2.5.1.2 Algorithm: Parametric Distribution Analysis (Right Censoring)

Data are input in time and censoring indicator pairs \( (t_i, c_i) \).

The MLE (maximum likelihood estimation) and LS (least squares) methods are used to estimate parameters for each distribution.

Under the MLE method:

1. First calculate the likelihood function \(\mathcal{L}( \theta )\) for the distribution, which is the product of the Probability Density Function for failure time data and the Survival Function for censored time data.
2. Then maximize the logarithm of the likelihood function \(\ell ( \theta )\) by setting its partial derivatives equal to zero with respect to parameters.
3. Numerically maximize the log-likelihood function to estimate the parameters in step 2.
4. Alternative numerical methods are used for special cases, such as fixed parameters or non-regular MLE.

For most distributions, parameter estimates are obtained using the Newton-Raphson method. The 2-parameter Exponential uses a closed-form non-regular MLE approach. The following table summarizes the parameter estimation methods for the 3-parameter Weibull, 3-parameter Lognormal, and 3-parameter Loglogistic.

Case	3-parameter Weibull	3-parameter Lognormal	3-parameter Loglogistic
No Fixed	Regular MLE: NAG simplex evaluations. Fallback Algorithm: Bisection iterations for shape.	Regular MLE: NAG simplex evaluations. Fallback Algorithm: NAG simplex evaluations with fixed threshold.	Regular MLE: NAG simplex evaluations. Fallback Algorithm: Bias-correction method.
Fix Shape / Scale	Regular MLE: NAG simplex evaluations. Fallback Algorithm: Iterations to converge to corrected threshold when shape < 1.	Regular MLE: NAG simplex evaluations. Fallback algorithm not required.	Regular MLE: NAG simplex evaluations. Fallback Algorithm: Bias-correction method.
Fix Threshold	Bisection iterations for shape.	NAG simplex evaluations for \(\mu\) and \(\sigma\).	NAG simplex evaluations for \(\mu\) and \(\sigma\).
Fix Shape / Scale and Threshold	Closed-form scale estimate.	Bisection iterations for \(\mu\).	Bisection iterations for \(\mu\).
Fix All	No parameter estimation required.	No parameter estimation required.	No parameter estimation required.

Under the LS method:

1. First plot the failure times (or log failure times) against the transformed cumulative probabilities on a probability plot.
2. Then fit a straight line by minimizing the sum of squared deviations
3. Use the slope and scale from the fitted line to calculate parameter estimates

Distributions

Supported distributions include:

Distribution	Parameters	Probability density function (PDF)	Cumulative distribution function (CDF)	Inverse cumulative distribution function
Normal	location: \(\mu\) scale: \(\sigma\)	\[f(x)=\frac{1}{ \sqrt{2 \pi} \sigma } e^{ -\frac{( x - \mu )^2}{2 \sigma^2} }, \; \sigma > 0\]	\[P(x) = \Phi \left( \frac{x - \mu}{\sigma} \right)\]	\[x = \mu + \sigma \Phi^{-1}(p)\]
Lognormal	location: \(\mu\) scale: \(\sigma\)	\(f(x)=\frac{1}{ \sqrt{2 \pi} \sigma x } e^{ -\frac{( \text{ln}(x) - \mu )^2}{2 \sigma^2} }, \) \[ x>0 , \; \sigma > 0\]	\[P(x) = \Phi \left( \frac{\text{ln}(x) - \mu}{\sigma} \right)\]	\[\text{ln}(x) = \mu + \sigma \Phi^{-1}(p)\]
3-parameter Lognormal	location: \(\mu\) scale: \(\sigma\) threshold: \(\lambda\)	\(f(x)=\frac{1}{\sqrt{2\pi}\sigma(x-\lambda)}e^{-\frac{(\ln(x-\lambda)-\mu)^2}{2\sigma^2}},\) \(x>\lambda,\sigma>0\)	\[P(x)=\Phi\left(\frac{\ln(x-\lambda)-\mu}{\sigma}\right)\]	\[\ln(x-\lambda)=\mu+\sigma\Phi^{-1}(p)\]
Exponential	scale: \(\theta\)	\[f(x) = \frac{ 1 }{ \theta } e^{-\frac{x}{\theta}}, \; x > 0, \; \theta > 0\]	\[P(x) = 1- e^{ -\frac{x}{\theta} }\]	\[ x = \theta ( - \text{ln}(1-p) ) \]
2-parameter Exponential	scale: \(\theta\) threshold: \(\lambda\)	\[f(x) = \frac{1}{\theta} e^{-\frac{x-\lambda}{\theta}}, x > \lambda, \theta > 0\]	\[P(x) = 1-e^{-\frac{x-\lambda}{\theta}}\]	\[x = \lambda + \theta\left(-\text{ln}(1-p)\right)\]
Smallest Extreme Value	location: \(\alpha\) scale: \(\beta\)	\(f(x) = \frac{1}{\beta } e^{ \frac{x- \alpha}{\beta}} e^{ - e^{ \frac{x- \alpha}{\beta} } }, \) \[\beta > 0\]	\[P(x) = 1- e^{-e^{\frac{x-\alpha}{\beta}}}\]	\[ x = \alpha + \beta \text{ln}( - \text{ln}(1-p) ) \]
Weibull	scale: \(a\) shape: \(b\)	\(f(x) = \frac{b}{a} \left(\frac{ x }{ a }\right)^{b - 1} e^{ -(\frac{ x }{ a })^b }, \) \[x > 0, \; a > 0, \; b > 0\]	\[P(x) = 1 - e^{ -(\frac{ x }{ a })^b }\]	\[ \text{ln}(x) = \text{ln}(a) + \frac{1}{b} \text{ln}( - \text{ln}(1-p) ) \]
3-parameter Weibull	scale: \(a\) shape: \(b\) threshold: \(\lambda\)	\(f(x) = \frac{b}{a} \left(\frac{x-\lambda}{a}\right)^{b-1} e^{-\left(\frac{x-\lambda}{a}\right)^b},\) \(x > \lambda, a > 0, b > 0\)	\[P(x) = 1-e^{-\left(\frac{x-\lambda}{a}\right)^b}\]	\[\ln(x-\lambda) = \ln(a) + \frac{1}{b}\ln\left(-\ln(1-p)\right)\]
Logistic	location: \(\mu\) scale: \(\sigma\)	\[f(x) = \frac{e^{\frac{x- \mu}{\sigma}}}{\sigma(1+e^{\frac{x- \mu}{\sigma}})^2 },\sigma > 0\]	\[P(x) = \frac{1}{1+e^{-\frac{x- \mu}{\sigma}}} \]	\[x =\mu+ \sigma\text{ln}(\frac{p}{1-p}) \]
Loglogistic	location: \(\mu\) scale: \(\sigma\)	\[\begin{array}{l}f(x) = \frac{e^{\frac{\text{ln}(x)- \mu}{\sigma}}}{x\sigma(1+e^{\frac{\text{ln}(x)- \mu}{\sigma}})^2 } , \\ x>0, \sigma > 0\end{array}\]	\[P(x) = \frac{1}{1+e^{-\frac{\text{ln}(x)- \mu}{\sigma}}} \]	\[x =e^{\mu+ \sigma\text{ln}(\frac{p}{1-p})} \]
3-parameter Loglogistic	location: \(\mu\) scale: \(\sigma\) threshold: \(\lambda\)	\[\begin{array}{l}f(x) = \frac{e^{\frac{\ln(x-\lambda)-\mu}{\sigma}}}{(x-\lambda)\sigma\left(1+e^{\frac{\ln(x-\lambda)-\mu}{\sigma}}\right)^2}, \ x>\lambda,\sigma>0\end{array}\]	\[P(x) = \frac{1}{1+e^{-\frac{\ln(x-\lambda)-\mu}{\sigma}}}\]	\[x = \lambda + e^{\mu+\sigma\ln\left(\frac{p}{1-p}\right)}\]

