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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_correg_quantile_linreg_easy (g02qf)

## Purpose

nag_correg_quantile_linreg_easy (g02qf) performs a multiple linear quantile regression, returning the parameter estimates and associated confidence limits based on an assumption of Normal, independent, identically distributed errors. nag_correg_quantile_linreg_easy (g02qf) is a simplified version of nag_correg_quantile_linreg (g02qg).

## Syntax

[df, b, bl, bu, info, ifail] = g02qf(x, y, tau, 'n', n, 'm', m, 'ntau', ntau)
[df, b, bl, bu, info, ifail] = nag_correg_quantile_linreg_easy(x, y, tau, 'n', n, 'm', m, 'ntau', ntau)

## Description

Given a vector of n$n$ observed values, y = {yi : i = 1,2,,n} $y=\left\{{y}_{i}:i=1,2,\dots ,n\right\}$, an n × p$n×p$ design matrix X$X$, a column vector, x$x$, of length p$p$ holding the i$i$th row of X$X$ and a quantile τ (0,1) $\tau \in \left(0,1\right)$, nag_correg_quantile_linreg_easy (g02qf) estimates the p$p$-element vector β$\beta$ as the solution to
 n minimize ∑ ρτ (yi − xiTβ) β ∈ ℝp i = 1
$minimize β ∈ ℝ p ∑ i=1 n ρ τ ( y i - xiT β )$
(1)
where ρτ ${\rho }_{\tau }$ is the piecewise linear loss function ρτ (z) = z (τI(z < 0)) ${\rho }_{\tau }\left(z\right)=z\left(\tau -I\left(z<0\right)\right)$, and I (z < 0) $I\left(z<0\right)$ is an indicator function taking the value 1$1$ if z < 0$z<0$ and 0$0$ otherwise.
nag_correg_quantile_linreg_easy (g02qf) assumes Normal, independent, identically distributed (IID) errors and calculates the asymptotic covariance matrix from
 Σ = ( τ (1 − τ) )/n (s(τ))2 (XTX) − 1 $Σ = τ ( 1 - τ ) n ( s( τ ) ) 2 ( XT X )-1$
where s$s$ is the sparsity function, which is estimated from the residuals, ri = yi xiT β̂ ${r}_{i}={y}_{i}-{x}_{i}^{\mathrm{T}}\stackrel{^}{\beta }$ (see Koenker (2005)).
Given an estimate of the covariance matrix, Σ̂$\stackrel{^}{\Sigma }$, lower, β̂L${\stackrel{^}{\beta }}_{L}$, and upper, β̂U${\stackrel{^}{\beta }}_{U}$, limits for a 95%$95%$ confidence interval are calculated for each of the p$p$ parameters, via
 β̂Li = β̂i − t n − p , 0.975 sqrt( Σ̂ii ) , β̂Ui = β̂i + t n − p , 0.975 sqrt( Σ̂ii ) $β^ Li = β^ i - t n-p , 0.975 Σ^ ii , β^ Ui = β^ i + t n-p , 0.975 Σ^ ii$
where tnp,0.975${t}_{n-p,0.975}$ is the 97.5$97.5$ percentile of the Student's t$t$ distribution with nk$n-k$ degrees of freedom, where k$k$ is the rank of the cross-product matrix XTX${X}^{\mathrm{T}}X$.
Further details of the algorithms used by nag_correg_quantile_linreg_easy (g02qf) can be found in the documentation for nag_correg_quantile_linreg (g02qg).

