We then used the bootstrapping algorithm and obtained the following means and standard errors for \( M \)= 1000 which, as reported below, com-pare quite well with the asymptotic theory estimates. The corresponding distributions are shown in Figures 14 and 15.

<table><tr><td></td><td>\( k _ { I } ^ { + } ( b o o t ) ( G L S ) \)</td><td>\( k _ { I } ^ { - } ( b o o t ) ( G L S ) \)</td><td>\( k _ { I } ^ { + } ( a s y m p ) ( G L S ) \)</td><td>\( \overline { { k _ { I } ^ { - } ( a s y m p ) ( G L S ) } } \)</td></tr><tr><td>mean</td><td>2.158</td><td>10.911</td><td>2.157</td><td>10.911</td></tr><tr><td>SE</td><td>0.0044</td><td>0.0247</td><td>0.00396</td><td>0.0225</td></tr></table>

Figure 14: Two parameters estimation \( ( k _ { I } ^ { + } , \, k _ { I } ^ { - } ) \). Bootstrapping distribution for \( k _ { I } ^ { + } \). We use GLS and M=1000 runs.

# 5.3 GLS Estimation of 3 Parameters

We tried next to estimate 3 parameters. We again used the GLS formulation with \( \gamma \) = 0.6. Once again we fixed all the parameters describing the domain and the polymerization function \( k _ { o n } \) and we also fixed either \( k _ { o f f } ^ { N } \) or \( k _ { o n } ^ { N } \) in the corresponding inverse problems.

# 5.4 GLS Estimation for \( k _ { I } ^ { + } \),\( k _ { I } ^ { - } \) and \( k _ { o n } ^ { N } \)

We fixed values as follows:

<table><tr><td>\( \overline { { k _ { o f f } ^ { N } } } \)</td><td>\( k _ { o n } ^ { m i n } \)</td><td>\( k _ { o n } ^ { m a x } \)</td><td>\( x _ { 1 } \)</td><td>\( x _ { 2 } \)</td><td>\( i _ { m a x } \)</td></tr><tr><td>93.33</td><td>1684.38</td><td>\( 1 . 5 \cdot 1 0 ^ { 9 } \)</td><td>0.062</td><td>0.859</td><td>\( 3 . 5 \cdot 1 0 ^ { 5 } \)</td></tr></table>

Figure 15: Two parameters estimation \( ( k _ { I } ^ { + } , \, k _ { I } ^ { - } ) \). Bootstrapping distribution for \( k _ { I } ^ { - } \). We use GLS and M=1000 runs.

We used as initial parameter values:

<table><tr><td></td><td>\( \underline { { k _ { I } ^ { + } } } \)</td><td>\( \underline { { k _ { I } ^ { - } } } \)</td><td>\( k _ { o n } ^ { N } \)</td></tr><tr><td>\( q _ { 0 } \)</td><td>2.1600</td><td>10.9270</td><td>4616.962</td></tr></table>

We obtained the estimated parameters together with the corresponding standard errors, variances and the condition numbers \( \kappa \) of the corresponding sensitivity matrices for the four data sets as reported below. The 95% con-fidence results based on the asymptotic theory are also depicted for DS 4 in Figure 16.

<table><tr><td></td><td>\( \overline { { k _ { I } ^ { + } } } \)</td><td>\( k _ { I } ^ { - } \)</td><td>\( \mathit { k } _ { o n } ^ { N } \)</td><td>SE</td><td>\( \sigma ^ { 2 } \)</td><td>\( \kappa \)</td></tr><tr><td>\( D S \) 1</td><td>2.26</td><td>13.49</td><td>4616.96</td><td>\( ( . 0 1 2 , . 0 9 9 , 5 3 . 9 2 5 ) \)</td><td>\( 8 . 5 2 \cdot 1 0 ^ { - 6 } \)</td><td>\( 8 . 8 9 \cdot 1 0 ^ { 1 0 } \)</td></tr><tr><td>\( D S \) 2</td><td>2.99</td><td>16.20</td><td>4616.96</td><td>\( ( . 0 2 1 , . 1 5 1 , 5 6 . 6 9 1 ) \)</td><td>\( 9 . 6 7 \cdot 1 0 ^ { - 6 } \)</td><td>\( 4 . 3 7 \cdot 1 0 ^ { 1 0 } \)</td></tr><tr><td>\( D S \) 3</td><td>2.18</td><td>15.76</td><td>9840.31</td><td>\( ( . 0 1 1 , . 1 0 3 , 9 0 . 4 6 6 ) \)</td><td>\( 6 . 4 5 \cdot 1 0 ^ { - 6 } \)</td><td>\( 3 . 9 4 \cdot 1 0 ^ { 1 1 } \)</td></tr><tr><td>\( D S \) 4</td><td>2.16</td><td>10.91</td><td>4616.96</td><td>\( ( 0 . 0 0 8 9 , 0 . 0 6 4 9 , 4 5 . 2 6 2 ) \)</td><td>\( 6 . 3 6 \cdot 1 0 ^ { - 6 } \)</td><td>\( 7 . 1 4 \cdot 1 0 ^ { 1 0 } \)</td></tr></table>

To compare these asymptotic results with bootstrapping, we carried out bootstrapping with Data Set (DS) 4 for the estimation of \( k _ { I } ^ { + } \), \( k _ { I } ^ { - } \) and \( k _ { o n } ^ { N } \) with the same initial values as above. We then obtained the following means and standard errors for a run with \( M \)= 1000, in comparison to the asymptotic theory.