Singhing with confidence: visualising the performance of confidence procedures

Figures & data

Figure 1. (a) A single example of a proposed CD from Equation (2(2) $\mathrm{C}(\mu ,\mathit{x})=\mathrm{T}(\frac{\mu -{\mu }_x}{{\sigma }_{\mathit{x}}/\sqrt{n}};n-1),$(2) ) generated from $\mathit{x}=\{x_1,\dots ,x_{10}\}\sim \mathrm{N}({\mu }_0=4,\sigma =3)$. The confidence level required to bound the true mean ${\mu }_0$ in this example is shown as 0.65. (

, ${\mathrm{C}}^{\ast }(\mathit{\mu },\mathit{x})$;

, ${\mathrm{C}}^{\ast }({\mu }_0,\mathit{x})$). (b) Singh plot for the proposed CD about the same target distribution, generated from $m={10}^4$ samples $\mathit{X}=\{{\mathit{x}}_1,\dots ,{\mathit{x}}_m\}$ (

, $\mathrm{S}(\mathit{\alpha };{\mu }_0)$;

, $\mathrm{U}(0,1)$;

, $\mathrm{S}(\alpha =0.7;{\mu }_0)$).

Figure 2. (a) A single example of a proposed CD from Equation (6(6) ${\mathrm{C}}^{\ast }(\theta ,\mathit{x})=\mathrm{B}(\theta ;a=\sum \mathit{x}+0.5,b=n-\sum \mathit{x}+0.5),$(6) ) generated from $\mathit{x}=\{x_1,\dots ,x_{10}\}\sim \mathrm{Ber}(p={\theta }_0)$. The confidence level required to bound the true rate ${\theta }_0$ is shown as 0.75. (

, ${\mathrm{C}}^{\ast }(\mathit{\theta },\mathit{x})$;

, ${\mathrm{C}}^{\ast }({\theta }_0,\mathit{x})$). (b) Singh plot for the proposed CD about the same target distribution, generated from $m={10}^4$ samples $\mathit{X}=\{{\mathit{x}}_1,\dots ,{\mathit{x}}_m\}$. (

, $\mathrm{S}(\mathit{\alpha };{\theta }_0)$;

, $\mathrm{U}(0,1)$;

, $\mathrm{S}(\alpha =0.7;{\theta }_0)$).

Figure 3. (a) A single example of a proposed CD from Equation (13) generated from $\mathit{x}=\{x_1,\dots ,x_{10}\}\sim \mathrm{Ber}(p={\theta }_0)$. The confidence level required to bound the true rate ${\theta }_0$ is shown as [0.05, 0.17]. (

, ${\mathrm{C}}_U^{\ast }(\mathit{\theta },\mathit{x})$;

, ${\mathrm{C}}_L^{\ast }(\mathit{\theta },\mathit{x})$;

, ${\mathrm{C}}^{\ast }({\theta }_0,\mathit{x})$). (b) Singh plot for the proposed CD about the same target distribution, generated from $m={10}^4$ samples $\mathit{X}=\{{\mathit{x}}_1,\dots ,{\mathit{x}}_m\}$. (

, ${\mathrm{S}}_U(\mathit{\alpha };{\theta }_0)$;

, ${\mathrm{S}}_L(\mathit{\alpha };{\theta }_0)$;

, $\mathrm{U}(0,1)$;

, $\mathrm{S}(\alpha =0.7;{\theta }_0)$).

