nag_estim_weibull (g07bec) : NAG Library, Mark 25

nag_estim_weibull (g07bec) computes maximum likelihood estimates for arguments of the Weibull distribution from data which may be right-censored.

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

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.

$cens = Nag_NoCensored$: 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

On entry:

n

, the number of observations.

Constraint:

n \geq 1

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

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

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 9 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.

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

Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.

On entry, argument

〈value〉

had an illegal value.

Iterations have failed to converge in

〈value〉

iterations.

Iterations have diverged.

Unable to calculate initial values.

On entry,

n = 〈value〉

.
Constraint:

n \geq 1

On entry, element

〈value〉

of ic was not valid.

ic [I] = 〈value〉

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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.

Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.

On entry, there are no exact observations.

Potential overflow detected.

On entry, tol is invalid:

tol = 〈value〉

On entry, observation

〈value〉

\leq 0.0

x [I] = 〈value〉

Hessian matrix is singular.

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.

nag_estim_weibull (g07bec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

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 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)