Comparison among simultaneous confidence regions for nonlinear diffusion models

Abstract

Accuracy measures for parameter estimates represent a tricky issue in nonlinear models. Practitioners often use separate marginal confidence intervals for each parameter in place of a simultaneous confidence region (sCR). However, this can be extremely misleading due to the curvature of the parameter space of the nonlinear model. For low parameter dimensions, routines for evaluating approximate sCRs are available in the most common software programs; however, the degree of accuracy depends on the curvature of the parameter space and the sample size. Exact sCRs are computationally intensive, and for this reason, in the past, they did not receive much attention. In this paper, we perform a comparison among exact, asymptotic exact, approximate sCRs, and marginal confidence intervals. More modern regions based on bootstrap are also examined as an alternative approach (both parametric and nonparametric). Their degree of accuracy is compared with both real data and simulation results. Among the nonlinear models, in this paper, the focus is on two of the most widespread diffusion models of products and technologies, that is, the Bass and Generalized Bass models. Three different empirical studies are analyzed here. Simulation studies are also performed for lifecycles with the same diffusion characteristics as those of the empirical studies. Our results show that, as the parameter dimension increases, overlapping among the alternative sCRs reduces. The approximate sCR shows inadequate values of overlapping with the exact sCR, even for moderate parameter dimension. Bootstrap regions also exhibit good performance in describing the shape of the exact region when curvature is present, but they fail to spread up to its boundary. The coverage probability of each region is assessed with simulations. We observe that the coverage probability of the approximate sCR decreases rapidly, even for moderate parameter dimension, and it is smaller than the nominal level for bootstrap regions.

Appendix

1.1 BM and GBM: model and derivatives

In Sect. 2, the diffusion models examined in this paper are summarized. As above explained, GBM$_{exp}$ and GBM$_{rect}$ denote model (3) when w(t) equals expression (4) and (5), respectively. The final expression of the GBM$_{exp}$ is

$$\begin{aligned} z(t) = \left\{ \begin{array}{ll} \displaystyle m \, \frac{1- e^{-(p+q)t}}{1+ \frac{q}{p} e^{-(p+q)t}} &{}\quad 0\le t < a_1 \\ \displaystyle m \, \frac{1- e^{-(p+q)\left\{ t + \frac{c_1}{b_1}\left[ e^{b_1 \left( t-a_1\right) }-1\right] \right\} }}{1+ \frac{q}{p} e^{-(p+q)\left\{ t + \frac{c_1}{b_1}\left[ e^{b_1 \left( t-a_1\right) -1}\right] \right\} }} &{}\quad t \ge a_1, \\ \end{array} \right. \end{aligned}$$

(A.1)

and the complete formulation for the GBM$_{rect}$ is

$$\begin{aligned} z(t) = \left\{ \begin{array}{ll} \displaystyle m \, \frac{1- e^{-(p+q)t}}{1+ \frac{q}{p} e^{-(p+q)t}} &{}\quad 0\le t < a_1 \\ \displaystyle m \, \frac{1- e^{-(p+q)\left[ t + c_1 (t-a_1)\right] }}{1+ \frac{q}{p} e^{-(p+q)\left[ t + c_1 (t-a_1)\right] }} &{}\quad a_1 \le t \le b_1 \\ \displaystyle m \, \frac{1- e^{-(p+q)\left[ t + c_1 (b_1-a_1)\right] }}{1+ \frac{q}{p} e^{-(p+q)\left[ t + c_1 (b_1-a_1)\right] }} &{}\quad t > b_1. \\ \end{array} \right. \end{aligned}$$

(A.2)

For simplicity, the expression of both the GBM$_{exp}$ and GBM$_{rect}$ in Eqs. (A.1) and (A.2) can be denoted as

$$\begin{aligned} z(t) = m \, \frac{1- e^{-(p+q)A}}{1+ \frac{q}{p} e^{-(p+q)A}}, \end{aligned}$$

(A.3)

where for the GBM$_{exp}$

$$\begin{aligned} A = \left\{ \begin{array}{ll} \displaystyle t &{}\quad 0\le t < a_1 \\ \displaystyle t + \frac{c_1}{b_1}\left[ e^{b_1 (t-a_1)}-1\right] &{}\quad t \ge a_1, \\ \end{array} \right. \end{aligned}$$

(A.4)

and for the GBM$_{rect}$

$$\begin{aligned} A = \left\{ \begin{array}{ll} \displaystyle t &{}\quad 0\le t < a_1 \\ \displaystyle t + c_1 (t-a_1) &{}\quad a_1 \le t \le b_1 \\ \displaystyle t + c_1 (b_1-a_1) &{}\quad t > b_1. \\ \end{array} \right. \end{aligned}$$

