g08ra:: Nonparametric Statistics (NAG Toolbox)

For random variables

Y_{1}, Y_{2}, \dots, Y_{n}

we assume that, after an arbitrary monotone increasing differentiable transformation,

h (.)

, the model

h (Y_{i}) = x_{i}^{T} β + ε_{i}

(1)

holds, where

x_{i}

is a known vector of explanatory variables and

β

is a vector of

p

unknown regression coefficients. The

ε_{i}

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

β

is proposed as

\hat{β} = M X^{T} a

with estimated variance-covariance matrix

M

. The statistics

a

and

M

depend on the ranks

r_{i}

of the observations

Y_{i}

and the density chosen for

ε_{i}

The matrix

X

is the

n

p

matrix of explanatory variables. It is assumed that

X

is of rank

p

and that a column or a linear combination of columns of

X

is not equal to the column vector of

1

or a multiple of it. This means that a constant term cannot be included in the model (1). The statistics

a

and

M

are found as follows. Let

ε_{i}

have pdf

f (ε)

and let

g = - f^{'} / f

. Let

W_{1}, W_{2}, \dots, W_{n}

be order statistics for a random sample of size

n

with the density

f (.)

. Define

Z_{i} = g (W_{i})

, then

a_{i} = E (Z_{r_{i}})

. To define

M

we need

M^{- 1} = X^{T} (B - A) X

, where

B

is an

n

n

diagonal matrix with

B_{i i} = E (g^{'} (W_{r_{i}}))

and

A

is a symmetric matrix with

A_{i j} = cov (Z_{r_{i}}, Z_{r_{j}})

. In the case of the Normal distribution, the

Z_{1} < \dots < Z_{n}

are standard Normal order statistics and

E (g^{'} (W_{i})) = 1

, for

i = 1, 2, \dots, n

Various statistics can be found from the analysis:

(a)	The score statistic $X^{T} a$ . This statistic is used to test the hypothesis $H_{0} : β = 0$ , see (e).
(b)	The estimated variance-covariance matrix $X^{T} (B - A) X$ of the score statistic in (a).
(c)	The estimate $\hat{β} = M X^{T} a$ .
(d)	The estimated variance-covariance matrix $M = {(X^{T} (B - A) X)}^{- 1}$ of the estimate $\hat{β}$ .
(e)	The $χ^{2}$ statistic $Q = {\hat{β}}^{T} M^{- 1} \hat{β} = a^{T} X {(X^{T} (B - A) X)}^{- 1} X^{T} a$ used to test $H_{0} : β = 0$ . Under $H_{0}$ , $Q$ has an approximate $χ^{2}$ -distribution with $p$ degrees of freedom.
(f)	The standard errors $M_{i i}^{1 / 2}$ of the estimates given in (c).
(g)	Approximate $z$ -statistics, i.e., $Z_{i} = {\hat{β}}_{i} / s e ({\hat{β}}_{i})$ for testing $H_{0} : β_{i} = 0$ . For $i = 1, 2, \dots, n$ , $Z_{i}$ has an approximate $N (0, 1)$ distribution.

In many situations, more than one sample of observations will be available. In this case we assume the model

h_{k} (Y_{k}) = X_{k}^{T} β + e_{k}, k = 1, 2, \dots, ns,

where ns is the number of samples. In an obvious manner,

Y_{k}

and

X_{k}

are the vector of observations and the design matrix for the

k

th sample respectively. Note that the arbitrary transformation

h_{k}

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

β

\hat{β} = D d

, where

D^{- 1} = \sum_{k = 1}^{ns} X_{k}^{T} (B_{k} - A_{k}) X_{k}

and

d = \sum_{k = 1}^{ns} X_{k}^{T} a_{k},

with

a_{k}

B_{k}

and

A_{k}

defined as

a

B

and

A

above but for the

k

th sample.

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Example

function g08ra_example


fprintf('g08ra example results\n\n');

% Single sample observations and design matrix
y = [1;  1;  3;  4;  2;  4;  1;  5;  4;  4;
     4;  4;  4;  1;  4;  5;  5;  4;  4;  3];
x = [1, 23;  1, 32;  1, 37;  1, 41;  1, 41;
     1, 48;  1, 48;  1, 55;  1, 55;  0, 56;
     1, 57;  1, 57;  1, 57;  0, 58;  1, 59;
     0, 59;  0, 60;  1, 61;  1, 62;  1, 62];

ns    = size(y,2);
ip    = size(x,2);
idist = int64(2);
nv    = int64(numel(y));
nmax  = nv;
tol   = 1e-05;

fprintf('Number of samples           = %3d\n', ns);
fprintf('Number of parameters fitted = %3d\n', ip);
fprintf('Distribution                = %3d\n', idist);
fprintf('Tolerance for ties          = %8.1e\n', tol);

[parvar, irank, zin, eta, vapvec, parest, ifail] = ...
  g08ra( ...
         nv, y, x, idist, nmax, tol);

% Display results
fprintf('\nScore statistic\n');
fprintf('%9.3f%9.3f\n', parest(1:ip));
fprintf('\nCovariance matrix of score statistic\n');
for j = 1:ip
  fprintf('%9.3f', parvar(1:j,j));
  fprintf('\n');
end
fprintf('\nParameter estimates\n');
fprintf('%9.3f', parest(ip+1:ip+ip));
fprintf('\n\nCovariance matrix of parameter estimates\n');
for j = 1:ip
  fprintf('%9.3f', parvar(j+1,1:j));
  fprintf('\n');
end

chisq = parest(2*ip+1);
fprintf('\nChi-squared statistic = %8.3f with %2d d.f.\n\n', chisq, ip);

sterr = reshape(parest(2*ip+2:end),[ip,2]);
fprintf('Standard errors of estimates and approximate z-statistics\n');
disp(sterr);

g08ra example results

Number of samples           =   1
Number of parameters fitted =   2
Distribution                =   2
Tolerance for ties          =  1.0e-05

Score statistic
   -1.048   64.333

Covariance matrix of score statistic
    0.673
   -4.159  533.670

Parameter estimates
   -0.852    0.114

Covariance matrix of parameter estimates
    1.560
    0.012    0.002

Chi-squared statistic =    8.221 with  2 d.f.

Standard errors of estimates and approximate z-statistics
    1.2492   -0.6824
    0.0444    2.5673

On entry,	$ns < 1$ ,
or	$tol \leq 0.0$ ,
or	$nmax \leq ip$ ,
or	$ldprvr < ip + 1$ ,
or	$ldx < nsum$ ,
or	$nmax \neq \max_{1 \leq i \leq ns} (nv (i))$ ,
or	$nv (i) \leq 0$ , for some $i$ , $nv (i)$ ,
or	$nsum \neq \sum_{i = 1}^{ns} nv (i)$ ,
or	$ip < 1$ ,
or	$lwork < nmax \times (ip + 1)$ .

On entry,	$idist < 1$ ,
or	$idist > 4$ .

NAG Toolbox: nag_nonpar_rank_regsn (g08ra)

▸▿ Contents

Purpose

Syntax

Description