g07be:: Univariate Estimation (NAG Toolbox)

nag_univar_estim_weibull (g07be) computes maximum likelihood estimates for arguments of the Weibull distribution from data which may be right-censored.

nag_univar_estim_weibull (g07be) computes maximum likelihood estimates of the arguments of the Weibull distribution from exact or right-censored data.

For

n

realizations,

y_{i}

, from a Weibull distribution a value

x_{i}

is observed such that

x_{i} \leq y_{i} .

There are two situations:

(a)	exactly specified observations, when $x_{i} = y_{i}$
(b)	right-censored observations, known by a lower bound, when $x_{i} < y_{i}$ .

The probability density function of the Weibull distribution, and hence the contribution of an exactly specified observation to the likelihood, is given by:

f (x; λ, γ) = λ γ x^{γ - 1} \exp (- λ x^{γ}), x > 0,   for ​ λ, γ > 0;

while the survival function of the Weibull distribution, and hence the contribution of a right-censored observation to the likelihood, is given by:

S (x; λ, γ) = \exp (- λ x^{γ}), x > 0,   for ​ λ, γ > 0 .

d

of the

n

observations are exactly specified and indicated by

i \in D

and the remaining

(n - d)

are right-censored, then the likelihood function,

Like ​ (λ, γ)

is given by

Like (λ, γ) \propto {(λ γ)}^{d} (\prod_{i \in D} x_{i}^{γ - 1}) \exp (- λ \sum_{i = 1}^{n} x_{i}^{γ}) .

To avoid possible numerical instability a different parameterisation

β, γ

is used, with

β = \log (λ)

. The kernel log-likelihood function,

L (β, γ)

, is then:

L (β, γ) = d \log (γ) + d β + (γ - 1) \sum_{i \in D} \log (x_{i}) - e^{β} \sum_{i = 1}^{n} x_{i}^{γ} .

If the derivatives

\frac{\partial L}{\partial β}

\frac{\partial L}{\partial γ}

\frac{\partial^{2} L}{{\partial β}^{2}}

\frac{\partial^{2} L}{\partial β \partial γ}

and

\frac{\partial^{2} L}{{\partial γ}^{2}}

are denoted by

L_{1}

L_{2}

L_{11}

L_{12}

and

L_{22}

, respectively, then the maximum likelihood estimates,

\hat{β}

and

\hat{γ}

, are the solution to the equations:

L_{1} (\hat{β}, \hat{γ}) = 0

(1)

and

L_{2} (\hat{β}, \hat{γ}) = 0

(2)

Estimates of the asymptotic standard errors of

\hat{β}

and

\hat{γ}

are given by:

se (\hat{β}) = \sqrt{\frac{- L_{22}}{L_{11} L_{22} - L_{12}^{2}}}, se (\hat{γ}) = \sqrt{\frac{- L_{11}}{L_{11} L_{22} - L_{12}^{2}}} .

An estimate of the correlation coefficient of

\hat{β}

and

\hat{γ}

is given by:

\frac{L_{12}}{\sqrt{L_{12} L_{22}}} .

Note: if an estimate of the original argument

λ

is required, then

\hat{λ} = \exp (\hat{β}) and se (\hat{λ}) = \hat{λ} se (\hat{β}) .

The equations (1) and (2) are solved by the Newton–Raphson iterative method with adjustments made to ensure that

\hat{γ} > 0.0

Gross A J and Clark V A (1975) Survival Distributions: Reliability Applications in the Biomedical Sciences Wiley

Kalbfleisch J D and Prentice R L (1980) The Statistical Analysis of Failure Time Data Wiley

Indicates whether the data is censored or non-censored.

$cens ='N'$: Each observation is assumed to be exactly specified. ic is not referenced.
$cens ='C'$: Each observation is censored according to the value contained in $ic (i)$ , for $i = 1, 2, \dots, n$ .

Constraint:

cens ='N'

'C'

x (i)

contains the

i

th observation,

x_{i}

, for

i = 1, 2, \dots, n

Constraint:

x (i) > 0.0

, for

i = 1, 2, \dots, n

The dimension of the array ic must be at least

n

cens ='C'

, and at least

1

otherwise

cens ='C'

, then

ic (i)

contains the censoring codes for the

i

th observation, for

i = 1, 2, \dots, n

ic (i) = 0

, the

i

th observation is exactly specified.

ic (i) = 1

, the

i

th observation is right-censored.

cens ='N'

, then ic is not referenced.