where Φ is the CDF function for the standard normal distribution.

Hazard Function

The hazard function is calculated as follows:

where \(f(x)\) is the PDF function and \(P(x)\) is the CDF function.

Estimate Parameter Standard Errors

Once parameters are estimated by MLE method, the Fisher information matrix (FIM) can be calculated by Hessian matrix:

Covariance matrix of parameters can be expressed as:

where n is the number of points.

Parameter's standard error is the square root of the diagonal elements in the covariance matrix C.

Non-regular MLE may occur for data estimated using a distribution with a threshold parameter. For the 2-parameter exponential distribution, the threshold is a non-regular parameter. For other distributions, non-regular MLE may result in the threshold being treated as fixed. In these cases, the standard error of the threshold is set to 0, and its lower and upper confidence limits are equal to the estimated threshold.

Distribution	Parameters	Lower Confidence Limit	Upper Confidence Limit
Normal, Lognormal, Smallest Extreme Value, Logistic, Loglogistic	location: \(\mu\)	\[ \hat{\mu} - Z_{\alpha}{SE_{\hat{\mu}}} \]	\[ \hat{\mu} + Z_{\alpha}{SE_{\hat{\mu}}} \]
	scale: \(\sigma\)	\[ \frac{\hat{\sigma}}{\exp\!\left(\dfrac{Z_{\alpha}\,SE_{\hat{\sigma}}}{\hat{\sigma}}\right)} \]	\[ \hat{\sigma}{\exp\!\left( \, \frac{Z_{\alpha}SE_{\hat{\sigma}}}{\hat{\sigma}} \right)} \]
3-parameter Lognormal, 3-parameter Loglogistic	location: \(\mu\)	\[ \hat{\mu} - Z_{\alpha}{SE_{\hat{\mu}}} \]	\[ \hat{\mu} + Z_{\alpha}{SE_{\hat{\mu}}} \]
	scale: \(\sigma\)	\[ \frac{\hat{\sigma}}{\exp\!\left(\dfrac{Z_{\alpha}\,SE_{\hat{\sigma}}}{\hat{\sigma}}\right)} \]	\[ \hat{\sigma}{\exp\!\left( \, \frac{Z_{\alpha}SE_{\hat{\sigma}}}{\hat{\sigma}} \right)} \]
	threshold: \(\lambda\)	\[ \hat{\lambda} - Z_{\alpha}{SE_{\hat{\lambda}}} \]	\[ \hat{\lambda} + Z_{\alpha}{SE_{\hat{\lambda}}} \]
Exponential	scale: \(\theta\)	\[\frac{ \hat{\theta} }{ \exp\!\left( \, \frac{Z_{\alpha}SE_{\hat{\theta}}}{\hat{\theta}} \right) }\]	\[ \hat{\theta}{\exp\!\left( \, \frac{Z_{\alpha}SE_{\hat{\theta}}}{\hat{\theta}} \right)}\]
2-parameter Exponential	scale: \(\theta\)	\[\frac{ \hat{\theta} }{ \exp\!\left( \, \frac{Z_{\alpha}SE_{\hat{\theta}}}{\hat{\theta}} \right) }\]	\[ \hat{\theta}{\exp\!\left( \, \frac{Z_{\alpha}SE_{\hat{\theta}}}{\hat{\theta}} \right)}\]
	threshold: \(\lambda\)	\[ \hat{\lambda} - Z_{\alpha}{SE_{\hat{\lambda}}} \]	\[ \hat{\lambda} + Z_{\alpha}{SE_{\hat{\lambda}}} \]
Weibull	scale: \(a\)	\[ \frac{\hat{a}}{\exp\!\left(\dfrac{Z_{\alpha}\,SE_{\hat{a}}}{\hat{a}}\right)} \]	\[ \hat{a}{\exp\!\left( \, \frac{Z_{\alpha}SE_{\hat{a}}}{\hat{a}} \right)} \]
	shape: \(b\)	\[ \frac{\hat{b}}{\exp\!\left(\dfrac{Z_{\alpha}\,SE_{\hat{b}}}{\hat{b}}\right)} \]	\[ \hat{b}{\exp\!\left( \, \frac{Z_{\alpha}SE_{\hat{b}}}{\hat{b}} \right)} \]
3-parameter Weibull	scale: \(a\)	\[ \frac{\hat{a}}{\exp\left(\dfrac{Z_{\alpha}SE_{\hat{a}}}{\hat{a}}\right)} \]	\[ \hat{a}{\exp\left( \frac{Z_{\alpha}SE_{\hat{a}}}{\hat{a}} \right)} \]
	shape: \(b\)	\[ \frac{\hat{b}}{\exp\left(\dfrac{Z_{\alpha}SE_{\hat{b}}}{\hat{b}}\right)} \]	\[ \hat{b}{\exp\left( \frac{Z_{\alpha}SE_{\hat{b}}}{\hat{b}} \right)} \]
	threshold: \(\lambda\)	\[ \hat{\lambda} - Z_{\alpha}{SE_{\hat{\lambda}}} \]	\[ \hat{\lambda} + Z_{\alpha}{SE_{\hat{\lambda}}} \]

