The mean, variance, and bias of the OLS based estimator of the extremum of a quadratic regression model for small samples

Figures & data

Table 1. Values of β₁, β₂, and β₃ used in the simulation study for values of θ corresponding to the 50th, 75th, and 95th percentile.

	θ = 0	$\theta =0.6744898$	$\theta =1.644854$
	(50th perc.)	(75th perc.)	(95th perc.)
β1	β2	β2	β2	β3
1	0	−0.1348980	−0.3289708	0.1
1	0	−0.6744898	−1.6448540	0.5
1	0	−1.2140816	−2.9607372	0.9

Figure 1. Graph of $\widehat{\mathit{bias}}(\widehat{\theta })$ for the case ${\beta }_3=0.1$ and ${\sigma }^2=1.$

Figure 2. Graph of $\widehat{\mathit{bias}}(\widehat{\theta })$ for the case ${\beta }_3=0.5$ and ${\sigma }^2=1.$

Figure 3. Graph of $\widehat{\mathit{bias}}(\widehat{\theta })$ for the case ${\beta }_3=0.9$ and ${\sigma }^2=1.$

Table 2. Results of $\widehat{\mathit{bias}}(\widehat{\theta })$ for the 50th (θ = 0), 75th ($\theta =0.6744898$), and 95th ($\theta =1.644854$) percentile with ${\sigma }^2=1$ and ${\sigma }^2=4.$

		${\sigma }^2=1$			${\sigma }^2=4$
		50th	75th	95th	50th	75th	95th
β3	n	perc.	perc.	perc.	perc.	perc.	perc.
0.1	25	0.0872	−0.2850	−0.8205	1.7594	5.0535	9.7925
0.1	50	−0.3881	0.5269	1.8434	0.9437	0.7268	0.4147
0.1	75	−0.2434	0.7980	2.2962	−0.1036	−0.7137	−1.5914
0.1	100	0.0176	−0.7367	−1.8219	−0.0888	−0.3370	−0.6941
0.1	150	2.2235	3.8669	6.2310	1.2390	1.3685	1.5548
0.1	200	−0.0862	0.0668	0.2868	−0.5010	−1.3501	−2.5716
0.1	300	−0.0295	0.1152	0.3234	−0.4402	−0.8896	−1.5361
0.1	400	0.0943	0.5279	1.1516	−0.3006	0.1043	0.6868
0.1	500	−0.0071	0.1272	0.3205	−1.3287	0.8078	3.8814
0.1	750	0.0003	0.0609	0.1480	0.0114	0.2632	0.6253
0.1	1000	−0.0005	0.0366	0.0900	0.0830	0.1406	0.2234
0.5	25	−0.0041	0.0946	0.2365	0.2258	0.6895	1.3565
0.5	50	0.0023	0.0484	0.1147	0.0948	0.4322	0.9176
0.5	75	−0.0002	0.0254	0.0623	−0.1151	−1.1370	−2.6071
0.5	100	−0.0005	0.0153	0.0380	−0.0043	0.1030	0.2575
0.5	150	0.0003	0.0115	0.0276	−0.0019	0.0485	0.1210
0.5	200	−0.0006	0.0076	0.0195	−0.0011	0.0351	0.0871
0.5	300	−0.0012	0.0036	0.0105	−0.0025	0.0182	0.0481
0.5	400	−0.0003	0.0044	0.0113	−0.0007	0.0167	0.0417
0.5	500	0.0000	0.0031	0.0077	−0.0001	0.0125	0.0307
0.5	750	−0.0001	0.0014	0.0035	−0.0002	0.0067	0.0166
0.5	1000	−0.0001	0.0008	0.0019	−0.0001	0.0044	0.0109
0.9	25	−0.0014	0.0292	0.0731	0.0608	−0.0050	−0.0998
0.9	50	0.0006	0.0123	0.0293	−0.0034	0.0561	0.1417
0.9	75	−0.0001	0.0072	0.0176	−0.0002	0.0328	0.0802
0.9	100	−0.0003	0.0040	0.0101	−0.0005	0.0196	0.0486
0.9	150	0.0001	0.0036	0.0087	0.0003	0.0143	0.0345
0.9	200	−0.0004	0.0023	0.0062	−0.0007	0.0095	0.0242
0.9	300	−0.0006	0.0008	0.0029	−0.0013	0.0047	0.0132
0.9	400	−0.0002	0.0016	0.0041	−0.0004	0.0054	0.0137
0.9	500	0.0000	0.0010	0.0025	0.0000	0.0038	0.0095
0.9	750	−0.0001	0.0003	0.0008	−0.0001	0.0018	0.0045
0.9	1000	0.0000	0.0001	0.0002	−0.0001	0.0010	0.0026

Figure 4. Graph of $\widehat{\mathit{bias}}(\widehat{\theta })$ for the case ${\beta }_3=0.1$ and ${\sigma }^2=4.$

Figure 5. Graph of $\widehat{\mathit{bias}}(\widehat{\theta })$ for the case ${\beta }_3=0.5$ and ${\sigma }^2=4.$

Figure 6. Graph of $\widehat{\mathit{bias}}(\widehat{\theta })$ for the case ${\beta }_3=0.9$ and ${\sigma }^2=4.$