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 realizations,
yi, from a Weibull distribution a value
xi is observed such that
There are two situations:
| (a) |
exactly specified observations, when xi=yi |
| (b) |
right-censored observations, known by a lower bound, when xi<yi. |
The probability density function of the Weibull distribution, and hence the contribution of an exactly specified observation to the likelihood, is given by:
while the survival function of the Weibull distribution, and hence the contribution of a right-censored observation to the likelihood, is given by:
If
d of the
n observations are exactly specified and indicated by
i∈D and the remaining
n-d are right-censored, then the likelihood function,
Like λ,γ is given by
To avoid possible numerical instability a different parameterization
β,γ is used, with
β=logλ. The kernel log-likelihood function,
Lβ,γ, is then:
If the derivatives
∂L
∂β
,
∂L
∂γ
,
∂2L
∂β2
,
∂2L
∂β
∂γ
and
∂2L
∂γ2
are denoted by
L1,
L2,
L11,
L12 and
L22, respectively, then the maximum likelihood estimates,
β^ and
γ^, are the solution to the equations:
and
Estimates of the asymptotic standard errors of
β^ and
γ^ are given by:
An estimate of the correlation coefficient of
β^ and
γ^ is given by:
Note: if an estimate of the original parameter
λ is required, then
The equations
(1) and
(2) are solved by the Newton–Raphson iterative method with adjustments made to ensure that
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
- 1: CENS – CHARACTER*1Input
On entry: indicates whether the data is censored or non-censored.
- If CENS='N', then each observation is assumed to be exactly specified. IC is not referenced.
- If CENS='C', then each observation is censored according to the value contained in ICi, for i=1,2,…,n.
Constraint:
CENS='C' or 'N'.
- 2: N – INTEGERInput
On entry:
n, the number of observations.
Constraint:
N≥1.
- 3: X(N) – double precision arrayInput
On entry: Xi contains the ith observation, xi, for i=1,2,…,n.
Constraint:
Xi>0.0, for i=1,2,…,n.
- 4: IC(*) – INTEGER arrayInput
Note: the dimension of the array
IC
must be at least
N if
CENS='C', and at least
1 otherwise.
On entry: if
CENS='C', then
ICi contains the censoring codes for the
ith observation, for
i=1,2,…,n.
If ICi=0, the ith observation is exactly specified.
If ICi=1, the ith observation is right-censored.
If
CENS='N', then
IC is not referenced.
Constraint:
if CENS='C', then ICi=0 or 1, for i=1,2,…,n.
- 5: BETA – double precisionOutput
On exit: the maximum likelihood estimate, β^, of β.
- 6: GAMMA – double precisionInput/Output
On entry: indicates whether an initial estimate of
γ is provided.
If 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.
If
GAMMA≤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, γ^, of γ.
- 7: TOL – double precisionInput
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.
If TOL=0.0, then a relative precision of 0.000005 is used.
Constraint:
machine precision≤TOL≤1.0 or TOL=0.0.
- 8: MAXIT – INTEGERInput
On entry: the maximum number of iterations allowed.
If MAXIT≤0, then a value of 25 is used.
- 9: SEBETA – double precisionOutput
On exit: an estimate of the standard error of β^.
- 10: SEGAM – double precisionOutput
On exit: an estimate of the standard error of γ^.
- 11: CORR – double precisionOutput
On exit: an estimate of the correlation between β^ and γ^.
- 12: DEV – double precisionOutput
On exit: the maximized kernel log-likelihood, Lβ^,γ^.
- 13: NIT – INTEGEROutput
-
On exit: the number of iterations performed.
- 14: WK(N) – double precision arrayWorkspace
- 15: IFAIL – INTEGERInput/Output
-
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.
On exit:
IFAIL=0 unless the routine detects an error (see
Section 6).
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.
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
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.
In a study,
20 patients receiving an analgesic to relieve headache pain had the following recorded relief times (in hours):
(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.