nag_estim_weibull (g07bec) : NAG Library, Mark 24

2 Specification

#include <nag.h>

#include <nagg07.h>

void	nag_estim_weibull (Nag_CestMethod cens, Integer n, const double x[], const Integer ic[], double beta, double gamma, double tol, Integer maxit, double sebeta, double segam, double corr, double dev, Integer nit, NagError fail)

3 Description

nag_estim_weibull (g07bec) 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

5 Arguments

1: cens – Nag_CestMethodInput

On entry: indicates whether the data is censored or non-censored.

$cens = Nag_NoCensored$: Each observation is assumed to be exactly specified. ic is not referenced.
$cens = Nag_Censored$: Each observation is censored according to the value contained in $ic [i - 1]$ , for $i = 1, 2, \dots, n$ .

Constraint:

cens = Nag_NoCensored

Nag_Censored

2: n – IntegerInput

On entry:

n

, the number of observations.

Constraint:

n \geq 1

3: x[n] – const doubleInput

On entry:

x [i - 1]

contains the

i

th observation,

x_{i}

, for

i = 1, 2, \dots, n

Constraint:

x [i - 1] > 0.0

, for

i = 1, 2, \dots, n

4: ic[

\dim

] – const IntegerInput

Note: the dimension, dim, of the array ic must be at least

$n$ when $cens = Nag_Censored$ ;
$1$ otherwise.

On entry: if

cens = Nag_Censored

, then

ic [i - 1]

contains the censoring codes for the

i

th observation, for

i = 1, 2, \dots, n

ic [i - 1] = 0

, the

i

th observation is exactly specified.

ic [i - 1] = 1

, the

i

th observation is right-censored.

cens = Nag_NoCensored

, then ic is not referenced.

Constraint: if

cens = Nag_Censored

, then

ic [i - 1] = 0

1

, for

i = 1, 2, \dots, n

5: beta – double *Output

On exit: the maximum likelihood estimate,

\hat{β}

, of

β

6: gamma – double *Input/Output

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 9 for further details.

On exit: contains the maximum likelihood estimate,

\hat{γ}

, of

γ

7: tol – doubleInput

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

8: maxit – IntegerInput

On entry: the maximum number of iterations allowed.

maxit \leq 0

, then a value of

25

is used.

9: sebeta – double *Output

On exit: an estimate of the standard error of

\hat{β}

10: segam – double *Output

On exit: an estimate of the standard error of

\hat{γ}

11: corr – double *Output

On exit: an estimate of the correlation between

\hat{β}

and

\hat{γ}

12: dev – double *Output

On exit: the maximized kernel log-likelihood,

L (\hat{β}, \hat{γ})

13: nit – Integer *Output

On exit: the number of iterations performed.

14: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_BAD_PARAM

On entry, argument

⟨value⟩

had an illegal value.

NE_CONVERGENCE

Iterations have failed to converge in

⟨value⟩

iterations.

NE_DIVERGENCE

Iterations have diverged.

NE_INITIALIZATION

Unable to calculate initial values.

NE_INT

On entry,

n = ⟨value⟩

.
Constraint:

n \geq 1

NE_INT_ARRAY_ELEM_CONS

On entry, element

⟨value⟩

of ic was not valid.

ic [I] = ⟨value⟩

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

NE_OBSERVATIONS

On entry, there are no exact observations.

NE_OVERFLOW

Potential overflow detected.

NE_REAL

On entry, tol is invalid:

tol = ⟨value⟩

NE_REAL_ARRAY_ELEM_CONS

On entry, observation

⟨value⟩

\leq 0.0

x [I] = ⟨value⟩

NE_SINGULAR

Hessian matrix is singular.

9 Further Comments

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}}}) .

10 Example

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.

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_estim_weibull (g07bec)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Documentnag_estim_weibull (g07bec)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_estim_weibull (g07bec)