(A.5)

In order to evaluate some of the sCRs compared in this study, partial derivatives of the model with respect to the parameters are required. The partial derivatives of z(t) of Eq. (A.3), with respect to the diffusion parameters (m, p, q),  are equal for the GBM$_{rect}$ and GBM$_{rect}$, since Eqs. (A.4) and (A.5) do not depend upon (m, p, q). The common partial derivatives are

$$\begin{aligned} \displaystyle \frac{\partial z(t)}{\partial m} = \frac{1- e^{-(p+q)A}}{1+ \frac{q}{p} e^{-(p+q)A}}, \end{aligned}$$

(A.6)

$$\begin{aligned} \displaystyle&\frac{\partial z(t)}{\partial p} \nonumber \\&\quad = m \frac{\left[ A e^{-(p+q)A}\right] \left[ 1\!+\! \frac{q}{p} e^{-(p+q)A} \right] -\left[ 1\!-\! e^{-(p+q)A} \right] \left[ -\frac{q}{p^2} e^{-(p+q)A}- \frac{q}{p} A e^{-(p+q)A}\right] }{\left[ 1+ \frac{q}{p} e^{-(p+q)A} \right] ^2} \nonumber \\&\quad = m \frac{\left[ A e^{-(p+q)A}\right] \left[ 1\!+\! \frac{q}{p} e^{-(p+q)A} \right] +\left[ 1\!-\! e^{-(p+q)A} \right] \left[ \frac{q}{p^2} e^{-(p+q)A}+ \frac{q}{p} A e^{-(p+q)A}\right] }{\left[ 1+ \frac{q}{p} e^{-(p+q)A} \right] ^2}, \nonumber \\ \end{aligned}$$

(A.7)

and

$$\begin{aligned} \displaystyle&\frac{\partial z(t)}{\partial q}\nonumber \\&\quad = m \frac{\left[ A e^{-(p+q)A}\right] \left[ 1\!+\! \frac{q}{p} e^{-(p+q)A} \right] -\left[ 1\!-\! e^{-(p+q)A} \right] \left[ \frac{1}{p} e^{-(p+q)A}- \frac{q}{p} A e^{-(p+q)A}\right] }{\left[ 1+ \frac{q}{p} e^{-(p+q)A} \right] ^2} \nonumber \\&\quad = m \frac{\left[ A e^{-(p+q)A}\right] \left[ 1\!+\! \frac{q}{p} e^{-(p+q)A} \right] +\left[ 1\!-\! e^{-(p+q)A} \right] \left[ -\frac{1}{p} e^{-(p+q)A}+ \frac{q}{p} A e^{-(p+q)A}\right] }{\left[ 1+ \frac{q}{p} e^{-(p+q)A} \right] ^2}. \nonumber \\ \end{aligned}$$

(A.8)

Let us now consider the partial derivative of z(t) with respect to $a_1, b_1,$ and $c_1$, for the GBM$_{exp}$ (Eqs. (A.3) and (A.4)). For $0\le t < a_1,$ the partial derivatives are zero, while for $t>a_1$, we obtain

$$\begin{aligned} \displaystyle&\frac{\partial z(t)}{\partial a_1} = m \frac{1}{\partial a_1} \partial \frac{1- e^{-(p+q)A}}{1+ \frac{q}{p} e^{-(p+q)A}} \nonumber \\&\quad = m \frac{\left[ (p+q) e^{-(p+q)A} \frac{\partial A}{\partial a_1} \right] \left[ 1\!+\! \frac{q}{p} e^{-(p+q)A} \right] +\left[ 1\!-\! e^{-(p+q)A} \right] \left[ \frac{q}{p} (p+q) e^{-(p+q)A} \frac{\partial A}{\partial a_1} \right] }{\left[ 1+ \frac{q}{p} e^{-(p+q)A}\right] ^2}, \nonumber \\ \end{aligned}$$

(A.9)

where

$$\begin{aligned} \frac{\partial A}{\partial a_1}= & {} \displaystyle \frac{\partial \left\{ t + \frac{c_1}{b_1}\left[ e^{b_1 (t-a_1)}-1\right] \right\} }{\partial a_1} = \displaystyle \frac{\partial \left[ \frac{c_1}{b_1} e^{b_1 (t-a_1)} \right] }{\partial a_1} \nonumber \\= & {} \displaystyle \frac{c_1}{b_1} e^{b_1 (t-a_1)} (-b_1) = \displaystyle - {c_1} e^{b_1 (t-a_1)}; \end{aligned}$$

