NAG Library Function Document
nag_rank_regsn (g08rac)
1
Purpose
nag_rank_regsn (g08rac) calculates the parameter estimates, score statistics and their variance-covariance matrices for the linear model using a likelihood based on the ranks of the observations.
2
Specification
#include <nag.h> |
#include <nagg08.h> |
void |
nag_rank_regsn (Nag_OrderType order,
Integer ns,
const Integer nv[],
const double y[],
Integer p,
const double x[],
Integer pdx,
Integer idist,
Integer nmax,
double tol,
double prvr[],
Integer pdparvar,
Integer irank[],
double zin[],
double eta[],
double vapvec[],
double parest[],
NagError *fail) |
|
3
Description
Analysis of data can be made by replacing observations by their ranks. The analysis produces inference for regression arguments arising from the following model.
For random variables
we assume that, after an arbitrary monotone increasing differentiable transformation,
, the model
holds, where
is a known vector of explanatory variables and
is a vector of
unknown regression coefficients. The
are random variables assumed to be independent and identically distributed with a completely known distribution which can be one of the following: Normal, logistic, extreme value or double-exponential. In
Pettitt (1982) an estimate for
is proposed as
with estimated variance-covariance matrix
. The statistics
and
depend on the ranks
of the observations
and the density chosen for
.
The matrix
is the
by
matrix of explanatory variables. It is assumed that
is of rank
and that a column or a linear combination of columns of
is not equal to the column vector of
or a multiple of it. This means that a constant term cannot be included in the model
(1). The statistics
and
are found as follows. Let
have pdf
and let
. Let
be order statistics for a random sample of size
with the density
. Define
, then
. To define
we need
, where
is an
by
diagonal matrix with
and
is a symmetric matrix with
. In the case of the Normal distribution, the
are standard Normal order statistics and
, for
.
The analysis can also deal with ties in the data. Two observations are adjudged to be tied if
, where
tol is a user-supplied tolerance level.
Various statistics can be found from the analysis:
(a) |
The score statistic . This statistic is used to test the hypothesis , see (e). |
(b) |
The estimated variance-covariance matrix of the score statistic in (a). |
(c) |
The estimate . |
(d) |
The estimated variance-covariance matrix of the estimate . |
(e) |
The statistic used to test . Under , has an approximate -distribution with degrees of freedom. |
(f) |
The standard errors of the estimates given in (c). |
(g) |
Approximate -statistics, i.e., for testing . For , has an approximate distribution. |
In many situations, more than one sample of observations will be available. In this case we assume the model
where
ns is the number of samples. In an obvious manner,
and
are the vector of observations and the design matrix for the
th sample respectively. Note that the arbitrary transformation
can be assumed different for each sample since observations are ranked within the sample.
The earlier analysis can be extended to give a combined estimate of
as
, where
and
with
,
and
defined as
,
and
above but for the
th sample.
The remaining statistics are calculated as for the one sample case.
4
References
Pettitt A N (1982) Inference for the linear model using a likelihood based on ranks J. Roy. Statist. Soc. Ser. B 44 234–243
5
Arguments
- 1:
– Nag_OrderTypeInput
-
On entry: the
order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by
. See
Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– IntegerInput
-
On entry: the number of samples.
Constraint:
.
- 3:
– const IntegerInput
-
On entry: the number of observations in the
th sample, for .
Constraint:
, for .
- 4:
– const doubleInput
-
Note: the dimension,
dim, of the array
y
must be at least
.
On entry: the observations in each sample. Specifically, must contain the th observation in the th sample.
- 5:
– IntegerInput
-
On entry: the number of parameters to be fitted.
Constraint:
.
- 6:
– const doubleInput
-
Note: the dimension,
dim, of the array
x
must be at least
- when ;
- when .
Where
appears in this document, it refers to the array element
- when ;
- when .
On entry: the design matrices for each sample. Specifically, must contain the value of the th explanatory variable for the th observation in the th sample.
Constraint:
must not contain a column with all elements equal.
- 7:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
x.
Constraints:
- if ,
;
- if , .
- 8:
– IntegerInput
-
On entry: the error distribution to be used in the analysis.
- Normal.
- Logistic.
- Extreme value.
- Double-exponential.
Constraint:
.
- 9:
– IntegerInput
-
On entry: the value of the largest sample size.
Constraint:
and .
- 10:
– doubleInput
-
On entry: the tolerance for judging whether two observations are tied. Thus, observations and are adjudged to be tied if .
Constraint:
.
- 11:
– doubleOutput
-
Note: the dimension,
dim, of the array
prvr
must be at least
- when ;
- when .
Where
appears in this document, it refers to the array element
- when ;
- when .
On exit: the variance-covariance matrices of the score statistics and the parameter estimates, the former being stored in the upper triangle and the latter in the lower triangle. Thus for , contains an estimate of the covariance between the th and th score statistics. For , contains an estimate of the covariance between the th and th parameter estimates.
- 12:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
prvr.
Constraints:
- if ,
;
- if , .
- 13:
– IntegerOutput
-
On exit: for the one sample case,
irank contains the ranks of the observations.
- 14:
– doubleOutput
-
On exit: for the one sample case,
zin contains the expected values of the function
of the order statistics.
- 15:
– doubleOutput
-
On exit: for the one sample case,
eta contains the expected values of the function
of the order statistics.
- 16:
– doubleOutput
-
On exit: for the one sample case,
vapvec contains the upper triangle of the variance-covariance matrix of the function
of the order statistics stored column-wise.
- 17:
– doubleOutput
-
On exit: the statistics calculated by the function.
The first
p components of
parest contain the score statistics.
The next
p elements contain the parameter estimates.
contains the value of the statistic.
The next
p elements of
parest contain the standard errors of the parameter estimates.
Finally, the remaining
p elements of
parest contain the
-statistics.
- 18:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry,
idist is outside the range
to
:
.
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and sum .
Constraint: the sum of .
- NE_INT_ARRAY
-
On entry, .
Constraint: , for .
- NE_INT_ARRAY_ELEM_CONS
-
On entry .
Constraint: elements of array .
- 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.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_MAT_ILL_DEFINED
-
The matrix is either singular or non-positive definite.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
- NE_OBSERVATIONS
-
All the observations were adjudged to be tied.
- NE_REAL
-
On entry, .
Constraint: .
- NE_REAL_ARRAY_ELEM_CONS
-
On entry, all elements in column of are equal to .
- NE_SAMPLE
-
The largest sample size is
which is not equal to
nmax,
.
7
Accuracy
The computations are believed to be stable.
8
Parallelism and Performance
nag_rank_regsn (g08rac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_rank_regsn (g08rac) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
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 time taken by nag_rank_regsn (g08rac) depends on the number of samples, the total number of observations and the number of arguments fitted.
In extreme cases the parameter estimates for certain models can be infinite, although this is unlikely to occur in practice. See
Pettitt (1982) for further details.
10
Example
A program to fit a regression model to a single sample of observations using two explanatory variables. The error distribution will be taken to be logistic.
10.1
Program Text
Program Text (g08race.c)
10.2
Program Data
Program Data (g08race.d)
10.3
Program Results
Program Results (g08race.r)