0.01	0.17
0.75	${\widehat{Q}}_{\widehat{\tau }}$	0.08	0.39	0.04	0.25	0.02	0.17
	${\widehat{Q}}_S$	−0.01	0.42	−0.03	0.28	−0.01	0.19
	${\widehat{Q}}_L$	0.07	0.41	0.03	0.27	0.01	0.19
	${\widehat{Q}}_H$	0.07	0.41	0.03	0.27	0.01	0.19
	${\widehat{Q}}_{HD}$	0.05	0.38	0.02	0.25	0.01	0.18
	${\widehat{Q}}_K$	0.00	0.40	0.00	0.26	0.00	0.18
0.90	${\widehat{Q}}_{\widehat{\tau }}$	0.13	0.48	0.08	0.30	0.03	0.21
	${\widehat{Q}}_S$	0.27	0.60	0.00	0.33	0.05	0.24
	${\widehat{Q}}_L$	0.21	0.57	0.11	0.34	0.04	0.24
	${\widehat{Q}}_H$	0.27	0.60	0.11	0.34	0.03	0.24
	${\widehat{Q}}_{HD}$	−0.01	0.48	0.08	0.31	0.03	0.22
	${\widehat{Q}}_K$	−0.01	0.48	0.03	0.32	0.00	0.23
0.99	${\widehat{Q}}_{\widehat{\tau }}$	0.02	0.69	−0.04	0.50	0.00	0.43
	${\widehat{Q}}_S$	−0.79	0.59	−0.36	0.52	−0.08	0.42
	${\widehat{Q}}_L$	−0.79	0.59	−0.36	0.52	−0.08	0.46
	${\widehat{Q}}_H$	−0.79	0.59	−0.36	0.52	−0.08	0.46
	${\widehat{Q}}_{HD}$	−0.81	0.58	−0.39	0.50	−0.15	0.42
	${\widehat{Q}}_K$	−0.79	0.59	−0.36	0.52	−0.08	0.46
	${\widehat{Q}}_{HB}$	0.09	0.80	−0.03	0.57	−0.02	0.45
0.999	${\widehat{Q}}_{\widehat{\tau }}$	−0.01	0.84	−0.33	0.55	−0.39	0.46
	${\widehat{Q}}_S$	−1.55	0.59	−1.13	0.51	−0.84	0.46
	${\widehat{Q}}_L$	−1.55	0.59	−1.13	0.51	−0.84	0.46
	${\widehat{Q}}_H$	−1.55	0.59	−1.13	0.51	−0.84	0.46
	${\widehat{Q}}_{HD}$	−1.55	0.59	−1.13	0.51	−0.85	0.46
	${\widehat{Q}}_K$	−1.55	0.59	−1.13	0.51	−0.84	0.46
	${\widehat{Q}}_{HB}$	0.05	0.98	−0.22	0.68	−0.26	0.58

Table 3. Monte Carlo bias and standard deviation (SD) estimates across competing quantile estimators for $X\sim \mathit{Logistic}(0,1).$

		n = 10		n = 25		n = 50
u	Estimator	bias	SD	bias	SD	bias	SD
0.50	${\widehat{Q}}_{\widehat{\tau }}$	−0.01	0.56	−0.01	0.35	−0.01	0.26
	${\widehat{Q}}_S$	0.19	0.63	−0.01	0.40	0.04	0.29
	${\widehat{Q}}_L$	−0.01	0.60	−0.01	0.40	0.00	0.28
	${\widehat{Q}}_H$	−0.01	0.60	−0.01	0.39	0.00	0.28
	${\widehat{Q}}_{HD}$	0.01	0.57	−0.01	0.36	−0.01	0.27
	${\widehat{Q}}_K$	−0.01	0.58	−0.01	0.38	0.00	0.28
0.75	${\widehat{Q}}_{\widehat{\tau }}$	0.21	0.66	0.08	0.41	0.04	0.30
	${\widehat{Q}}_S$	0.01	0.71	−0.05	0.44	−0.01	0.33
	${\widehat{Q}}_L$	0.16	0.71	0.06	0.47	0.02	0.33
	${\widehat{Q}}_H$	0.17	0.71	0.06	0.47	0.02	0.33
	${\widehat{Q}}_{HD}$	0.14	0.67	0.05	0.42	0.02	0.31
	${\widehat{Q}}_K$	0.04	0.67	0.01	0.43	0.00	0.32
0.90	${\widehat{Q}}_{\widehat{\tau }}$	0.38	0.96	0.15	0.61	0.08	0.45
	${\widehat{Q}}_S$	0.68	1.32	−0.01	0.66	0.11	0.51
	${\widehat{Q}}_L$	0.57	1.24	0.21	0.70	0.10	0.50
	${\widehat{Q}}_H$	0.68	1.32	0.21	0.70	0.10	0.49
	${\widehat{Q}}_{HD}$	0.14	0.99	0.18	0.65	0.09	0.46
	${\widehat{Q}}_K$	0.12	0.97	0.01	0.64	0.01	0.47
0.99	${\widehat{Q}}_{\widehat{\tau }}$	−0.29	1.52	−0.29	1.29	−0.01	1.25
	${\widehat{Q}}_S$	−1.75	1.32	−0.82	1.30	−0.12	1.30
	${\widehat{Q}}_L$	−1.75	1.32	−0.82	1.30	−0.12	1.30
	${\widehat{Q}}_H$	−1.75	1.32	−0.82	1.30	−0.12	1.30
	${\widehat{Q}}_{HD}$	−1.78	1.30	−0.90	1.25	−0.31	1.17
	${\widehat{Q}}_K$	−1.75	1.32	−0.82	1.30	−0.12	1.30
	${\widehat{Q}}_{HB}$	−0.12	1.83	−0.21	1.43	−0.05	1.22
0.999	${\widehat{Q}}_{\widehat{\tau }}$	−1.32	1.87	−1.71	1.42	−1.63	1.29
	${\widehat{Q}}_S$	−4.06	1.33	−3.12	1.31	−2.44	1.27
	${\widehat{Q}}_L$	−4.06	1.33	−3.12	1.31	−2.44	1.27
	${\widehat{Q}}_H$	−4.06	1.33	−3.12	1.31	−2.44	1.27
	${\widehat{Q}}_{HD}$	−4.07	1.33	−3.12	1.31	−2.45	1.26
	${\widehat{Q}}_K$	−4.07	1.33	−3.12	1.31	−2.44	1.27
	${\widehat{Q}}_{HB}$	−1.14	2.14	−1.39	1.73	−1.28	1.57