## References

Koenker R (2005) Quantile Regression Econometric Society Monographs, Cambridge University Press, New York

## Parameters

### Compulsory Input Parameters

1:     x(n,m) – double array
n, the first dimension of the array, must satisfy the constraint n2${\mathbf{n}}\ge 2$.
X$X$, the design matrix, with the i$\mathit{i}$th value for the j$\mathit{j}$th variate supplied in x(i,j)${\mathbf{x}}\left(\mathit{i},\mathit{j}\right)$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$ and j = 1,2,,m$\mathit{j}=1,2,\dots ,{\mathbf{m}}$.
2:     y(n) – double array
n, the dimension of the array, must satisfy the constraint n2${\mathbf{n}}\ge 2$.
y$y$, observations on the dependent variable.
3:     tau(ntau) – double array
ntau, the dimension of the array, must satisfy the constraint ntau1${\mathbf{ntau}}\ge 1$.
The vector of quantiles of interest. A separate model is fitted to each quantile.
Constraint: sqrt(ε) < tau(l) < 1sqrt(ε)$\sqrt{\epsilon }<{\mathbf{tau}}\left(\mathit{l}\right)<1-\sqrt{\epsilon }$ where ε$\epsilon$ is the machine precision returned by nag_machine_precision (x02aj), for l = 1,2,,ntau$\mathit{l}=1,2,\dots ,{\mathbf{ntau}}$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array y and the first dimension of the array x. (An error is raised if these dimensions are not equal.)
n$n$, the number of observations in the dataset.
Constraint: n2${\mathbf{n}}\ge 2$.
2:     m – int64int32nag_int scalar
Default: The second dimension of the array x.
p$p$, the number of variates in the model.
Constraint: 1m < n$1\le {\mathbf{m}}<{\mathbf{n}}$.
3:     ntau – int64int32nag_int scalar
Default: The dimension of the array tau.
The number of quantiles of interest.
Constraint: ntau1${\mathbf{ntau}}\ge 1$.

None.

### Output Parameters

1:     df – double scalar
The degrees of freedom given by nk$n-k$, where n$n$ is the number of observations and k$k$ is the rank of the cross-product matrix XTX${X}^{\mathrm{T}}X$.
2:     b(m,ntau) – double array
β̂$\stackrel{^}{\beta }$, the estimates of the parameters of the regression model, with b(j,l)${\mathbf{b}}\left(j,l\right)$ containing the coefficient for the variable in column j$j$ of x, estimated for τ = tau(l)$\tau ={\mathbf{tau}}\left(l\right)$.
3:     bl(m,ntau) – double array
β̂L${\stackrel{^}{\beta }}_{L}$, the lower limit of a 95%$95%$ confidence interval for β̂$\stackrel{^}{\beta }$, with bl(j,l)${\mathbf{bl}}\left(j,l\right)$ holding the lower limit associated with b(j,l)${\mathbf{b}}\left(j,l\right)$.
4:     bu(m,ntau) – double array
β̂U${\stackrel{^}{\beta }}_{U}$, the upper limit of a 95%$95%$ confidence interval for β̂$\stackrel{^}{\beta }$, with bu(j,l)${\mathbf{bu}}\left(j,l\right)$ holding the upper limit associated with b(j,l)${\mathbf{b}}\left(j,l\right)$.
5:     info(ntau${\mathbf{ntau}}$) – int64int32nag_int array
info(l)${\mathbf{info}}\left(l\right)$ holds additional information concerning the model fitting and confidence limit calculations when τ = tau(l)$\tau ={\mathbf{tau}}\left(l\right)$.
 Code Warning 0$0$ Model fitted and confidence limits calculated successfully. 1$1$ The function did not converge whilst calculating the parameter estimates. The returned values are based on the estimate at the last iteration. 2$2$ A singular matrix was encountered during the optimization. The model was not fitted for this value of τ$\tau$. 8$8$ The function did not converge whilst calculating the confidence limits. The returned limits are based on the estimate at the last iteration. 16$16$ Confidence limits for this value of τ$\tau$ could not be calculated. The returned upper and lower limits are set to a large positive and large negative value respectively.
It is possible for multiple warnings to be applicable to a single model. In these cases the value returned in info is the sum of the corresponding individual nonzero warning codes.
6:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 11${\mathbf{ifail}}=11$
Constraint: n2${\mathbf{n}}\ge 2$.
ifail = 21${\mathbf{ifail}}=21$
Constraint: 1m < n$1\le {\mathbf{m}}<{\mathbf{n}}$. Constraint: 1m < n$1\le {\mathbf{m}}<{\mathbf{n}}$.
ifail = 51${\mathbf{ifail}}=51$
Constraint: ntau1${\mathbf{ntau}}\ge 1$.
ifail = 61${\mathbf{ifail}}=61$
On entry is invalid.
ifail = 111${\mathbf{ifail}}=111$
A potential problem occurred whilst fitting the model(s).
Additional information has been returned in info.
ifail = 999${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

Not applicable.