(A.10)

$$\begin{aligned}&\displaystyle \frac{\partial z(t)}{\partial b_1} = m \frac{1}{\partial b_1} \partial \frac{1- e^{-(p+q)A}}{1+ \frac{q}{p} e^{-(p+q)A}} \nonumber \\&\quad = m \frac{\left[ (p+q) e^{-(p+q)A} \frac{\partial A}{\partial b_1} \right] \left[ 1\!+\! \frac{q}{p} e^{-(p+q)A} \right] +\left[ 1\!-\! e^{-(p+q)A} \right] \left[ \frac{q}{p} (p+q) e^{-(p+q)A} \frac{\partial A}{\partial b_1} \right] }{\left[ 1+ \frac{q}{p} e^{-(p+q)A}\right] ^2}, \nonumber \\ \end{aligned}$$

(A.11)

where

$$\begin{aligned} \frac{\partial A}{\partial b_1}= & {} \displaystyle \frac{\partial \left\{ t + \frac{c_1}{b_1}\left[ e^{b_1 (t-a_1)}-1\right] \right\} }{\partial b_1} = \displaystyle \frac{\partial \left[ \frac{c_1}{b_1} e^{b_1 (t-a_1)} - \frac{c_1}{b_1}\right] }{\partial b_1} \nonumber \\= & {} \displaystyle - \frac{c_1}{b_1^2} e^{b_1 (t-a_1)} + \frac{c_1}{b_1} (t-a_1) e^{b_1 (t-a_1)} + \frac{c_1}{b_1^2}; \end{aligned}$$

(A.12)

and

$$\begin{aligned}&\displaystyle \frac{\partial z(t)}{\partial c_1} = m \frac{1}{\partial c_1} \partial \frac{1- e^{-(p+q)A}}{1+ \frac{q}{p} e^{-(p+q)A}}\nonumber \\&\quad = m \frac{\left[ (p+q) e^{-(p+q)A} \frac{\partial A}{\partial c_1} \right] \left[ 1\!+\! \frac{q}{p} e^{-(p+q)A} \right] +\left[ 1\!-\! e^{-(p+q)A} \right] \left[ \frac{q}{p} (p+q) e^{-(p+q)A} \frac{\partial A}{\partial c_1} \right] }{\left( 1+ \frac{q}{p} e^{-(p+q)A}\right) ^2}, \nonumber \\ \end{aligned}$$

(A.13)

where

$$\begin{aligned} \frac{\partial A}{\partial c_1}= & {} \displaystyle \frac{\partial \left\{ t + \frac{c_1}{b_1}\left[ e^{b_1 (t-a_1)}-1\right] \right\} }{\partial c_1} = \displaystyle \frac{\partial \left[ \frac{c_1}{b_1} e^{b_1 (t-a_1)} - \frac{c_1}{b_1}\right] }{\partial c_1} \nonumber \\= & {} \displaystyle \frac{1}{b_1} \left( e^{b_1 (t-a_1)}-1 \right) . \end{aligned}$$

(A.14)

In the case of the GBM$_{rect}$, the partial derivatives of z(t) (Eqs. (A.3) and (A.5)), with respect to $(a_1, b_1, c_1)$, are zero for $0\le t < a_1.$ For $t>a_1$, they correspond to those found in Eqs. (A.9), (A.11), and (A.13) for the GBM$_{exp}$, where now,

$$\begin{aligned} \frac{\partial A}{\partial a_1} = \left\{ \begin{array}{ll} \displaystyle \frac{1}{{\partial a_1}} \partial \left[ t+c_1 (t-a_1) \right] = -c_1 &{}\quad a_1 \le t \le b_1 \\ \displaystyle \frac{1}{{\partial a_1}} \partial \left[ t+c_1 (b_1-a_1) \right] = -c_1 &{}\quad t > b_1, \\ \end{array} \right. \end{aligned}$$

(A.15)

$$\begin{aligned} \frac{\partial A}{\partial b_1} = \left\{ \begin{array}{ll} \displaystyle \frac{1}{{\partial b_1}} \partial \left[ t+c_1 (t-a_1) \right] = 0 &{}\quad a_1 \le t \le b_1 \\ \displaystyle \frac{1}{{\partial b_1}} \partial \left[ t+c_1 (b_1-a_1) \right] = c_1 &{}\quad t > b_1, \\ \end{array} \right. \end{aligned}$$

(A.16)

$$\begin{aligned} \frac{\partial A}{\partial c_1} = \left\{ \begin{array}{ll} \displaystyle \frac{1}{{\partial c_1}} \partial \left[ t+c_1 (t-a_1) \right] = t-a_1 &{}\quad a_1 \le t \le b_1 \\ \displaystyle \frac{1}{{\partial c_1}} \partial \left[ t+c_1 (b_1-a_1) \right] = b_1 -a_1 &{}\quad t > b_1. \\ \end{array} \right. \end{aligned}$$