where \( Z_\alpha = Z_{1-\alpha/2} = \Phi^{-1}(1 - \tfrac{\alpha}{2}) \)

Quantities of Distribution

Once parameters are estimated using either the MLE or LS method, quantities of interest can be calculated

1. Mean
2. Standard Deviation
3. Q1, Median, Q3
4. IQR = Q3 - Q1

Formulas to calculate the distribution Mean

Distribution	Mean	Lower Confidence Level	Upper Confidence Level
Normal	\[\hat{\mu}\]	\[\hat{\mu} - Z_{\alpha}SE_{\hat{\mu}}\]	\[\hat{\mu} + Z_{\alpha}SE_{\hat{\mu}}\]
Lognormal	\[ \exp(\hat{\mu} + \frac{1}{2}\hat{\sigma}^2)\]	\[\hat{(Mean)} / {\exp}(Z_{\alpha}{SE_{\hat{(Mean)}}})\]	\[\hat{(Mean)}{\exp}(Z_{\alpha}{SE_{\hat{(Mean)}}}) \]
3-parameter Lognormal	\[\hat{\lambda} + \exp\left(\hat{\mu} + \frac{1}{2}\hat{\sigma}^2\right)\]	\(\lambda < 0:\) \(\hat{(Mean)} - Z_{\alpha}SE_{\hat{(Mean)}}\) \(\lambda \ge 0:\) \(\exp\left(\log(\hat{(Mean)}) - \dfrac{Z_{\alpha}SE_{\hat{(Mean)}}}{\hat{(Mean)}}\right)\)	\(\lambda < 0:\) \(\hat{(Mean)} + Z_{\alpha}SE_{\hat{(Mean)}}\) \(\lambda \ge 0:\) \(\exp\left(\log(\hat{(Mean)}) + \dfrac{Z_{\alpha}SE_{\hat{(Mean)}}}{\hat{(Mean)}}\right)\)
Exponential	\[\hat{\theta}\]	\[\hat{\theta} / \exp({\frac{Z_{\alpha} SE_{\hat{\theta}}}{\hat{\theta}}})\]	\[\hat{\theta} \exp({\frac{Z_{\alpha} SE_{\hat{\theta}}}{\hat{\theta}}})\]
2-parameter Exponential	\[\hat{\theta} + \hat{\lambda}\]	\(\lambda < 0:\) \(\hat{(Mean)} - Z_{\alpha}SE_{\hat{(Mean)}}\) \(\lambda \ge 0:\) \(\exp\left(\log(\hat{(Mean)}) - \dfrac{Z_{\alpha}SE_{\hat{(Mean)}}}{\hat{(Mean)}}\right)\)	\(\lambda < 0:\) \(\hat{(Mean)} + Z_{\alpha}SE_{\hat{(Mean)}}\) \(\lambda \ge 0:\) \(\exp\left(\log(\hat{(Mean)}) + \dfrac{Z_{\alpha}SE_{\hat{(Mean)}}}{\hat{(Mean)}}\right)\)
Smallest Extreme Value	\[\hat{\mu} - EC\,\hat{\sigma}\]	\[\hat{(Mean)} - Z_{\alpha}SE_{\hat{(Mean)}}\]	\[\hat{(Mean)} + Z_{\alpha}SE_{\hat{(Mean)}}\]
Weibull	\[\hat{a}\,\Gamma\!\left(1+\tfrac{1}{\hat{b}}\right)\]	\[ \exp\!\left( \log(\hat{\text{Mean}}) - Z_{\alpha}\, SE_{\log(\hat{\text{Mean}})} \right) \]	\[ \exp\!\left( \log(\hat{\text{Mean}}) + Z_{\alpha}\, SE_{\log(\hat{\text{Mean}})} \right) \]
3-parameter Weibull	\[\hat{\lambda} + \hat{a}\Gamma\left(1+\frac{1}{\hat{b}}\right)\]	\(\lambda < 0:\) \(\hat{(Mean)} - Z_{\alpha}SE_{\hat{(Mean)}}\) \(\lambda \ge 0:\) \(\exp\left(\log(\hat{(Mean)}) - \dfrac{Z_{\alpha}SE_{\hat{(Mean)}}}{\hat{(Mean)}}\right)\)	\(\lambda < 0:\) \(\hat{(Mean)} + Z_{\alpha}SE_{\hat{(Mean)}}\) \(\lambda \ge 0:\) \(\exp\left(\log(\hat{(Mean)}) + \dfrac{Z_{\alpha}SE_{\hat{(Mean)}}}{\hat{(Mean)}}\right)\)
Logistic	\[\hat{\mu}\]	\[\hat{\mu} - Z_{\alpha}SE_{\hat{\mu}}\]	\[\hat{\mu} + Z_{\alpha}SE_{\hat{\mu}}\]
Loglogistic	\[ \exp(\hat{\mu})\Gamma(1+\hat{\sigma})\Gamma(1-\hat{\sigma}) \]	\[\hat{(Mean)} / {\exp}(Z_{\alpha}{SE_{\hat{(Mean)}}})\]	\[\hat{(Mean)}{\exp}(Z_{\alpha}{SE_{\hat{(Mean)}}}) \]
3-parameter Loglogistic	\[\hat{\lambda} + \exp(\hat{\mu})\Gamma(1+\hat{\sigma})\Gamma(1-\hat{\sigma}), \quad \hat{\sigma}<1\]	\(\lambda < 0:\) \(\hat{(Mean)} - Z_{\alpha}SE_{\hat{(Mean)}}\) \(\lambda \ge 0:\) \(\exp\left(\log(\hat{(Mean)}) - \dfrac{Z_{\alpha}SE_{\hat{(Mean)}}}{\hat{(Mean)}}\right)\)	\(\lambda < 0:\) \(\hat{(Mean)} + Z_{\alpha}SE_{\hat{(Mean)}}\) \(\lambda \ge 0:\) \(\exp\left(\log(\hat{(Mean)}) + \dfrac{Z_{\alpha}SE_{\hat{(Mean)}}}{\hat{(Mean)}}\right)\)