Calling nag_correg_quantile_linreg_easy (g02qf) is equivalent to calling nag_correg_quantile_linreg (g02qg) with
• rcord = '2'${\mathbf{rcord}}=\text{'2'}$,
• intcpt = 'N'${\mathbf{intcpt}}=\text{'N'}$,
• weight = 'U'${\mathbf{weight}}=\text{'U'}$,
• ${\mathbf{lddat}}={\mathbf{n}}$,
• setting each element of isx to 1$1$,
• ip = m${\mathbf{ip}}={\mathbf{m}}$,
• Interval Method = IID${\mathbf{Interval Method}}=\mathrm{IID}$, and
• Significance Level = 0.95${\mathbf{Significance Level}}=0.95$.

## Example

```function nag_correg_quantile_linreg_easy_example
x = [1,    420.16; 1,     800.8; 1,    643.36; 1,    541.41; 1,    1245.7;
1,    2551.7; 1,    901.16; 1,      1201; 1,    1795.3; 1,    639.08;
1,     634.4; 1,    1165.8; 1,    750.88; 1,    956.23; 1,    815.62;
1,     945.8; 1,    1148.6; 1,    1264.2; 1,     829.4; 1,    1768.8;
1,    1095.4; 1,    979.16; 1,    2822.5; 1,    447.45; 1,    1309.9;
1,    922.35; 1,      1179; 1,    1492.4; 1,    2293.2; 1,     975.8;
1,    502.84; 1,    627.47; 1,    1017.9; 1,    616.72; 1,    889.98;
1,    423.88; 1,    790.92; 1,    1162.2; 1,    558.78; 1,    555.88;
1,    1197.1; 1,    943.25; 1,    713.44; 1,     530.8; 1,    1348.3;
1,    838.76; 1,    1142.2; 1,    2340.6; 1,    535.08; 1,      1088;
1,    587.18; 1,    596.44; 1,    484.66; 1,      1541; 1,    924.56;
1,      1536; 1,    1115.8; 1,    487.76; 1,     678.9; 1,    1044.7;
1,    692.64; 1,    671.88; 1,    1389.8; 1,    997.88; 1,    690.47;
1,    2497.8; 1,       507; 1,    860.69; 1,    1585.4; 1,    654.16;
1,    873.31; 1,      1862; 1,    933.92; 1,    894.46; 1,    2008.9;
1,    433.68; 1,    1148.6; 1,    697.31; 1,     587.6; 1,    926.88;
1,    571.25; 1,    896.47; 1,    839.04; 1,    598.35; 1,    454.48;
1,     829.5; 1,     461.1; 1,       585; 1,      1264; 1,    977.11;
1,     800.8; 1,      1938; 1,    883.98; 1,    502.44; 1,    698.83;
1,    718.36; 1,    713.52; 1,    920.42; 1,     543.9; 1,       906;
1,    1897.6; 1,    1587.3; 1,     880.6; 1,    891.68; 1,    4957.8;
1,    796.83; 1,    889.68; 1,    969.68; 1,    854.88; 1,    1221.5;
1,       420; 1,    1167.4; 1,     544.6; 1,       562; 1,     523.8;
1,    1031.4; 1,     689.6; 1,    670.78; 1,    1462.9; 1,    1398.5;
