Bayesian autoregressive adaptive refined descriptive sampling algorithm in the Monte Carlo simulation

Figures & data

Figure 1. Sequences of $\widehat{{\rho }_j}$ estimates for different true values of ρ. (a) $\rho =0.1$. (b) $\rho =0.3$. (c) $\rho =0.6$.

Figure 2. Sequences of $\widehat{{\theta }_j}$ estimates for $\theta =2.5$ and different values of ρ. (a) $\rho =0.1$. (b) $\rho =0.3$. (c) $\rho =0.6$.

Table 1. Subsets of prime numbers size for a Gumbel distribution with mean $E\left(X\right)=0.577$, $p_1=7$, $p_2=11$ and $p_3=13$.

$\mathit{i}$	$r_i^1$	$x_i^1$	$\mathit{i}$	$r_i^2$	$x_i^2$	$\mathit{i}$	$r_i^3$	$x_i^3$
1	0.071	$-0$.970	1	0.045	$-1.129$	1	0.038	$-1.181$
2	0.214	$-0$.432	2	0.136	$-0$.689	2	0.115	$-0$.770
3	0.357	$-0$.029	3	0.227	$-0$.393	3	0.192	$-0$.501
4	0.500	0.367	4	0.318	$-0$.136	4	0.269	$-0$.272
5	0.643	0.817	5	0.409	0.112	5	0.346	$-0$.059
6	0.786	1.422	6	0.500	0.367	6	0.423	0.151
7	0.929	2.602	7	0.591	0.642	7	0.500	0.367
	${\hat{\theta }}_1$	0.539	8	0.682	0.960	8	0.577	0.601
			9	0.773	1.355	9	0.654	0.856
			10	0.864	1.920	10	0.731	1.159
			11	0.955	3.068	11	0.808	1.544
				${\hat{\theta }}_2$	0.553	12	0.885	2.099
						13	0.962	3.239
							${\hat{\theta }}_3$	0.556

Table 2. The observed mean of a Gumbel distribution using three regular subsets of prime number size.

j	1	2	3	Overall mean ($T_{\mathrm{R}\mathrm{D}\mathrm{S}}$)
${\hat{\theta }}_j$	0.539	0.553	0.556	0.551

Table 3. The simulated estimator ${\hat{\rho }}_{\mathrm{R}\mathrm{D}\mathrm{S}}$ and its variance for $\theta =2.5$ and for various values of ρ, n and β. Variances in parentheses.

$\rho =0.1$
		${\hat{\rho }}_{\mathrm{R}\mathrm{D}\mathrm{S}}$
n	${\rho }_0$	$\text{β}=2$	$\text{β}=2.5$	$\text{β}=3$	$\text{β}=3.5$
20	0.1067	0.0798 (0.0032)	0.1188 (0.0009)	0.1886 (0.0021)	0.2284 (0.0032)
30	0.1075	0.0445 (0.00008)	0.0812 (0.0002)	0.1144 (0.0005)	0.1499 (0.0009)
40	0.1062	0.0306 (0.00002)	0.0581 (0.00009)	0.0846 (0.0002)	0.1107 (0.0003)
$\rho =0.3$
		${\hat{\rho }}_{\mathrm{R}\mathrm{D}\mathrm{S}}$
n	${\rho }_0$	$\text{β}=7.5$	$\text{β}=8$	$\text{β}=8.5$	$\text{β}=9$
20	0.3035	0.2815 (0.0078)	0.3047 (0.0084)	0.3009 (0.0056)	0.3232 (0.0063)
n	${\rho }_0$	$\text{β}=10.5$	$\text{β}=11$	$\text{β}=11.5$	$\text{β}=12$
30	0.3030	0.2633 (0.0030)	0.2736 (0.0032)	0.2947 (0.0034)	0.3034 (0.0036)
n	${\rho }_0$	$\text{β}=15$	$\text{β}=15.5$	$\text{β}=16.0$	$\text{β}=16.5$
40	0.3080	0.2926 (0.0026)	0.3006 (0.0026)	0.3125 (0.0028)	0.3236 (0.0031)
$\rho =0.6$
		${\hat{\rho }}_{\mathrm{R}\mathrm{D}\mathrm{S}}$
n	${\rho }_0$	$\text{β}=50$	$\text{β}=50.5$	$\text{β}=55$	$\text{β}=55.5$
20	0.6435	0.5980(0.0559)	0.6017 (0.0516)	0.6191 (0.0487)	0.6365 (0.0423)
n	${\rho }_0$	$\text{β}=75.5$	$\text{β}=80$	$\text{β}=80.5$	$\text{β}=85.5$
30	0.6204	0.5600 (0.0199)	0.5960 (0.0051)	0.6057 (0.0209)	0.6283 (0.0201)
n	${\rho }_0$	$\text{β}=110$	$\text{β}=110.5$	$\text{β}=111$	$\text{β}=111.5$
40	0.6136	0.6002 (0.0124)	0.6009 (0.0127)	0.6014 (0.0123)	0.6013 (0.0121)

Table 4. The simulated estimator ${\hat{\theta }}_{\mathrm{R}\mathrm{D}\mathrm{S}}$ and its variance for $\theta =2.5$ and for different values of ρ and n. Variances are given in parentheses.

n	ρ	${\theta }_0$	${\hat{\theta }}_{\mathrm{R}\mathrm{D}\mathrm{S}}$
20	0.1	2.5252	2.4405 (0.3438)
	0.3	2.5216	2.4534 (0.3347)
	0.6	2.5140	2.4970 (0.3742)
30	0.1	2.5063	2.3841 (0.2141)
	0.3	2.5551	2.4140 (0.2073)
	0.6	2.5840	2.4293 (0.2189)
40	0.1	2.4949	2.3522 (0.1453)
	0.3	2.6455	2.3609 (0.1588)
	0.6	2.4982	2.3906 (0.1608)