where \(EC\) is Euler's constant and \( Z_\alpha = Z_{1-\alpha/2} = \Phi^{-1}(1 - \tfrac{\alpha}{2}) \)

The lower confidence interval for mean cannot be below the threshold for distributions with a threshold parameter.

Formulas to calculate the distribution Standard Deviation

Distribution	Standard Deviation	Lower Confidence Level	Upper Confidence Level
Normal	\[\hat{\sigma}\]	\[\hat{\sigma}/\exp\!\left( \,\dfrac{Z_{\alpha}}{\hat{\sigma}SE_{\hat{\sigma}}} \right) \]	\[\hat{\sigma}\exp\!\left( \,\dfrac{Z_{\alpha}}{\hat{\sigma}SE_{\hat{\sigma}}} \right)\]
Lognormal	\[ \sqrt{ \exp\left( \hat{\mu} + \tfrac{1}{2}\hat{\sigma}^{2} \right) \left[ \exp\left( \hat{\sigma}^{2} \right) - 1 \right] } \]	\[\hat{(SD)} / {\exp}(Z_{\alpha}{SE_{\hat{(SD)}}})\]	\[\hat{(SD)}{\exp}(Z_{\alpha}{SE_{\hat{(SD)}}}) \]
3-parameter Lognormal	\[\exp\left(\hat{\mu} + \frac{1}{2}\hat{\sigma}^2\right)\sqrt{\exp(\hat{\sigma}^2)-1}\]	\[\hat{(SD)} / \exp\left(\dfrac{Z_{\alpha}SE_{\hat{(SD)}}}{\hat{(SD)}}\right)\]	\[\hat{(SD)}\exp\left(\dfrac{Z_{\alpha}SE_{\hat{(SD)}}}{\hat{(SD)}}\right)\]
Exponential	\[\hat{\theta}\]	\[\hat{\theta} / \exp({\frac{Z_{\alpha} SE_{\hat{\theta}}}{\hat{\theta}}})\]	\[\hat{\theta} \exp({\frac{Z_{\alpha} SE_{\hat{\theta}}}{\hat{\theta}}})\]
2-parameter Exponential	\[\hat{\theta}\]	\[\exp\left(\log(\hat{SD}) - \dfrac{Z_{\alpha}SE_{\hat{SD}}}{\hat{SD}}\right)\]	\[\exp\left(\log(\hat{SD}) + \dfrac{Z_{\alpha}SE_{\hat{SD}}}{\hat{SD}}\right)\]
Smallest Extreme Value	\[\hat{\sigma}\sqrt{\tfrac{\pi^{2}}{6}}\]	\[\hat{(SD)}/\exp\!\left( \,\dfrac{Z_{\alpha}}{\hat{(SD)}SE_{\hat{(SD)}}} \right) \]	\[\hat{(SD)}\exp\!\left( \,\dfrac{Z_{\alpha}}{\hat{(SD)}SE_{\hat{(SD)}}} \right)\]
Weibull	\[\hat{a}\sqrt{\!\left[\Gamma\!\left(1+\tfrac{2}{\hat{b}}\right)-\Gamma^{2}\!\left(1+\tfrac{1}{\hat{b}}\right)\right]}\]	\[ \exp\!\Big(\log \hat{\sigma} - Z_{\alpha}\,SE_{\log \hat{\sigma}}\Big) \]	\[ \exp\!\Big(\log \hat{\sigma} + Z_{\alpha}\,SE_{\log \hat{\sigma}}\Big) \]
3-parameter Weibull	\[\hat{a}\sqrt{\Gamma\left(1+\frac{2}{\hat{b}}\right)-\Gamma^2\left(1+\frac{1}{\hat{b}}\right)}\]	\[\exp\left(\log(\hat{SD}) - \dfrac{Z_{\alpha}SE_{\hat{SD}}}{\hat{SD}}\right)\]	\[\exp\left(\log(\hat{SD}) + \dfrac{Z_{\alpha}SE_{\hat{SD}}}{\hat{SD}}\right)\]
Logistic	\[\hat{\sigma}\sqrt{\tfrac{\pi^{2}}{3}}\]	\[\hat{(SD)}/\exp\!\left( \,\dfrac{Z_{\alpha}}{\hat{(SD)}SE_{\hat{(SD)}}} \right) \]	\[\hat{(SD)}\exp\!\left( \,\dfrac{Z_{\alpha}}{\hat{(SD)}SE_{\hat{(SD)}}} \right)\]
Loglogistic	\[\sqrt{\,\exp(2\hat{\mu})\Big[\Gamma(1+2\hat{\sigma})\,\Gamma(1-2\hat{\sigma}) - \Gamma^2(1+\hat{\sigma})\,\Gamma^2(1-\hat{\sigma})\Big]\,}\]	\[\hat{(SD)} / {\exp}(Z_{\alpha}{SE_{\hat{(SD)}}})\]	\[\hat{(SD)}{\exp}(Z_{\alpha}{SE_{\hat{(SD)}}}) \]
3-parameter Loglogistic	\[\exp(\hat{\mu})\sqrt{\Gamma(1+2\hat{\sigma})\Gamma(1-2\hat{\sigma})-\left[\Gamma(1+\hat{\sigma})\Gamma(1-\hat{\sigma})\right]^2}, \quad \hat{\sigma}<\frac{1}{2}\]	\[\exp\left(\log(\hat{SD}) - \dfrac{Z_{\alpha}SE_{\hat{SD}}}{\hat{SD}}\right)\]	\[\exp\left(\log(\hat{SD}) + \dfrac{Z_{\alpha}SE_{\hat{SD}}}{\hat{SD}}\right)\]

