[Forum SIS] Analisi statistica dati COVID-19: carattere subesponenziale Agosto-Giudici

Ven 13 Mar 2020 10:50:44 CET

Buongiorno,

mi permetto di inserirmi nella discussione relativa allo spam sulla 
mailing list, producendone un po'.

Stavo pensando stamane che è possibile caratterizzare meglio la natura 
subesponenziale del processo Agosto-Giudici per $\alpha+\beta<1$. I 
calcoli seguenti andrebbero controllati per bene, ma l'idea è che 
l'intero processo sia maggiorabile con una quantità che può essere 
stimata. L'utilità del calcolo, come si vedrà, è limitata per fini 
pratici, tuttavia chiarisce un aspetto teorico del processo.

A tale scopo mi serve richiamare

     1. la disuguaglianza di Jensen, che per una funzione *concava* 
$x\to\phi(x)$  si scrive

     (1) $E[\phi(X)] \le \phi(E[X])$;

     2. la funzione $x\to x^{\alpha+\beta}$  è concava per 
$\alpha+\beta<1$ e $x>0$.

Considero un processo di crescita caratterizzato dalla condizione

     (2) $E[\lambda_t | \lambda_{t-1}] = C\lambda_{t-1}^{\alpha+\beta}$, 
     dove $\alpha+\beta<1$.

Voglio stimare il valore atteso non-condizionato $E[\lambda_t]$. Uso 
(spero correttamente stavolta) la tecnica della media iterata

     (3) $E[\lambda_t] = E_{\lambda_{t-1}}[E[\lambda_t|\lambda_{t-1}]] = 
CE_{\lambda_{t-1}}[\lambda_{t-1}^{\alpha+\beta}|\lambda_{t-1}] \le 
C\bigl(E[\lambda_{t-1}])^{\alpha+\beta}$

Itero esplicitamente una sola volta,

     (4) $= 
C\left(E_{\lambda_{t-2}}[E[\lambda_{t-1}|\lambda_{t-2}]\right)^{\alpha+\beta} 
= 
C^{1+(\alpha+\beta)}\left(E[\lambda_{t-2}^{\alpha+\beta}]\right)^{\alpha+\beta}\le 
C^{1+(\alpha+\beta)}\left(E[\lambda_{t-2}]\right)^{(\alpha+\beta)^2}$;

quindi induco

     (5) $E[\lambda_t]\le 
C^{\sum_{k=0}^{t-1}(\alpha+\beta)^k}E[\lambda_0] = 
C^{\sum_{k=0}^{t-1}(\alpha+\beta)^k} $

assumendo $\lambda_0=1$. La somma ad esponente è una somma geometrica 
(di ragione $<1$), dunque

     (6) $\sum_{k=0}^{t-1}(\alpha+\beta)^k = 
\frac{1-(\alpha+\beta)^{t}}{1-(\alpha+\beta)}$.

Concludo che

     (7) $\lim_{t\to\infty}E[\lambda_t] \le C^{1/[1-(\alpha+\beta)]} 
\equiv K$.

Ovviamente, più $\alpha+\beta$ è prossimo ad 1, più la costante di 
maggiorazione sarà divergente, ancorché finita... Questo è chiaramente 
diverso rispetto ad un processo esponenziale crescente, per il quale

     (8) $\lim_{t\to\infty}E[\lambda_t] = \infty$.

Faccio solo qualche esempio. Per $C=1.5$, si ha

$(\alpha+\beta)=0.9$ $K=1.024$,

$(\alpha+\beta)=0.95$ $K\sim 10^6$,

$(\alpha+\beta)=0.99$ $K \sim 10^{30}$.

Fine.

Metto le mani avanti, è un calcolo fatto di fretta... Spero di non aver 
preso cantonate macroscopiche (in tal caso mi prenderò di buon grado le 
vs. bastonate)... Forse qualcuno con buona volonta' e tempo da perdere 
potrebbe controllare.

In ogni caso, cordiali saluti.

f. p.

PS: metto in allegato una versione pdf.

--

Filippo Palombi, PhD

ENEA - Italian Agency for New Technologies,
Energy and Sustainable Economic Development

Via E. Fermi 45
00044 Frascati - Italy

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