1,    377.06; 1,    830.44; 1,    820.82; 1,    851.54; 1,    975.04;
1,    875.17; 1,    1121.1; 1,      1338; 1,    1392.4; 1,    625.52;
1,    867.64; 1,    1256.3; 1,    805.54; 1,    725.75; 1,    1362.9;
1,    558.58; 1,    989.01; 1,    1999.3; 1,     884.4; 1,      1525;
1,    1209.5; 1,    1257.5; 1,     672.2; 1,      1125; 1,    2051.2;
1,     923.4; 1,    1827.4; 1,    1466.3; 1,    472.32; 1,    1014.2;
1,     730.1; 1,    590.76; 1,    880.39; 1,    2432.4; 1,     831.8;
1,    873.74; 1,    940.92; 1,    1139.5; 1,    951.44; 1,    1177.9;
1,    507.52; 1,       473; 1,    1222.6; 1,     576.2; 1,       601;
1,    1519.6; 1,     696.6; 1,       714; 1,    687.66; 1,    650.82;
1,     829.3; 1,    953.12; 1,    949.58; 1,     959.8; 1,    953.12;
1,    497.12; 1,      1213; 1,    953.12; 1,    570.17; 1,    958.87;
1,    939.04; 1,    724.73; 1,    1129.4; 1,    1283.4; 1,    408.34;
1,      1943; 1,    1511.6; 1,    638.67; 1,    539.64; 1,    1342.6;
1,    1225.8; 1,     463.6; 1,     511.8; 1,    715.37; 1,    562.64;
1,     689.8; 1,    800.47; 1,    736.76; 1,    1532.3; 1,     975.6;
1,    1415.4; 1,    1056.1; 1,    1613.8; 1,    2208.8; 1,    387.32;
1,     608.5; 1,       636; 1,    387.32; 1,    958.66; 1,     759.4;
1,       411; 1,    835.94; 1,    1078.8; 1,    499.75; 1,    1024.8;
1,    748.64; 1,    832.76; 1,    1006.4; 1,    987.64; 1,       615;
1,       726; 1,     788.1; 1,    887.47; 1,    494.42; 1,      1020;
1,    1595.2; 1,     776.6; 1,    1230.9; 1,      1808; 1,    415.44;
1,    440.52; 1,     541.2; 1,    581.36; 1,    743.08; 1,    1057.7];
y = [ 255.84; 572.08; 459.82; 310.96; 907.4; 863.92; 485.68; 811.58;
831.44; 403; 427.8; 534.76; 495.56; 650; 392.05; 633.8; 860.6;
934.98; 630.76; 1143.4; 813.31; 700.44; 2032.7; 263.71; 830.96;
590.62; 769.08; 815.36; 1570.4; 630.59; 338; 483.48; 645.99; 412.36;
600.48; 319.56; 520; 696.2; 348.45; 452.4; 774.8; 614.51; 512.72;
390.6; 662.01; 658.84; 612.56; 1504.4; 392.6; 708.76; 406.22;
443.56; 296.92; 692.17; 640.12; 1071.5; 588.14; 333.84; 496.6;
511.26; 466.96; 503.4; 700.56; 543.4; 357.64; 1301.1; 317.72;
430.34; 879.07; 424.32; 624.7; 912.89; 518.96; 582.54; 1509.8;
338; 580.22; 484.06; 419.64; 543.88; 399.67; 476.32; 588.64;
444.1; 386.36; 628; 248.81; 423.28; 712.1; 527.8; 503.36; 968.39;