where \( Z_\alpha = Z_{1-\alpha/2} = \Phi^{-1}(1 - \tfrac{\alpha}{2}) \)

Percentiles

Calculate the time at which a prespecified percent of the population fails.

Distribution	Percentile (\( \hat{x_p} \))	Variance \( \left( \text{Var} \left( \hat{x_p} \right ) \right )\)
Normal	\[ \hat{\mu} + z_p\hat{\sigma} \]	\[ \text{Var}(\hat{\mu}) + 2z_p\text{Cov}(\hat{\mu},\hat{\sigma}) + z_p^2\text{Var}(\hat{\sigma}) \]
Lognormal	\[ \exp(\hat{\mu} + z_p\hat{\sigma}) \]	\[ \hat{x}_p^2\left[\text{Var}(\hat{\mu}) + 2z_p\text{Cov}(\hat{\mu},\hat{\sigma}) + z_p^2\text{Var}(\hat{\sigma})\right] \]
3-parameter Lognormal	\[ \hat{\lambda} + \exp(\hat{\mu} + z_p\hat{\sigma}) \]	\( \hat{y}_p^2 \left[ \text{Var}(\hat{\mu}) +2z_p\text{Cov}(\hat{\mu},\hat{\sigma}) +z_p^2\text{Var}(\hat{\sigma}) \right] +\text{Var}(\hat{\lambda}) +2\hat{y}_p\text{Cov}(\hat{\mu},\hat{\lambda}) +2z_p\hat{y}_p\text{Cov}(\hat{\sigma},\hat{\lambda}) \) where \(\hat{y}_p=\exp(\hat{\mu}+z_p\hat{\sigma})=\hat{x}_p-\hat{\lambda}\)
Exponential	\[ -\ln(1-p)\hat{\theta} \]	\[ [-\ln(1-p)]^2\text{Var}(\hat{\theta}) \]
2-parameter Exponential	\[ \hat{\lambda} - \ln(1-p)\hat{\theta} \]	\[ [-\ln(1-p)]^2\text{Var}(\hat{\theta}) +\text{Var}(\hat{\lambda}) +2[-\ln(1-p)]\text{Cov}(\hat{\theta},\hat{\lambda}) \]
Smallest Extreme Value	\[ \hat{\mu} + z_p\hat{\sigma} \]	\[ \text{Var}(\hat{\mu}) + 2z_p\text{Cov}(\hat{\mu},\hat{\sigma}) + z_p^2\text{Var}(\hat{\sigma}) \]
Weibull	\[ \hat{a}\big[-\ln(1-p)\big]^{1/\hat{b}} \]	\[ \left(\frac{\hat{x}_p}{\hat{a}}\right)^2\text{Var}(\hat{a}) -2\frac{z_p\hat{x}_p^2}{\hat{a}\hat{b}^2}\text{Cov}(\hat{a},\hat{b}) +\left(\frac{z_p\hat{x}_p}{\hat{b}^2}\right)^2\text{Var}(\hat{b}) \]
3-parameter Weibull	\[ \hat{\lambda} + \hat{a}\big[-\ln(1-p)\big]^{1/\hat{b}} \]	\( \left(\frac{\hat{y}_p}{\hat{a}}\right)^2\text{Var}(\hat{a}) -2\frac{z_p\hat{y}_p^2}{\hat{a}\hat{b}^2}\text{Cov}(\hat{a},\hat{b}) +\left(\frac{z_p\hat{y}_p}{\hat{b}^2}\right)^2\text{Var}(\hat{b}) +\text{Var}(\hat{\lambda}) +2\frac{\hat{y}_p}{\hat{a}}\text{Cov}(\hat{a},\hat{\lambda}) -2\frac{z_p\hat{y}_p}{\hat{b}^2}\text{Cov}(\hat{b},\hat{\lambda}) \) where \(\hat{y}_p=\hat{x}_p-\hat{\lambda}\)
Logistic	\[ \hat{\mu} + z_p\hat{\sigma} \]	\[ \text{Var}(\hat{\mu}) + 2z_p\text{Cov}(\hat{\mu},\hat{\sigma}) + z_p^2\text{Var}(\hat{\sigma}) \]
Loglogistic	\[ \exp(\hat{\mu} + z_p\hat{\sigma}) \]	\[ \hat{x}_p^2\left[ \text{Var}(\hat{\mu}) +2z_p\text{Cov}(\hat{\mu},\hat{\sigma}) +z_p^2\text{Var}(\hat{\sigma}) \right] \]
3-parameter Loglogistic	\[ \hat{\lambda} + \exp(\hat{\mu}+z_p\hat{\sigma}) \]	\( \hat{y}_p^2 \left[ \text{Var}(\hat{\mu}) +2z_p\text{Cov}(\hat{\mu},\hat{\sigma}) +z_p^2\text{Var}(\hat{\sigma}) \right] +\text{Var}(\hat{\lambda}) +2\hat{y}_p\text{Cov}(\hat{\mu},\hat{\lambda}) +2z_p\hat{y}_p\text{Cov}(\hat{\sigma},\hat{\lambda}) \) where \(\hat{y}_p=\exp(\hat{\mu}+z_p\hat{\sigma})=\hat{x}_p-\hat{\lambda}\)