Table 4. Bootstrap percentile confidence interval coverage probabilites for location and scale parameters across quantile function estimators for $\alpha =0.05.$

	Laplace(0,1)				Normal(0,1)				Logistic(0,1)
	n = 10		n = 25		n = 10		n = 25		n = 10		n = 25
Estimator	μ	σ	μ	σ	μ	σ	μ	σ	μ	σ	μ	σ
${\widehat{Q}}_{\widehat{\tau }}$	0.970	0.895	0.970	0.902	0.941	0.925	0.945	0.929	0.949	0.912	0.936	0.913
${\widehat{Q}}_S$	0.929	0.816	0.917	0.890	0.906	0.779	0.927	0.870	0.920	0.786	0.917	0.877
${\widehat{Q}}_L$	0.949	0.845	0.939	0.916	0.925	0.809	0.937	0.901	0.945	0.830	0.930	0.905
${\widehat{Q}}_H$	0.949	0.824	0.940	0.909	0.914	0.792	0.939	0.895	0.918	0.798	0.928	0.902
${\widehat{Q}}_{HD}$	0.958	0.795	0.956	0.939	0.901	0.726	0.929	0.856	0.926	0.734	0.923	0.874
${\widehat{Q}}_K$	0.936	0.807	0.942	0.897	0.909	0.757	0.927	0.864	0.923	0.771	0.914	0.875
${\widehat{Q}}_{HB}$	0.936	0.895	0.935	0.907	0.932	0.930	0.939	0.923	0.933	0.906	0.919	0.906

Figure 4. Comparison of six quantile function estimators for example data of n = 20 mouse spleen to brain weight ratios.

Table 5. Select estimated quantiles for spleen to brain weight ratios for n = 20 mice.

	u
Estimator	0.01	0.1	0.25	0.5	0.75	0.9	0.99	0.999
${\widehat{Q}}_{\widehat{\tau }}$	0.012	0.036	0.052	0.070	0.092	0.113	0.175	0.192
${\widehat{Q}}_S$	0.025	0.042	0.057	0.070	0.098	0.110	0.165	0.165
${\widehat{Q}}_L$	0.025	0.037	0.056	0.069	0.094	0.110	0.165	0.165
${\widehat{Q}}_H$	0.025	0.036	0.056	0.069	0.094	0.114	0.165	0.165
${\widehat{Q}}_{HD}$	0.026	0.038	0.054	0.070	0.091	0.120	0.162	0.165
${\widehat{Q}}_K$	0.025	0.039	0.055	0.070	0.090	0.110	0.165	0.165
${\widehat{Q}}_{HB}$	0.013	0.038	0.055	0.070	0.090	0.110	0.186	0.211

Table 6. 95% Percentile confidence intervals for the E(X) and VAR(X) based on each quantile estimator for spleen to brain weight ratios for n = 20 mice.

	E(X)		STD(X)
Estimator	lower	upper	lower	upper
${\widehat{Q}}_{\widehat{\tau }}$	0.059	0.089	0.020	0.046
${\widehat{Q}}_S$	0.061	0.088	0.018	0.043
${\widehat{Q}}_L$	0.061	0.090	0.020	0.044
${\widehat{Q}}_H$	0.061	0.089	0.020	0.044
${\widehat{Q}}_{HD}$	0.061	0.089	0.020	0.041
${\widehat{Q}}_K$	0.061	0.089	0.019	0.042
${\widehat{Q}}_{HB}$	0.060	0.089	0.019	0.047