Figure 4. (a) A single example of a proposed CD from Equation (14) generated from $\mathit{x}=\{x_1,\dots ,x_{10}\}\sim \mathrm{F}([{\mu }_1,{\mu }_2],[{\sigma }_1,{\sigma }_2])$ where $\mathrm{F}([{\mu }_1,{\mu }_2],[{\sigma }_1,{\sigma }_2])=0.5\cdot \mathrm{N}({\mu }_1=4,{\sigma }_1=3)+0.5\cdot \mathrm{N}({\mu }_2=5,{\sigma }_2=1.5)$. The confidence level required to bound the true value $x_{n+1}$ is shown as $\mathrm{C}({\mu }_0,\mathit{x})=[0.55,0.64]$. (

, ${\mathrm{C}}_U^{\ast }(\mathit{\theta },\mathit{x})$;

, ${\mathrm{C}}_L^{\ast }(\mathit{\theta },\mathit{x})$;

, ${\mathrm{C}}^{\ast }({\theta }_0,\mathit{x})$). (b) Singh plot for the proposed imprecise CD about the same target distribution, generated from $m={10}^4$ samples $\mathit{X}=\{{\mathit{x}}_1,\dots ,{\mathit{x}}_m\}$ (

, ${\mathrm{S}}_U(\mathit{\alpha };{\theta }_0)$;

, ${\mathrm{S}}_L(\mathit{\alpha };{\theta }_0)$;

, $\mathrm{U}(0,1)$;

, $\mathrm{S}(\alpha =0.7;\mathrm{F}([{\mu }_1,{\mu }_2],[{\sigma }_1,{\sigma }_2]))$).

Figure 5. A series of Singh plots used for inference about ${\theta }_0=0.4$ generated using Equation (13) from $m={10}^4$ samples of varying length n (

, ${\mathrm{S}}_U(\mathit{\alpha };{\theta }_0)$;

, ${\mathrm{S}}_L(\mathit{\alpha };{\theta }_0)$;

, $\mathrm{U}(0,1)$;

, $\mathrm{S}(\alpha =0.7;{\theta }_0)$).

Figure 6. A series of Singh plots used for inference about a varying ${\theta }_0$ generated using Equation (13) from $m={10}^4$ samples of length n = 10 (

, ${\mathrm{S}}_U(\mathit{\alpha };{\theta }_0)$;

, ${\mathrm{S}}_L(\mathit{\alpha };{\theta }_0)$;

, $\mathrm{U}(0,1)$;

, $\mathrm{S}(\alpha =0.7;{\theta }_0)$).

Figure 7. A series of Singh plots used for inference about ${\theta }_0=0.4$ generated using Equation (13) from $m={10}^4$ samples of length n = 10 sample with varying degrees of confidence demonstrated by altering the c parameter in Equation (15) (

, ${\mathrm{S}}_U(\mathit{\alpha };{\theta }_0)$;

, ${\mathrm{S}}_L(\mathit{\alpha };{\theta }_0)$;

, $\mathrm{U}(0,1)$;

, $\mathrm{S}(\alpha =0.7;{\theta }_0)$).

Figure 8. Global Singh plot produced using k = 100 θ values drawn from $[0,1]$, each producing $m={10}^4$ Monte Carlo samples using Equation (13) for inference about θ with a sample size of n = 10 (

, ${\mathrm{S}}_U(\mathit{\alpha };\mathit{\theta })$;

, ${\mathrm{S}}_L(\mathit{\alpha };\mathit{\theta })$;

, $\mathrm{U}(0,1)$;

, $\mathrm{S}(\alpha =0.7;{\theta }_0)$).

Figure 9. Singh plots representing the coverage probability for a desired α confidence level interval using Equation (19(19) ${\overline{{\mathrm{C}}^{\ast }}}^{-1}(\mu ,\mathit{x})=1-{({\left(\frac{\sqrt{n}(\overline{\mu }-{\mu }_{\mathit{x}})}{{\sigma }_{\mathit{x}}}\right)}^2+1)}^{-1}.$(19) ) for inference about data generated from Bernoulli distributions with varying θ-parameters. Two plots are shown, for sample sizes of n = 5 (left) and n = 30 (right), each produced from $m={10}^4$ samples (

, $\mathrm{S}(\mathit{\alpha };{\theta }_0=0.05)$;

, $\mathrm{S}(\mathit{\alpha };{\theta }_0=0.2)$;

, $\mathrm{S}(\mathit{\alpha };{\theta }_0=0.5)$;

, $\mathrm{U}(0,1)$).