Assessment of model risk due to the use of an inappropriate parameter estimator

Figures & data

Figure 1. Plot of a Cost Function$\mathfrak{C}(\mathit{\gamma })$for a simple binary LRM

Figure 2. The plot of the cost function from five parameter estimators against number of iterations

Notes: The plots of the cost function values $\mathfrak{C}(\gamma )$ against the number of iterations $\mathfrak{I}$ per 100 iterations computed on the simulated dataset. The five parameter estimation methods are BGD, SGD, MBGD, IRLS and EM.

Figure 3. The plot of the cost function from six parameter estimators against number of iterations

Notes: The plots of the cost function values $\mathfrak{C}(\gamma )$ against the number of iterations $\mathfrak{I}$ per 100 iterations computed on the simulated dataset. The six parameter estimation methods are NM, PW, CG, TN, BFGS and LM-BFGS.

Figure 4. The plot of the cost function from five parameter estimators against number of iterations for real dataset

Notes: The plots of the cost function values $\mathfrak{C}(\gamma )$ against the number of iterations $\mathfrak{I}$ per 100 iterations computed on the real given dataset. The five parameter estimation methods are BGD, SGD, MBGD, IRLS and EM.

Figure 5. The plot of the cost function from six parameter estimators against a number of iterations for real dataset

Notes: The plots of the cost function values $\mathfrak{C}(\gamma )$ against the number of iterations $\mathfrak{I}$ per 100 iterations computed on the real given dataset. The six parameter estimation methods are NM, PW, CG, TN, BFGS and LM-BFGS.

Table 1. Parameter estimation method results for PD using Binary LRM

Parameter estimator	$\mathfrak{I}$	$\mathit{b}$	$\mathit{\kappa }(\epsilon )$	${\hat{\mathit{\gamma }}}_0$	${\hat{\mathit{\gamma }}}_1$	$\mathfrak{A}$	$\mathfrak{C}(\gamma )$
BGD	100	1	2(0.01)	0.0051	0.4914	0.8294	0.3881
SGD	100	1	10(0.01)	0.0075	0.4898	0.8294	0.3881
MBGD	100	1	10(0.01)	0.0099	0.4996	0.8292	0.2936
IRLS	100	1	7(1e-08)	0.0134	0.4930	0.8291	0.3881
EM	100	1	39 (1e-08)	0.0134	0.4930	0.8291	0.3881
NM	100	1	88 (1e-08)	0.0239	0.4977	0.8236	0.3853
PW	100	1	2 (1e-08)	0.0239	0.4977	0.8236	0.3853
CG	100	1	9 (1e-08)	0.0239	0.4977	0.8236	0.3853
TN	100	1	8 (1e-08)	0.0239	0.4977	0.8236	0.3853
BFGS	100	1	10 (1e-08)	0.0239	0.4977	0.8236	0.3853
LM-BFGS	100	1	9 (1e-08)	0.0239	0.4977	0.8236	0.3853

Notes: The results $\mathfrak{I}$ is the maximum number of iteration the method was set to, $b$ is the random mini-batch size, $\kappa$ is the number of iteration the method converged to and $\epsilon$ the tolerance level in brackets. For the gradient descent methods, the tolerance level is 0.01 and in all other methods it is smaller, i.e., 1e-08 = 0.00000008. ${\hat{\gamma }}_0$ and ${\hat{\gamma }}_1$ are, respectively, the intercept and slope estimated parameters. $\mathfrak{A}$ is the accuracy classification score given by Equation (31)(31) $\mathfrak{A}=\frac{{\sum }_{i=1}^n[I({\hat{\pi }}_i\ge 0.5)\equiv (Y_i=1)]+{\sum }_{i=1}^n[I({\hat{\pi }}_i<0.5)\equiv (Y_i=0)]}{n},$(31) . $\mathfrak{C}(\gamma )$ is the cost function for LRM defined in Equation (5)(5) $\mathfrak{C}(\mathit{\gamma })=-\frac{1}{n}\sum _{i=1}^n\left[Y_{i}ln({\pi }_i)+(1-Y_i)ln(1-{\pi }_i)\right].$(5) . NB for MBGD 40 sub-batches were generated from the sample size.

Table 2. Parameter estimation method results for PD using Binary LRM on real dataset

Parameter estimator	$\mathfrak{I}$	$\mathit{b}$	$\mathit{\kappa }(\epsilon )$	${\hat{\mathit{\gamma }}}_0$	${\hat{\mathit{\gamma }}}_1$	$\mathfrak{A}$	$\mathfrak{C}(\gamma )$
BGD	100	1	10(0.01)	0.1426	0.0883	0.6244	1.3172
SGD	100	1	10(0.01)	2.1909	0.1115	0.6244	2.3416
MBGD	100	1	10(0.01)	0.2211	0.1369	0.6244	0.2564
IRLS	100	1	4 (1e-08)	1.1281	−0.0179	0.6339	0.6559
EM	100	1	8 (1e-08)	1.1281	−0.0179	0.6339	0.6559
NM	100	1	85 (1e-08)	1.1281	−0.0179	0.6339	0.6559
PW	100	1	60 (1e-08)	1.1281	−0.0179	0.6339	0.6559
CG	100	1	14 (1e-08)	1.1281	−0.0179	0.6339	0.6559
TN	100	1	15 (1e-08)	1.1281	−0.0179	0.6339	0.6559
BFGS	100	1	8 (1e-08)	1.1281	−0.0179	0.6339	0.6559
LM-BFGS	100	1	17 (1e-08)	1.1281	−0.0178	0.6339	0.6559

Notes: For MBGD 12 sub-batches were generated from the sample size.