Template:Normal Distribution fisher matrix confidence bounds

Bounds on the Parameters
The lower and upper bounds on the mean, $$\widehat{\mu }$$, are estimated from:


 * $$\begin{align}

& {{\mu }_{U}}= & \widehat{\mu }+{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (upper bound),} \\ & {{\mu }_{L}}= & \widehat{\mu }-{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (lower bound)}\text{.} \end{align}$$

Since the standard deviation, $$$$, must be positive,  $$\ln $$  is treated as normally distributed, and the bounds are estimated from:


 * $$\begin{align}

& {{\sigma }_{U}}= & \cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var}}}}\text{ (upper bound),} \\ & {{\sigma }_{L}}= & \frac\text{ (lower bound),} \end{align}$$

where $${{K}_{\alpha }}$$  is defined by:


 * $$\alpha =\frac{1}{\sqrt{2\pi }}\int_^{\infty }{{e}^{-\tfrac{2}}}dt=1-\Phi ({{K}_{\alpha }})$$

If $$\delta $$  is the confidence level, then  $$\alpha =\tfrac{1-\delta }{2}$$  for the two-sided bounds and  $$\alpha =1-\delta $$  for the one-sided bounds. The variances and covariances of $$\widehat{\mu }$$  and  $$$$  are estimated from the Fisher matrix, as follows:


 * $$\left( \begin{matrix}

\widehat{Var}\left( \widehat{\mu } \right) & \widehat{Cov}\left( \widehat{\mu }, \right) \\ \widehat{Cov}\left( \widehat{\mu }, \right) & \widehat{Var}\left( \right)  \\ \end{matrix} \right)=\left( \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\mu }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial } \\ {} & {} \\   -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma^{2}}  \\ \end{matrix} \right)_{\mu =\widehat{\mu },\sigma =\widehat{\sigma }}^{-1}$$

$$\Lambda $$ is the log-likelihood function of the normal distribution, described in Chapter Parameter Estimation and Appendix: Distribution Log-Likelihood Equations.

Bounds on Reliability
The reliability of the normal distribution is:


 * $$\widehat{R}(t;\hat{\mu },)=\int_{t}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\widehat{\mu }} \right)}^{2}}}}dt$$

Let $$\widehat{z}=\tfrac{t-\widehat{\mu }}{}$$, the above equation then becomes:


 * $$\hat{R}(\widehat{z})=\int_{\widehat{z}(t)}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz$$

The bounds on $$z$$  are estimated from:


 * $$\begin{align}

& {{z}_{U}}= & \widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \\ & {{z}_{L}}= & \widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \end{align}$$

where:


 * $$Var(\widehat{z})={{\left( \frac{\partial \hat{z}}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial \hat{z}}{\partial } \right)}^{2}}Var+2\left( \frac{\partial \hat{z}}{\partial \mu } \right)\left( \frac{\partial \hat{z}}{\partial } \right)Cov\left( \widehat{\mu }, \right)$$

or:


 * $$Var(\widehat{z})=\frac{1}{\widehat{\sigma }^{2}}\left[ Var(\widehat{\mu })+{{\widehat{z}}^{2}}Var+2\cdot \widehat{z}\cdot Cov\left( \widehat{\mu }, \right) \right]$$

The upper and lower bounds on reliability are:


 * $$\begin{align}

& {{R}_{U}}= & \int_^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (upper bound)} \\ & {{R}_{L}}= & \int_^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (lower bound)} \end{align}$$

Bounds on Time
The bounds around time for a given normal percentile (unreliability) are estimated by first solving the reliability equation with respect to time, as follows:


 * $$\hat{T}(\widehat{\mu },)=\widehat{\mu }+z\cdot $$

where:


 * $$z={{\Phi }^{-1}}\left[ F(T) \right]$$

and:


 * $$\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz$$

The next step is to calculate the variance of $$\hat{T}(\widehat{\mu },)$$  or:


 * $$\begin{align}

Var(\hat{T})= & {{\left( \frac{\partial \hat{T}}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial \hat{T}}{\partial } \right)}^{2}}Var \\ & +2\left( \frac{\partial \hat{T}}{\partial \mu } \right)\left( \frac{\partial \hat{T}}{\partial } \right)Cov\left( \widehat{\mu }, \right) \\ Var(\hat{T})= & Var(\widehat{\mu })+{{\widehat{z}}^{2}}Var+2\cdot z\cdot Cov\left( \widehat{\mu }, \right) \end{align}$$

The upper and lower bounds are then found by:


 * $$\begin{align}

& {{T}_{U}}= & \hat{T}+{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (upper bound)} \\ & {{T}_{L}}= & \hat{T}-{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (lower bound)} \end{align}$$