where

For Normal, Lognormal, and 3-parameter Lognormal distributions: \(z_p=\Phi^{-1}(p)\)

For Smallest Extreme Value, Weibull, and 3-parameter Weibull distributions: \(z_p=\ln[-\ln(1-p)]\)

For Logistic, Loglogistic, and 3-parameter Loglogistic distributions: \(z_p=\ln\left(\frac{p}{1-p}\right)\)

For Normal, Smallest Extreme Value, and Logistic distributions:

Lower confidence limit: \( \hat{x}_p - Z_{\alpha}\sqrt{ \text{Var}(\hat{x}_p)} \)

Upper confidence limit: \( \hat{x}_p + Z_{\alpha}\sqrt{ \text{Var}(\hat{x}_p)} \)

For Lognormal, Exponential, Weibull, Loglogistic distributions:

Lower confidence limit: \( \exp\!\left( \ln(\hat{x}_p) \;-\; Z_{\alpha}\,\frac{\sqrt{\, \text{Var}(\hat{x}_p)\,}}{\hat{x}_p} \right) \)

Upper confidence limit: \( \exp\!\left( \ln(\hat{x}_p) \;+\; Z_{\alpha}\,\frac{\sqrt{\, \text{Var}(\hat{x}_p)\,}}{\hat{x}_p} \right) \)

For 3-parameter Lognormal, 2-parameter Exponential, 3-parameter Weibull, and 3-parameter Loglogistic distributions:

If \(\hat{\lambda} < 0\):

Lower confidence limit: \( \hat{x}_p - Z_{\alpha}\sqrt{\text{Var}(\hat{x}_p)} \)

Upper confidence limit: \( \hat{x}_p + Z_{\alpha}\sqrt{\text{Var}(\hat{x}_p)} \)

If \(\hat{\lambda} \ge 0\):

Lower confidence limit: \( \exp\left( \ln(\hat{x}_p) - Z_{\alpha}\frac{\sqrt{\text{Var}(\hat{x}_p)}}{\hat{x}_p} \right) \)

Upper confidence limit: \( \exp\left( \ln(\hat{x}_p) + Z_{\alpha}\frac{\sqrt{\text{Var}(\hat{x}_p)}}{\hat{x}_p} \right) \)

For the threshold distributions, the lower confidence limit should not be less than \(\hat{\lambda}\).

Survival Probabilities

Calculate the percent of the population that has not yet failed at a pre-specified time point.