Constraint: if

cens ='C'

, then

ic (i) = 0

1

, for

i = 1, 2, \dots, n

Indicates whether an initial estimate of

γ

is provided.

gamma > 0.0

, it is taken as the initial estimate of

γ

and an initial estimate of

β

is calculated from this value of

γ

gamma \leq 0.0

, then initial estimates of

γ

and

β

are calculated, internally, providing the data contains at least two distinct exact observations. (If there are only two distinct exact observations, then the largest observation must not be exactly specified.) See Further Comments for further details.

The relative precision required for the final estimates of

β

and

γ

. Convergence is assumed when the absolute relative changes in the estimates of both

β

and

γ

are less than tol.

tol = 0.0

, then a relative precision of

0.000005

is used.

Constraint:

machine precision \leq tol \leq 1.0

tol = 0.0

The maximum number of iterations allowed.

maxit \leq 0

, then a value of

25

is used.

Default: the dimension of the array x.

n

, the number of observations.

Constraint:

n \geq 1

The maximum likelihood estimate,

\hat{β}

, of

β

Contains the maximum likelihood estimate,

\hat{γ}

, of

γ

An estimate of the standard error of

\hat{β}

An estimate of the standard error of

\hat{γ}

An estimate of the correlation between

\hat{β}

and

\hat{γ}

The maximized kernel log-likelihood,

L (\hat{β}, \hat{γ})

The number of iterations performed.

ifail = 0

unless the function detects an error (see Error Indicators and Warnings).

Errors or warnings detected by the function:

On entry, there are no exactly specified observations, or the function was requested to calculate initial values and there are either less than two distinct exactly specified observations or there are exactly two and the largest observation is one of the exact observations.

The method has failed to converge in maxit iterations. You should increase tol or maxit.

Process has diverged. The process is deemed divergent if three successive increments of

β

γ

increase or if the Hessian matrix of the Newton–Raphson process is singular. Either different initial estimates should be provided or the data should be checked to see if the Weibull distribution is appropriate.

A potential overflow has been detected. This is an unlikely exit usually caused by a large input estimate of

γ

An unexpected error has been triggered by this routine. Please contact NAG.

Your licence key may have expired or may not have been installed correctly.

Dynamic memory allocation failed.

Given that the Weibull distribution is a suitable model for the data and that the initial values are reasonable the convergence to the required accuracy, indicated by tol, should be achieved.

The initial estimate of

γ

is found by calculating a Kaplan–Meier estimate of the survival function,

\hat{S} (x)

, and estimating the gradient of the plot of

\log (- \log (\hat{S} (x)))

against

x

. This requires the Kaplan–Meier estimate to have at least two distinct points.

The initial estimate of

\hat{β}

, given a value of

\hat{γ}

, is calculated as

\hat{β} = \log (\frac{d}{\sum_{i = 1}^{n} x_{i}^{\hat{γ}}}) .

In a study,

20

patients receiving an analgesic to relieve headache pain had the following recorded relief times (in hours):

1.1 1.4 1.3 1.7 1.9 1.8 1.6 2.2 1.7 2.7 4.1 1.8 1.5 1.2 1.4 3.0 1.7 2.3 1.6 2.0

(See Gross and Clark (1975).) This data is read in and a Weibull distribution fitted assuming no censoring; the parameter estimates and their standard errors are printed.

NAG Toolbox: nag_univar_estim_weibull (g07be)

▸▿ Contents

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Example

On entry,	$cens \neq'N'$ or $'C'$ ,
or	$n < 1$ ,
or	$tol < 0.0$ ,
or	$0.0 < tol < machine precision$ ,
or	$tol > 1.0$ .

On entry,	the $i$ th observation, $x (i) \leq 0.0$ , for some $i = 1, 2, \dots, n$ ,
or	the $i$ th censoring code, $ic (i) \neq 0$ or $1$ , for some $i = 1, 2, \dots, n$ and $cens ='C'$ .