500.63; 354.64; 482.58; 436.81; 497.32; 593.17; 374.8; 588.52;
1033.6; 726.39; 654.6; 693.68; 1827.2; 550.73; 693.68; 523.49;
528.38; 761.28; 335; 640.48; 361.4; 473.2; 401.32; 628.45; 581.2;
436; 771.45; 929.75; 276.56; 757.12; 591.2; 588.35; 821.6; 637.55;
664.2; 1022.3; 674.95; 444.86; 679.44; 776.76; 462.9; 538.75;
959.52; 377.78; 680; 1251; 553.15; 977; 737.82; 810.9; 561.2;
810.68; 1068; 728.4; 983; 1049.9; 372.32; 708.9; 522.7; 361.52;
633.12; 1424.8; 620.8; 631.8; 517.92; 820; 608.64; 830.96; 360.88;
301; 925.58; 395.76; 378; 1162; 442; 397; 383.46; 404.04; 588.52;
621.12; 670.8; 681.76; 621.12; 297.57; 807.36; 621.12; 353.49;
696.8; 548.6; 383.94; 811.2; 745.24; 284.8; 1305.7; 837.8; 431.1;
442; 795.34; 801.35; 353.6; 418.6; 448.45; 468; 508.8; 577.91;
526.76; 883.28; 570.52; 890.24; 742.53; 865.32; 1318.8; 242.32;
444.56; 331; 242.32; 680.42; 416.4; 266; 576.28; 596.84; 408.5;
708.48; 429.04; 614.76; 734.24; 619.64; 385.32; 433; 400.8; 515.62;
327.42; 775.02; 1138.2; 485.52; 772.76; 993.96; 305.44; 306.52;
299.2; 468; 522.6; 750.32];
tau = [0.10; 0.25; 0.50; 0.75; 0.90];
% Call the model fitting routine
[df, b, bl, bu, info, ifail] = nag_correg_quantile_linreg_easy(x, y, tau);
if (ifail == 0)
% Display the parameter estimates
for l=1:numel(tau)
fprintf('\nQuantile: %6.3f\n\n', tau(l));
fprintf('        Lower   Parameter   Upper\n');
fprintf('        Limit   Estimate    Limit\n');
for j=1:2
fprintf('%3d   %7.3f   %7.3f   %7.3f\n', j, bl(j,l), b(j,l), bu(j,l));
end
fprintf('\n');
end
elseif (ifail == 111)
disp(info);
end
```
```

Quantile:  0.100

Lower   Parameter   Upper
Limit   Estimate    Limit
1    74.951   110.132   145.313
2     0.370     0.402     0.433

Quantile:  0.250

Lower   Parameter   Upper
Limit   Estimate    Limit
1    64.187    95.468   126.748
2     0.446     0.474     0.502

Quantile:  0.500

Lower   Parameter   Upper
Limit   Estimate    Limit
1    55.412    81.486   107.560
2     0.537     0.560     0.584

Quantile:  0.750

Lower   Parameter   Upper
Limit   Estimate    Limit
1    41.418    62.425    83.432
2     0.625     0.644     0.663

Quantile:  0.900

Lower   Parameter   Upper
Limit   Estimate    Limit
1    26.818    67.346   107.875
2     0.650     0.686     0.723