Distribution	Survival Probability (\( \hat{z} \))	Variance of Survival Probability (\( Var(\hat{z}) \))
Normal	\[ \frac{x - \hat{\mu}}{\hat{\sigma}} \]	\[ \frac{Var(\hat{\mu}) + 2\hat{z}Cov(\hat{\mu},\hat{\sigma}) + \hat{z}^{2}Var(\hat{\sigma})}{\hat{\sigma}^2} \]
Lognormal	\[ \frac{\ln(x) - \hat{\mu}}{\hat{\sigma}} \]	\[ \frac{Var(\hat{\mu}) + 2\hat{z}Cov(\hat{\mu},\hat{\sigma}) + \hat{z}^{2}Var(\hat{\sigma})}{\hat{\sigma}^2} \]
3-parameter Lognormal	\[ \frac{\ln(x-\hat{\lambda})-\hat{\mu}}{\hat{\sigma}} \]	\[ \frac{1}{\hat{\sigma}^2}\left[ Var(\hat{\mu}) +2\hat{z}Cov(\hat{\mu},\hat{\sigma}) +\hat{z}^2Var(\hat{\sigma}) +\frac{2Cov(\hat{\mu},\hat{\lambda})}{x-\hat{\lambda}} +\frac{2\hat{z}Cov(\hat{\sigma},\hat{\lambda})}{x-\hat{\lambda}} +\frac{Var(\hat{\lambda})}{(x-\hat{\lambda})^2} \right]\]
Exponential	\[ ln(x) - ln(\hat{\theta}) \]	\[ \frac{Var(\hat{\theta})}{\hat{\theta}^2} \]
2-parameter Exponential	\[ \ln(x-\hat{\lambda})-\ln(\hat{\theta}) \]	\[ \frac{Var(\hat{\theta})}{\hat{\theta}^2} +\frac{Var(\hat{\lambda})}{(x-\hat{\lambda})^2} +\frac{2Cov(\hat{\theta},\hat{\lambda})}{\hat{\theta}(x-\hat{\lambda})} \]
Smallest Extreme Value	\[ \frac{x - \hat{\mu}}{\hat{\sigma}} \]	\[ \frac{Var(\hat{\mu}) + 2\hat{z}Cov(\hat{\mu},\hat{\sigma}) + \hat{z}^{2}Var(\hat{\sigma})}{\hat{\sigma}^2} \]
Weibull	\[ \hat{b}\,(\ln x - \ln\hat{a}) \]	\[ \frac{\hat{z}^{2}\,Var(\hat{b})}{\hat{b}^{2}} \;-\; 2\,\frac{\hat{z}\,Cov(\hat{a},\hat{b})}{\hat{a}} \;+\; \frac{\hat{b}^{2}\,Var(\hat{a})}{\hat{a}^{2}} \]
3-parameter Weibull	\[ \hat{b}\,[\ln(x-\hat{\lambda})-\ln(\hat{a})] \]	\[ \frac{\hat{z}^{2}Var(\hat{b})}{\hat{b}^{2}} -2\frac{\hat{z}Cov(\hat{a},\hat{b})}{\hat{a}} +\frac{\hat{b}^{2}Var(\hat{a})}{\hat{a}^{2}} +\frac{\hat{b}^{2}Var(\hat{\lambda})}{(x-\hat{\lambda})^{2}} +2\frac{\hat{b}^{2}Cov(\hat{a},\hat{\lambda})}{\hat{a}(x-\hat{\lambda})} -2\frac{\hat{z}Cov(\hat{b},\hat{\lambda})}{x-\hat{\lambda}} \]
Logistic	\[ \frac{x - \hat{\mu}}{\hat{\sigma}} \]	\[ \frac{Var(\hat{\mu}) + 2\hat{z}Cov(\hat{\mu},\hat{\sigma}) + \hat{z}^{2}Var(\hat{\sigma})}{\hat{\sigma}^2} \]
Loglogistic	\[ \frac{\ln(x) - \hat{\mu}}{\hat{\sigma}} \]	\[ \frac{Var(\hat{\mu}) + 2\hat{z}Cov(\hat{\mu},\hat{\sigma}) + \hat{z}^{2}Var(\hat{\sigma})}{\hat{\sigma}^2} \]
3-parameter Loglogistic	\[ \frac{\ln(x-\hat{\lambda})-\hat{\mu}}{\hat{\sigma}} \]	\[ \frac{1}{\hat{\sigma}^2}\left[ Var(\hat{\mu}) +2\hat{z}Cov(\hat{\mu},\hat{\sigma}) +\hat{z}^2Var(\hat{\sigma}) +\frac{2Cov(\hat{\mu},\hat{\lambda})}{x-\hat{\lambda}} +\frac{2\hat{z}Cov(\hat{\sigma},\hat{\lambda})}{x-\hat{\lambda}} +\frac{Var(\hat{\lambda})}{(x-\hat{\lambda})^2} \right] \]

For distributions with a threshold parameter, the transformed value is defined for \(x>\hat{\lambda}\). For \(x\le\hat{\lambda}\), the survival probability is 1.

Lower bound = \( \hat{z} - Z_{\alpha}\sqrt{Var(\hat{z})} \)

Upper bound = \( \hat{z} + Z_{\alpha}\sqrt{Var(\hat{z})} \)

For Normal, Lognormal, and 3-parameter Lognormal distributions:

Lower confidence limit: \( 1 - \frac{1}{2}\left(1 + \mathrm{erf}\left(\frac{UB}{\sqrt{2}}\right)\right) \)

Upper confidence limit: \( 1 - \frac{1}{2}\left(1 + \mathrm{erf}\left(\frac{LB}{\sqrt{2}}\right)\right) \)

For Exponential, 2-parameter Exponential, Smallest Extreme Value, Weibull, and 3-parameter Weibull distributions:

Lower confidence limit: \( \exp(-\exp(UB)) \)

Upper confidence limit: \( \exp(-\exp(LB)) \)

For Logistic, Loglogistic, and 3-parameter Loglogistic distributions:

Lower confidence limit: \( \frac{1}{1 + \exp(UB)} \)

Upper confidence limit: \( \frac{1}{1 + \exp(LB)} \)

Probability Plot

Points in the probability plot are sorted in ascending order, for the ith point, its probability is calculated by the median rank (Benard's method):

The middle line is the expected percentiles for given probabilities in terms of the inverse cumulative distribution function using parameters from the MLE or LS method.

To estimate confidence limits of percentiles, variance of percentiles is calculated by propagation of error:

where x_p is the percentile for a given probability p, and (.)^T denotes the transpose of a vector.

• Confidence limits of percentiles can be calculated as below:

where \(Z_{\alpha} = \Phi^{-1}(1- \alpha / 2)\) .

