AMILLR

This functon evaluates Mill’s ratio (the ratio of the ordinate to the upper tail area of the standardized normal distribution).

Function Return Value

AMILLR ‑ Function value, Mill’s ratio. (Output)

Required Arguments

X ‑ Value at which Mill’s ratio is evaluated. (Input)
In order to avoid overflow, X must be less than a bound that is machine dependent. On most machines, the bound is greater than –13. The function underflows (and is set to 0.0) for small values of X. On most machines, the underflow does not occur unless X is less than ‑13.

FORTRAN 90 Interface

Generic: AMILLR (X)

Specific: The specific interface names are S_AMILLR and D_AMILLR.

FORTRAN 77 Interface

Single: AMILLR (X)

Double: The double precision name is DMILLR.

Description

Function AMILLR evaluates Mill’s ratio, the hazard rate for the standard normal distribution. It is computed as the ratio of the ordinate to the upper tail area of the standard normal distribution, that is, ɸ(x)/(1 ‑ Φ(x)), where ɸ(x) and Φ(x) are the standard normal density and cumulative distribution functions, respectively. The reciprocal of Mill’s ratio is called the failure rate in reliability and life testing applications. As x becomes small, the ratio goes to zero. For large x (how large is machine dependent), the ratio cannot be computed. Function AMILLR computes 1 ‑ Φ(x) using the complementary error function (IMSL 1991) rather than as one minus the normal distribution function, which would underflow sooner as x gets small.

Comments

Informational Error

 

Type	Code	Description
2	1	The function underflows because X is too small.

Example

In this example, we compute Mill’s ratio at x = ‑1.0.

 

USE UMACH_INT

USE AMILLR_INT

 

IMPLICIT NONE

INTEGER NOUT

REAL R, X

!

CALL UMACH (2, NOUT)

X = -1.0

R = AMILLR(X)

WRITE (NOUT,99999) R

99999 FORMAT (' Mill''s ratio at -1.0 is ', F8.5)

END

Output

 

Mill’s ratio at -1.0 is 0.28760