(A.17)

For the BM, the partial derivatives of z(t) of Eq. (6), with respect to the diffusion parameters (m, p, q), correspond to those of Eqs. (A.6), (A.7), and (A.8), where $A=t$, for $t\ge 0.$

Table 5 Music cassettes. Estimates, asymptotic standard errors, $95\%$ mCIs, and $R^2$ for the BM ($k=3$)

Table 6 Austrian solar thermal capacity. Estimates, asymptotic standard errors, and $95\%$ mCIs for the BM with fixed m ($k=2$), BM ($k=3$), and GBM$_{rect}$ ($k=6$). $R^2$ is provided for each model

Table 7 Algerian natural gas production (truncated time series). The estimates, asymptotic standard errors, and $95\%$ mCIs for the BM with fixed m ($k=2$), BM ($k=3$), and GBM$_{exp}$ ($k=6$). $R^2$ is provided for each model

Table 8 Algerian natural gas production (complete time series). The estimates, asymptotic standard errors, and $95\%$ mCIs for the BM with fixed m ($k=2$), BM ($k=3$), and GBM$_{exp}$ ($k=6$). $R^2$ is provided for each model

Table 9 Coverage probability of ${\mathfrak {C}(\Theta )}$, ${\mathfrak {L}(\Theta )}$, ${\mathfrak {I}(\Theta )}$, ${\mathfrak {B}_p(\Theta )}$, ${\mathfrak {B}_{np}(\Theta )}$, and mCIs of the restricted BM ($k=2$), for a set of different $(\sigma /m)^2\times 10^6$ values. $N=1000,~B=1000$. The true values consist of the estimates in Tables 7 ($k=2$), 6 ($k=2$), 8 ($k=2$), for the settings Gas.2, Sol.2, and TGas.2, respectively. The nominal level is $1-\alpha =0.95$

1.2 Fitted models

In this “Appendix”, we list estimation details for all the datasets described in Sect. 4 and the different models fitted. The parameter estimates, asymptotic standard errors, $95\%$ mCIs, and determination index $R^2$ are shown in Table 5 for the cassette sales, Table 6 ($k=2, 3, 6$) for solar thermal capacity, and Tables 7 and 8 ($k=2, 3, 6$) for the complete and truncated natural gas production time series, respectively.

1.3 Coverage probability tables

The tables in this section display in detail the numerical values for coverage probabilities for all the simulation settings examined in Sect. 6. Coverage probabilities were represented graphically in Figs. 8, 9, 10, 11, 12, 13, 14 and 16, 17, 18 and 19.

Table 10 Coverage probability of ${\mathfrak {C}(\Theta )}$, ${\mathfrak {L}(\Theta )}$, ${\mathfrak {I}(\Theta )}$, ${\mathfrak {B}_p(\Theta )}$, ${\mathfrak {B}_{np}(\Theta )}$, and mCIs of the BM ($k=3$), for a set of different $(\sigma /m)^2\times 10^6$ values. $N=1000,~B=1000$. The true values consist of the estimates in Tables 5, 7 ($k=3$), 6 ($k=3$), 8 ($k=3$), for the settings Cass.3, Gas.3, Sol.3, and TGas.3, respectively. The nominal level is $1-\alpha =0.95$

Table 11 Coverage probability of ${\mathfrak {C}(\Theta )}$, ${\mathfrak {L}(\Theta )}$, ${\mathfrak {I}(\Theta )}$, ${\mathfrak {B}_p(\Theta )}$, ${\mathfrak {B}_{np}(\Theta )}$, and mCIs of the GBM ($k=6$), for a set of different $(\sigma /m)^2\times 10^6$ values. $N=1000,~B=1000$. The true values consist of the estimates in Tables 6 ($k=6$), 7 ($k=6$), and 8 ($k=6$), for the settings Sol.6, Gas.6, and TGas.6, respectively. The nominal level is $1-\alpha =0.95$

Table 12 Cassette sales in the case of heteroscedasticity. $N=1000,~B=1000,~n=33$. Coverage probability of ${\mathfrak {C}(\Theta )}$, ${\mathfrak {L}(\Theta )}$, ${\mathfrak {I}(\Theta )}$, ${\mathfrak {B}_p(\Theta )}$, ${\mathfrak {B}_{np}(\Theta )}$, and mCIs, for a set of different $(\sigma _u/m)^2\times 10^{11}$ values, for the BM ($k=3$). The true values consist of the estimates in Table 5. The nominal level is $1-\alpha =0.95$