Statistical analysis and first-passage-time applications of a lognormal diffusion process with multi-sigmoidal logistic mean

Abstract

We consider a lognormal diffusion process having a multisigmoidal logistic mean, useful to model the evolution of a population which reaches the maximum level of the growth after many stages. Referring to the problem of statistical inference, two procedures to find the maximum likelihood estimates of the unknown parameters are described. One is based on the resolution of the system of the critical points of the likelihood function, and the other is on the maximization of the likelihood function with the simulated annealing algorithm. A simulation study to validate the described strategies for finding the estimates is also presented, with a real application to epidemiological data. Special attention is also devoted to the first-passage-time problem of the considered diffusion process through a fixed boundary.

Keywords: Asymptotic distribution; First-passage-time; First-passage-time location function; Lognormal diffusion process; Maximum likelihood estimation; Multi-sigmoidal growth.

Fig. 1

The multisigmoidal logistic function for some choices of the parameters: $t_0=0$, $l_0=\frac{10}{1+\eta }$, $\eta =e^{-1}$, ${\beta }_1=0.1$, a ${\beta }_2=-0.009$ and, from bottom to top, ${\beta }_3=0.0002,0.0003,0.0004$; b ${\beta }_2=-0.007$ and, from bottom to top, ${\beta }_3=0.0002,0.0003,0.0004$

Fig. 2

The multisigmoidal logistic function and the corresponding inflection points for $t_0=0$, $l_0=5$, $\eta =e^{-1}$, ${\beta }_1=0.1$, a ${\beta }_2=-0.009$ and ${\beta }_3=0.0002$; b ${\beta }_2=-0.007$ and ${\beta }_3=0.0001$

Fig. 3

100 simulated sample paths of the process X(t) with $p=3$, $Q_{\beta }(t)=0.1t-0.009t^2+0.0002t^3$, $\eta =e^{-1}$, $\sigma =0.01$, $t_0=0$ and $x_0=5$. The black line represents the sample mean of the process, while the red line represents the boundary $S=15$

Fig. 4

a The FPTL function and b the approximated FPT density of the process X(t) through the constant boundary $S=15$, for the same assumptions of Fig. 3

Fig. 5

100 simulated sample paths of the diffusion process for $\sigma =0.01$, $\eta =e^{-1}$ and a ${\beta }_1=0.1$, ${\beta }_2=-0.009$, ${\beta }_3=0.0002$ and b ${\beta }_1=0.1$, ${\beta }_2=-0.007$, ${\beta }_3=0.0003$ (simulation study)

Fig. 6

The resistor average distance between a–b the sample and the estimated distribution, and c–d the theoretical and estimated distribution for the case 1 of Table 2, for different degrees of the polynomial (simulation study)

Fig. 7

The RAE for a $\eta =e^{-1}$ and $\sigma \in [0.01,0.05]$, b $\sigma =0.05$ and $\eta \in \left[e^{-3},,,e^{-1}\right]$ and c $\sigma =0.01$, $\eta =e^{-1}$ and with respect of the number of replications. In all the cases $Q_{\beta }(t)=0.1t-0.009t^2+0.0002t^3$ (simulation study)

Fig. 8

The theoretical, sample and estimated means of the process X(t) for the parameters of the cases number 1 and 2 of Table 2 (from left to right). The results are obtained via Newton-Raphson method in (a) and (b) and via S.A. in (c) and (d). (Simulation study)

Fig. 9

a 50 simulated sample paths of the diffusion process X(t) for $\sigma =0.01$ , $\eta =e^{-1}$ and $Q_{\beta }(t)=0.1t-0.009t^2+0.0002t^3$. b Theoretical, sample and estimated means of the process X(t) for the FPT density approximation (simulation study—FPT problem)

Fig. 10

Resistor-average distance between a the theoretical and the estimated distributions and b between the sample and the estimated distributions for the FPT density approximation (simulation study - FPT problem)

Fig. 11

The approximated FPT density and the FPTL function of the process X(t) through the boundary $S=15$ (simulation study—FPT problem)

Fig. 12

a Number of infections in France, Italy, Spain and United Kingdom, the black line represents the sample mean. b Sample and the estimated means obtained by solving the system (23) (real application)

Fig. 13

a The resistor-average distances between the sample and the estimated distributions considering different degrees of the polynomial. b The $\alpha$-percentiles of the estimated diffusion process X(t) obtained for a degree $p=3$ and for $\alpha =95,90,75$ (real application)

Fig. 14

a Resistor-average distances between the sample and the estimated distributions in the restricted time range. b Sample and the estimated means obtained by solving the system (23) in the restricted time range (real application with $t_f=246$)

Fig. 15

a The sample and the forecasted means in the complete time range $I_C=[0,250]$ considering a degree $p=3$. b Resistor-average distances between the sample and the estimated distributions in the restricted time range $I_R$. c Approximated FPT density and d FPTL function in the restricted time range $I_R$ through the boundary $S=0.7$ (real application—FPT problem)