naginterfaces.library.univar.estim_​normal

naginterfaces.library.univar.estim_normal(method, x, xc, ic, xmu, xsig, tol, maxit)[source]

estim_normal computes maximum likelihood estimates and their standard errors for parameters of the Normal distribution from grouped and/or censored data.

For full information please refer to the NAG Library document for g07bb

https://www.nag.com/numeric/nl/nagdoc_28.7/flhtml/g07/g07bbf.html

Parameters
methodstr, length 1

Indicates whether the Newton–Raphson or algorithm should be used.

If , the Newton–Raphson algorithm is used.

If , the algorithm is used.

xfloat, array-like, shape

The observations , or , for .

If the observation is exactly specified – the exact value, .

If the observation is right-censored – the lower value, .

If the observation is left-censored – the upper value, .

If the observation is interval-censored – the lower or upper value, or , (see ).

xcfloat, array-like, shape

If the th observation, for is an interval-censored observation then should contain the complementary value to , that is, if , then contains upper value, , and if , then contains lower value, . Otherwise if the th observation is exact or right - or left-censored need not be set.

Note: if then the observation is ignored.

icint, array-like, shape

contains the censoring codes for the th observation, for .

If , the observation is exactly specified.

If , the observation is right-censored.

If , the observation is left-censored.

If , the observation is interval-censored.

xmufloat

If the initial estimate of the mean, ; otherwise need not be set.

xsigfloat

Specifies whether an initial estimate of and are to be supplied.

is the initial estimate of and must contain an initial estimate of .

Initial estimates of and are calculated internally from:

  1. the exact observations, if the number of exactly specified observations is ; or

  2. the interval-censored observations; if the number of interval-censored observations is ; or

  3. they are set to and respectively.

tolfloat

The relative precision required for the final estimates of and . Convergence is assumed when the absolute relative changes in the estimates of both and are less than .

If , a relative precision of is used.

maxitint

The maximum number of iterations.

If , a value of is used.

Returns
xmufloat

The maximum likelihood estimate, , of .

xsigfloat

The maximum likelihood estimate, , of .

sexmufloat

The estimate of the standard error of .

sexsigfloat

The estimate of the standard error of .

corrfloat

The estimate of the correlation between and .

devfloat

The maximized log-likelihood, .

nobsint, ndarray, shape

The number of the different types of each observation;

contains number of right-censored observations.

contains number of left-censored observations.

contains number of interval-censored observations.

contains number of exactly specified observations.

nitint

The number of iterations performed.

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: or .

(errno )

On entry, effective number of observations .

(errno )

On entry, and .

Constraint: , , or .

(errno )

On entry, .

Constraint: or .

(errno )

The chosen method has not converged in iterations.

(errno )

The process has diverged.

(errno )

The EM process has failed.

(errno )

Standard errors cannot be computed.

Notes

A sample of size is taken from a Normal distribution with mean and variance and consists of grouped and/or censored data. Each of the observations is known by a pair of values such that:

The data is represented as particular cases of this form:

exactly specified observations occur when ,

right-censored observations, known only by a lower bound, occur when ,

left-censored observations, known only by a upper bound, occur when ,

and interval-censored observations when .

Let the set identify the exactly specified observations, sets and identify the observations censored on the right and left respectively, and set identify the observations confined between two finite limits. Also let there be exactly specified observations, i.e., the number in . The probability density function for the standard Normal distribution is

and the cumulative distribution function is

The log-likelihood of the sample can be written as:

where and .

Let

and

then the first derivatives of the log-likelihood can be written as:

and

The maximum likelihood estimates, and , are the solution to the equations:

and

and if the second derivatives , and are denoted by , and respectively, then estimates of the standard errors of and are given by:

and an estimate of the correlation of and is given by:

To obtain the maximum likelihood estimates the equations (1) and (2) can be solved using either the Newton–Raphson method or the Expectation-maximization algorithm of Dempster et al. (1977).

Newton–Raphson Method

This consists of using approximate estimates and to obtain improved estimates and by solving

for the corrections and .

EM Algorithm

The expectation step consists of constructing the variable as follows:

the maximization step consists of substituting (3), (4), (5) and (6) into (1) and (2) giving:

and

where

and where , and are , and evaluated at and . Equations (3) and (8) are the basis of the iterative procedure for finding and . The procedure consists of alternately estimating and using (7) and (8) and estimating using (3) and (6).

In choosing between the two methods a general rule is that the Newton–Raphson method converges more quickly but requires good initial estimates whereas the algorithm converges slowly but is robust to the initial values. In the case of the censored Normal distribution, if only a small proportion of the observations are censored then estimates based on the exact observations should give good enough initial estimates for the Newton–Raphson method to be used. If there are a high proportion of censored observations then the algorithm should be used and if high accuracy is required the subsequent use of the Newton–Raphson method to refine the estimates obtained from the algorithm should be considered.

References

Dempster, A P, Laird, N M and Rubin, D B, 1977, Maximum likelihood from incomplete data via the algorithm (with discussion), J. Roy. Statist. Soc. Ser. B (39), 1–38

Swan, A V, 1969, Algorithm AS 16. Maximum likelihood estimation from grouped and censored normal data, Appl. Statist. (18), 110–114

Wolynetz, M S, 1979, Maximum likelihood estimation from confined and censored normal data, Appl. Statist. (28), 185–195