```
```function g02qf_example
x = [1,    420.16; 1,     800.8; 1,    643.36; 1,    541.41; 1,    1245.7;
1,    2551.7; 1,    901.16; 1,      1201; 1,    1795.3; 1,    639.08;
1,     634.4; 1,    1165.8; 1,    750.88; 1,    956.23; 1,    815.62;
1,     945.8; 1,    1148.6; 1,    1264.2; 1,     829.4; 1,    1768.8;
1,    1095.4; 1,    979.16; 1,    2822.5; 1,    447.45; 1,    1309.9;
1,    922.35; 1,      1179; 1,    1492.4; 1,    2293.2; 1,     975.8;
1,    502.84; 1,    627.47; 1,    1017.9; 1,    616.72; 1,    889.98;
1,    423.88; 1,    790.92; 1,    1162.2; 1,    558.78; 1,    555.88;
1,    1197.1; 1,    943.25; 1,    713.44; 1,     530.8; 1,    1348.3;
1,    838.76; 1,    1142.2; 1,    2340.6; 1,    535.08; 1,      1088;
1,    587.18; 1,    596.44; 1,    484.66; 1,      1541; 1,    924.56;
1,      1536; 1,    1115.8; 1,    487.76; 1,     678.9; 1,    1044.7;
1,    692.64; 1,    671.88; 1,    1389.8; 1,    997.88; 1,    690.47;
1,    2497.8; 1,       507; 1,    860.69; 1,    1585.4; 1,    654.16;
1,    873.31; 1,      1862; 1,    933.92; 1,    894.46; 1,    2008.9;
1,    433.68; 1,    1148.6; 1,    697.31; 1,     587.6; 1,    926.88;
1,    571.25; 1,    896.47; 1,    839.04; 1,    598.35; 1,    454.48;
1,     829.5; 1,     461.1; 1,       585; 1,      1264; 1,    977.11;
1,     800.8; 1,      1938; 1,    883.98; 1,    502.44; 1,    698.83;
1,    718.36; 1,    713.52; 1,    920.42; 1,     543.9; 1,       906;
1,    1897.6; 1,    1587.3; 1,     880.6; 1,    891.68; 1,    4957.8;
1,    796.83; 1,    889.68; 1,    969.68; 1,    854.88; 1,    1221.5;
1,       420; 1,    1167.4; 1,     544.6; 1,       562; 1,     523.8;
1,    1031.4; 1,     689.6; 1,    670.78; 1,    1462.9; 1,    1398.5;
1,    377.06; 1,    830.44; 1,    820.82; 1,    851.54; 1,    975.04;
1,    875.17; 1,    1121.1; 1,      1338; 1,    1392.4; 1,    625.52;
1,    867.64; 1,    1256.3; 1,    805.54; 1,    725.75; 1,    1362.9;
1,    558.58; 1,    989.01; 1,    1999.3; 1,     884.4; 1,      1525;
1,    1209.5; 1,    1257.5; 1,     672.2; 1,      1125; 1,    2051.2;
1,     923.4; 1,    1827.4; 1,    1466.3; 1,    472.32; 1,    1014.2;
1,     730.1; 1,    590.76; 1,    880.39; 1,    2432.4; 1,     831.8;
1,    873.74; 1,    940.92; 1,    1139.5; 1,    951.44; 1,    1177.9;
1,    507.52; 1,       473; 1,    1222.6; 1,     576.2; 1,       601;
1,    1519.6; 1,     696.6; 1,       714; 1,    687.66; 1,    650.82;
1,     829.3; 1,    953.12; 1,    949.58; 1,     959.8; 1,    953.12;
1,    497.12; 1,      1213; 1,    953.12; 1,    570.17; 1,    958.87;
1,    939.04; 1,    724.73; 1,    1129.4; 1,    1283.4; 1,    408.34;
1,      1943; 1,    1511.6; 1,    638.67; 1,    539.64; 1,    1342.6;
1,    1225.8; 1,     463.6; 1,     511.8; 1,    715.37; 1,    562.64;
1,     689.8; 1,    800.47; 1,    736.76; 1,    1532.3; 1,     975.6;
1,    1415.4; 1,    1056.1; 1,    1613.8; 1,    2208.8; 1,    387.32;
1,     608.5; 1,       636; 1,    387.32; 1,    958.66; 1,     759.4;
1,       411; 1,    835.94; 1,    1078.8; 1,    499.75; 1,    1024.8;
1,    748.64; 1,    832.76; 1,    1006.4; 1,    987.64; 1,       615;
