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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 nn observed values, y = {yi : i = 1,2,,n} y = { y i : i = 1, 2, , n } , an n × pn×p design matrix XX, a column vector, xx, of length pp holding the iith row of XX and a quantile τ (0,1) τ ( 0 , 1 ) , nag_correg_quantile_linreg_easy (g02qf) estimates the pp-element vector ββ as the solution to
n
minimize  ρτ (yixiTβ)
β p i = 1
minimize β p i=1 n ρ τ ( y i - xiT β )
(1)
where ρτ ρ τ  is the piecewise linear loss function ρτ (z) = z (τI(z < 0)) ρ τ ( z ) = z ( τ - I ( z < 0 ) ) , and I (z < 0) I ( z < 0 )  is an indicator function taking the value 11 if z < 0z<0 and 00 otherwise.
nag_correg_quantile_linreg_easy (g02qf) assumes Normal, independent, identically distributed (IID) errors and calculates the asymptotic covariance matrix from
Σ = ( τ (1τ) )/n (s(τ))2 (XTX)1
Σ = τ ( 1 - τ ) n ( s( τ ) ) 2 ( XT X )-1
where ss is the sparsity function, which is estimated from the residuals, ri = yi xiT β̂ ri = yi - xiT β^  (see Koenker (2005)).
Given an estimate of the covariance matrix, Σ̂Σ^, lower, β̂Lβ^L, and upper, β̂Uβ^U, limits for a 95%95% confidence interval are calculated for each of the pp parameters, via
β̂Li = β̂i t np , 0.975 sqrt( Σ̂ii ) , β̂Ui = β̂i + t np , 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,0.975tn-p,0.975 is the 97.597.5 percentile of the Student's tt distribution with nkn-k degrees of freedom, where kk is the rank of the cross-product matrix XTXXTX.
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 n2n2.
XX, the design matrix, with the iith value for the jjth variate supplied in x(i,j)xij, for i = 1,2,,ni=1,2,,n and j = 1,2,,mj=1,2,,m.
2:     y(n) – double array
n, the dimension of the array, must satisfy the constraint n2n2.
yy, observations on the dependent variable.
3:     tau(ntau) – double array
ntau, the dimension of the array, must satisfy the constraint ntau1ntau1.
The vector of quantiles of interest. A separate model is fitted to each quantile.
Constraint: sqrt(ε) < tau(l) < 1sqrt(ε)ε<taul<1-ε where εε is the machine precision returned by nag_machine_precision (x02aj), for l = 1,2,,ntaul=1,2,,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.)
nn, the number of observations in the dataset.
Constraint: n2n2.
2:     m – int64int32nag_int scalar
Default: The second dimension of the array x.
pp, the number of variates in the model.
Constraint: 1m < n1m<n.
3:     ntau – int64int32nag_int scalar
Default: The dimension of the array tau.
The number of quantiles of interest.
Constraint: ntau1ntau1.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     df – double scalar
The degrees of freedom given by nkn-k, where nn is the number of observations and kk is the rank of the cross-product matrix XTXXTX.
2:     b(m,ntau) – double array
β̂β^, the estimates of the parameters of the regression model, with b(j,l)bjl containing the coefficient for the variable in column jj of x, estimated for τ = tau(l)τ=taul.
3:     bl(m,ntau) – double array
β̂Lβ^L, the lower limit of a 95%95% confidence interval for β̂β^, with bl(j,l)bljl holding the lower limit associated with b(j,l)bjl.
4:     bu(m,ntau) – double array
β̂Uβ^U, the upper limit of a 95%95% confidence interval for β̂β^, with bu(j,l)bujl holding the upper limit associated with b(j,l)bjl.
5:     info(ntauntau) – int64int32nag_int array
info(l)infol holds additional information concerning the model fitting and confidence limit calculations when τ = tau(l)τ=taul.
Code Warning
00 Model fitted and confidence limits calculated successfully.
11 The function did not converge whilst calculating the parameter estimates. The returned values are based on the estimate at the last iteration.
22 A singular matrix was encountered during the optimization. The model was not fitted for this value of ττ.
88 The function did not converge whilst calculating the confidence limits. The returned limits are based on the estimate at the last iteration.
1616 Confidence limits for this value of ττ 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
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 11ifail=11
Constraint: n2n2.
  ifail = 21ifail=21
Constraint: 1m < n1m<n. Constraint: 1m < n1m<n.
  ifail = 51ifail=51
Constraint: ntau1ntau1.
  ifail = 61ifail=61
On entry is invalid.
  ifail = 111ifail=111
A potential problem occurred whilst fitting the model(s).
Additional information has been returned in info.
  ifail = 999ifail=-999
Dynamic memory allocation failed.

Accuracy

Not applicable.

Further Comments

Calling nag_correg_quantile_linreg_easy (g02qf) is equivalent to calling nag_correg_quantile_linreg (g02qg) with

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)
  fprintf('\nAdditional error information (info):\n');
  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)
  fprintf('\nAdditional error information (info):\n');
  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



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