Approximate Calculation Method for Noncentral t-Distribution Quantile

Table 1

Corollary results of coefficient k (C = 0.5).

	0.7
n	Exact solution	Method 1	Relative error (%)	Method 2	Relative error (%)	Method 3	Relative error (%)
3	0.597986	0.415034	−30.59	0.524401	−12.31	0.608001	1.67
4	0.572487	0.435142	−23.99	0.524401	−8.40	0.577322	0.84
5	0.560064	0.449987	−19.65	0.524401	−6.37	0.563115	0.54
6	0.552664	0.460839	−16.61	—	—	0.554921	0.41
7	0.547827	0.469046	−14.38	0.524401	−4.28	0.54959	0.32
8	0.544407	0.475423	−12.67	0.524401	−3.68	0.545844	0.26
9	0.541868	0.480519	−11.32	0.524401	−3.22	0.543068	0.22
10	0.539877	0.484634	−10.23	0.524401	−2.87	0.585111	0.19
20	0.53166	0.503979	−5.21	0.524401	−1.37	0.562683	0.08
30	0.529147	0.510691	−3.49	0.524401	−0.90	0.554224	0.05
40	0.527907	0.514103	−2.61	0.524401	−0.66	0.549534	0.04
Error range			−30.59∼−2.61		−12.31∼−0.66		0.04∼1.67
	0.8
n	Exact solution	Method 1	Relative error (%)	Method 2	Relative error (%)	Method 3	Relative error (%)
3	0.970715	0.88509	−8.82	0.841621	−13.3	0.990143	2
4	0.924547	0.859236	−7.06	0.841621	−9.0	0.935135	1.15
5	0.902606	0.849782	−5.85	—	—	0.909861	0.8
6	0.889844	0.845484	−4.99	—	—	0.895342	0.62
7	0.881479	0.843315	−4.33	0.841621	−4.52	0.885917	0.5
8	0.875564	0.842031	−3.83	0.841621	−3.88	0.879306	0.43
9	0.871202	0.841351	−3.43	0.841621	−3.4	0.874412	0.37
10	0.867798	0.840878	−3.1	0.841621	−3.02	0.870643	0.33
20	0.853853	0.840228	−1.6	0.841621	−1.43	0.855123	0.15
30	0.849585	0.840502	−1.07	0.841621	−0.94	0.850419	0.1
40	0.847534	0.84068	−0.81	0.841621	−0.7	0.848145	0.07
Error range			−8.82∼−0.81		−13.3∼−0.7		0.07∼2
	0.9
n	Exact solution	Method 1	Relative error (%)	Method 2	Relative error (%)	Method 3	Relative error (%)
3	1.498545	1.481939	−1.11	1.281552	−14.5	1.530211	2.11
4	1.418883	1.403378	−1.09	—	—	1.437254	1.29
5	1.381884	1.368009	−1	—	—	1.394886	0.94
6	1.360419	1.348175	−0.9	—	—	1.370643	0.75
7	1.346566	1.335561	−0.82	1.281552	−4.83	1.354944	0.62
8	1.336729	1.326928	−0.73	1.281552	−4.13	1.343949	0.54
9	1.32965	1.320672	−0.68	1.281552	−3.62	1.335819	0.46
10	1.324126	1.315862	−0.62	1.281552	−3.22	1.329564	0.41
20	1.301272	1.29683	−0.34	1.281552	−1.52	1.303854	0.2
30	1.294449	1.291352	−0.24	1.281552	−1	1.296077	0.13
40	1.291037	1.28876	−0.18	1.281552	−0.73	1.292321	0.1
Error range			−1.11∼−0.18		−14.5∼−0.73		0.1∼2.11
	0.95
n	Exact solution	Method 1	Relative error (%)	Method 2	Relative error (%)	Method 3	Relative error (%)
3	1.938445	1.954945	0.85	—	—	1.978771	2.08
4	1.829525	1.83853	0.49	—	—	1.853356	1.3
5	1.77926	1.784694	0.31	—	—	1.796427	0.96
6	1.75043	1.753874	0.20	—	—	1.763918	0.77
7	1.731816	1.734234	0.14	1.644854	−5.02	1.742891	0.64
8	1.718722	1.72031	0.09	1.644854	−4.3	1.728176	0.55
9	1.709037	1.710337	0.08	1.644854	−3.76	1.717302	0.48
10	1.701662	1.702589	0.05	1.644854	−3.34	1.708939	0.43
20	1.671163	1.67113	0	1.644854	−1.57	1.6746	0.21
30	1.661982	1.661913	0	1.644854	−1.03	1.664222	0.13
40	1.657536	1.657417	0	1.644854	−0.77	1.659212	0.1
Error range			0∼0.85		−5.02∼−0.77		0.1∼2.08