• For some positive random variables, e.g. in Lognormal, Exponential, Weibull, and Loglogistic distributions, confidence limits are:

The fitted line is constructed as follows for each distribution:

Distribution	x-axis	y-axis
Normal	\[ t_p \]	\[ \phi^{-1}(p) \]
Lognormal	\[ ln(t_p) \]	\[ \phi^{-1}(p) \]
3-parameter Lognormal	\[ \ln(t_p-\hat{\lambda}) \]	\[ \phi^{-1}(p) \]
Exponential	\[ ln(t_p) \]	\[ ln(-ln(1-p)) \]
2-parameter Exponential	\[ \ln(t_p-\hat{\lambda}) \]	\[ \ln(-\ln(1-p)) \]
Smallest Extreme Value	\[ t_p \]	\[ ln(-ln(1-p)) \]
Weibull	\[ ln(t_p) \]	\[ ln(-ln(1-p)) \]
3-parameter Weibull	\[ \ln(t_p-\hat{\lambda}) \]	\[ \ln(-\ln(1-p)) \]
Logistic	\[ t_p \]	\[ ln(\frac{p}{1-p}) \]
Loglogistic	\[ ln(t_p) \]	\[ ln(\frac{p}{1-p}) \]
3-parameter Loglogistic	\[ \ln(t_p-\hat{\lambda}) \]	\[ \ln\left(\frac{p}{1-p}\right) \]

where Φ is the CDF function for the standard normal distribution.

The probability plot is another tool that can be used to assess fit. The probability plot shows an approximately straight line if the assumed distribution is a good fit.

Goodness of Fit

Anderson-Darling Test

The hypotheses for the Anderson-Darling test are:

H₀: Input data follows the distribution.

H₁: Input data doesn't follow the distribution.

• Anderson-Darling Statistic

Input data are sorted first. The CDF function is used to calculate the probability Z_i for each point, and \(F_n(Z_i)\) is the probability for the ith point calculated by Benard method or Kaplan-Meier method. The Anderson-Darling statistic A² is calculated using two methods:

1. Benard method

   \( A^2 = n \sum_{i=1}^{n+1} \Bigl[ \text{PartA}_i + \text{PartB}_i + \text{PartC}_i \Bigr]\),

   where \( \text{PartA}_i = (Z_{i-1} - Z_i) \; + \; \ln\! \left(\frac{1 - Z_{i-1}}{1 - Z_i} \right) \), \( \text{PartB}_i = 2 F_n(Z_{i-1}) \, \ln \! \left(\dfrac{1 - Z_i}{1 - Z_{i-1}} \right) \),

       \( \text{PartC}_i = F_n(Z_{i-1})^2 \, \ln \! \left(\dfrac{Z_i(1 - Z_{i-1})}{Z_{i-1}(1 - Z_i)}\right) \), \(Z_0 = 0, \, F_n(Z_0) = 0, \, \ln(Z_0) = 0, Z_{n+1} = 1 - 10^{-12}\).

2. Kaplan-Meier method

\( A^2 = n \Bigl [ -1 - \ln(Z_n) -\ln(1 - Z_n) + \sum_{i=2}^n \bigl( \text{PartB}_i + \text{PartC}_i \bigr) \; \Bigr ]\),

3. When there is no right censoring in maximum likelihood estimation method,

   \[ A^2 = -n -(1/n) \sum_{i=1}^n \Bigl[ (2i-1) \text{ln} (Z_i) + ( 2n+1-i ) \text{ln} (1 - Z_i) \Bigr]\]

   A² value can be used to assess the fit. The smaller the value is, the better the model will be.

• P-value can be calculated for the Anderson-Darling Test using the Kaplan-Meier method

P-value is calculated using a bootstrap resampling approach. And the value can be compared with other software. Events are drawn from a distribution using the estimated parameters. Censoring indicators from the input data are retained. If P-value is less than the critical value, e.g. 0.05, H₀ will be rejected.

The P-value is the proportion of bootstrap samples whose A² statistic values are greater than or equal to input data's [4]. i.e.

where b is the number of test statistic values that are greater than or equal to input data's test statistic, and N is the number of bootstrap samples.

When there is no right censoring, we use another method [2] to calculate the P-value like Statistical Process Control app.

Pearson Correlation Coefficient

The Pearson correlation coefficient is calculated under the Least Squares estimation method. It measures how well the straight line fits the input data. A high value indicates the least squares regression lines explains most of the variation in Y through X.

Reference

1. R.A. Lockhart and M.A. Stephens (1994). "Estimation and Tests of Fit for the Three-parameter Weibull Distribution". Journal of the Royal Statistical Society, Vol. 56, No. 3, pp. 491-500.
2. Ralph B. D'Agostino, Michael A. Stephens (Eds.) (1986). Goodness-of-Fit Techniques. New York: Marcel Dekker
3. Hartigan J A (1975). Clustering Algorithms. Wiley
4. W. Stute, W. G. Manteiga, and M. P. Quindimil (1993). “Bootstrap based goodness-of-fit-tests”. Metrika, 40.1: pp. 243-256.
5. W. Q. Meeker, L. A. Escobar, and F. G. Pascual (2021). Statistical Methods for Reliability Data, 2nd Edition. New York: John Wiley & Sons.
6. F. Giesbrecht and O. Kempthorne (1976). "Maximum Likelihood Estimation in the Three-parameter Lognormal Distribution". Journal of the Royal Statistical Society, Series B, Vol. 38, No. 3, pp. 257-264.
7. H.L. Harter and A.H. Moore (1966). "Local-Maximum-Likelihood Estimation of the Parameters of Three-parameter Lognormal Populations from Complete and Censored Samples". Journal of the American Statistical Association, Vol. 61, No. 314, pp. 842-851.
8. J.F. Lawless (2003). Statistical Models and Methods for Lifetime Data, 2nd Edition. Hoboken, New Jersey: John Wiley & Sons.
9. R.C.H. Cheng (1990). "Embedded Models in Three-parameter Distributions and Their Estimation". Journal of the Royal Statistical Society, Series B, Vol. 52, No. 1, pp. 135-149.
10. R.L. Smith (1985). "Maximum Likelihood Estimation in a Class of Non-Regular Cases". Biometrika, Vol. 72, No. 1, pp. 67-90.