1,       726; 1,     788.1; 1,    887.47; 1,    494.42; 1,      1020;
1,    1595.2; 1,     776.6; 1,    1230.9; 1,      1808; 1,    415.44;
1,    440.52; 1,     541.2; 1,    581.36; 1,    743.08; 1,    1057.7];
y = [ 255.84; 572.08; 459.82; 310.96; 907.4; 863.92; 485.68; 811.58;
831.44; 403; 427.8; 534.76; 495.56; 650; 392.05; 633.8; 860.6;
934.98; 630.76; 1143.4; 813.31; 700.44; 2032.7; 263.71; 830.96;
590.62; 769.08; 815.36; 1570.4; 630.59; 338; 483.48; 645.99; 412.36;
600.48; 319.56; 520; 696.2; 348.45; 452.4; 774.8; 614.51; 512.72;
390.6; 662.01; 658.84; 612.56; 1504.4; 392.6; 708.76; 406.22;
443.56; 296.92; 692.17; 640.12; 1071.5; 588.14; 333.84; 496.6;
511.26; 466.96; 503.4; 700.56; 543.4; 357.64; 1301.1; 317.72;
430.34; 879.07; 424.32; 624.7; 912.89; 518.96; 582.54; 1509.8;
338; 580.22; 484.06; 419.64; 543.88; 399.67; 476.32; 588.64;
444.1; 386.36; 628; 248.81; 423.28; 712.1; 527.8; 503.36; 968.39;
500.63; 354.64; 482.58; 436.81; 497.32; 593.17; 374.8; 588.52;
1033.6; 726.39; 654.6; 693.68; 1827.2; 550.73; 693.68; 523.49;
528.38; 761.28; 335; 640.48; 361.4; 473.2; 401.32; 628.45; 581.2;
436; 771.45; 929.75; 276.56; 757.12; 591.2; 588.35; 821.6; 637.55;
664.2; 1022.3; 674.95; 444.86; 679.44; 776.76; 462.9; 538.75;
959.52; 377.78; 680; 1251; 553.15; 977; 737.82; 810.9; 561.2;
810.68; 1068; 728.4; 983; 1049.9; 372.32; 708.9; 522.7; 361.52;
633.12; 1424.8; 620.8; 631.8; 517.92; 820; 608.64; 830.96; 360.88;
301; 925.58; 395.76; 378; 1162; 442; 397; 383.46; 404.04; 588.52;
621.12; 670.8; 681.76; 621.12; 297.57; 807.36; 621.12; 353.49;
696.8; 548.6; 383.94; 811.2; 745.24; 284.8; 1305.7; 837.8; 431.1;
442; 795.34; 801.35; 353.6; 418.6; 448.45; 468; 508.8; 577.91;
526.76; 883.28; 570.52; 890.24; 742.53; 865.32; 1318.8; 242.32;
444.56; 331; 242.32; 680.42; 416.4; 266; 576.28; 596.84; 408.5;
708.48; 429.04; 614.76; 734.24; 619.64; 385.32; 433; 400.8; 515.62;
327.42; 775.02; 1138.2; 485.52; 772.76; 993.96; 305.44; 306.52;
299.2; 468; 522.6; 750.32];
tau = [0.10; 0.25; 0.50; 0.75; 0.90];
% Call the model fitting routine
[df, b, bl, bu, info, ifail] = g02qf(x, y, tau);
if (ifail == 0)
% Display the parameter estimates
for l=1:numel(tau)
fprintf('\nQuantile: %6.3f\n\n', tau(l));
fprintf('        Lower   Parameter   Upper\n');
fprintf('        Limit   Estimate    Limit\n');
for j=1:2
fprintf('%3d   %7.3f   %7.3f   %7.3f\n', j, bl(j,l), b(j,l), bu(j,l));
end
fprintf('\n');
end
elseif (ifail == 111)
disp(info);
end
```
```

Quantile:  0.100

Lower   Parameter   Upper
Limit   Estimate    Limit
1    74.951   110.132   145.313
2     0.370     0.402     0.433

Quantile:  0.250

Lower   Parameter   Upper
Limit   Estimate    Limit
1    64.187    95.468   126.748
2     0.446     0.474     0.502

Quantile:  0.500

Lower   Parameter   Upper
Limit   Estimate    Limit
1    55.412    81.486   107.560
2     0.537     0.560     0.584

Quantile:  0.750

Lower   Parameter   Upper
Limit   Estimate    Limit
1    41.418    62.425    83.432
2     0.625     0.644     0.663

Quantile:  0.900

Lower   Parameter   Upper
Limit   Estimate    Limit
1    26.818    67.346   107.875
2     0.650     0.686     0.723

```