G07BEF : NAG Library, Mark 24

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

G07BEF computes maximum likelihood estimates for parameters of the Weibull distribution from data which may be right-censored.

G07BEF computes maximum likelihood estimates of the parameters of the Weibull distribution from exact or right-censored data.

For

n

realisations,

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 parameter

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

On entry: 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'

On entry:

n

, the number of observations.

Constraint:

N \geq 1

On entry:

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

Note: the dimension of the array IC must be at least

N

CENS ='C'

, and at least

1

otherwise.

On entry: if

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

On exit: the maximum likelihood estimate,

\hat{β}

, of

β

On entry: 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 Section 8 for further details.

On exit: contains the maximum likelihood estimate,

\hat{γ}

, of

γ

On entry: 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

On entry: the maximum number of iterations allowed.

MAXIT \leq 0

, then a value of

25

is used.

On exit: an estimate of the standard error of

\hat{β}

On exit: an estimate of the standard error of

\hat{γ}

On exit: an estimate of the correlation between

\hat{β}

and

\hat{γ}

On exit: the maximized kernel log-likelihood,

L (\hat{β}, \hat{γ})

On exit: the number of iterations performed.

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

On entry, there are no exactly specified observations, or the routine 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

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 Library Routine Document

G07BEF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

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 $CENS ='C'$ .

NAG Library Routine DocumentG07BEF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G07BEF