| Sažetak | Difuzijski modeli rasta imaju važnu primjenu u biomedicinskim istraživanjima, posebno u modeliranju rasta tumora. Parametri modela se uobičajeno procjenjuju metodom najveće vjerodostojnosti iz diskretnih opservacija trajektorija. Budući da nije uvijek moguće eksplicitno izraziti funkciju vjerodostojnosti, a time i procjenitelj metodom maksimalne vjerodostojnosti, na osnovu diskretnih opservacija, parametri modela se procjenjuju drugim metodama. Posebno su zanimljivi aproksimativni procjenitelji najveće vjerodostojnosti (AMLE) parametara drifta: procjenitelji koji imaju svojstvo da po vjerojatnosti teže procjeniteljima najveće vjerodostojnosti na osnovi neprekidnih opservacija (MLE) duž ograničenog fiksnog vremenskog intervala kada očica podjele intervala teži ka nuli. Iako nije moguće konzistentno procjenjivati parametre drifta duž ograničenog fiksnog vremenskog intervala, moguće je istražiti asimptotsku distribuiranost AMLE kada očica podjele intervala teži ka nuli i njenu primjenu u statističkom zaključivanju o modelu. U radu se gleda stohastička diferencijalna jednadžba oblika: \(dX_t = \mu(X_t, \theta)dt + \sigma_0\nu(X_t)dW_t, X_0 = x_0 > 0\), gdje su \(\mu\) i \(\nu\) realne funkcije, \(\mu(\cdot,\theta)\) je funkcija drifta (funkcija pomaka) i \(\sigma_0\nu (\cdot)\) je difuzijski koeficijent, pri čemu se pretpostavlja da je parametar \(\sigma_0 > 0\) poznat. Neka je \(X\) rješenje dane SDJ uz pravu vrijednost parametra \(\theta_0\). Pretpostavljamo da nepoznati parametar \(\theta_0\) pripada prostoru \(\Theta\) koji je relativno kompaktan, konveksan podskup od \(\mathbb{R}^d\). Neka je zadan fiksan, realan broj \(T > 0\), i neka je \(0 =: t_0 < t_1 < \dots < t_n := T, n \in \mathbb{N}\) zadana subdivizija segmenta [0, T]. Neka je \(\Delta_n := \max_{1\leq i \leq n} (t_i-t_{i-1})\). Iz diskretnih opservacija \((X_{t_i}, 0 \leq i \leq n)\) trajektorije \((X_t, t \in [0, T])\), koristeći Eulerovu aproksimaciju zadane SDJ, procjenjujemo nepoznati parametar drifta (pomaka) \(\theta\), i dobijemo procjenitelj \(\Bar {\theta}_n\), kojeg zovemo AMLE parametra \(\theta\). Pomoću neprekidnih opservacija \((X_t, t \in [0, T])\), dobijemo procjenitelj \(\hat{\theta}_t\) za \(\theta\), kojeg zovemo MLE parametra \(\theta\). Postavljaju se uvjeti na funkcije \(\mu\) i \(\nu\) uz koje su slučajni vektori \(\frac{1}{\sqrt{\Delta_n}}(\Bar {\theta}_n-\hat{\theta}_T)\) asimptotski miješano normalno distribuirani, kada \(n \to +\infty\), pri čemu očica subdivizije \(\Delta_n\) konvergira u nulu. Napravljene su i simulacije koje potvrđuju rezultate. |
| Sažetak (engleski) | Diffusion models of growth have important applications in biomedical research, especially in tumor growth modeling. Model parameters are usually estimated by maximum likelihood method. Since tumors are observable in discrete time moments over bounded time interval, and since it is not possible to obtain the likelihood function in closed form for many diffusion models, model parameters are estimated by other methods. Approximate maximum likelihood estimators (AMLE) of drift parameters are especially interesting. These estimators converge in probability to the maximum likelihood estimators based on continuous observations (MLE) over bounded time interval and when the diameter of subdivision tends to zero. Although it is not possible to estimate drift parameters consistently over bounded time interval, it is possible to investigate asymptotic distribution of AMLE when the diameter of subdivision tends to zero. Let \((\Omega, \mathcal{F}, (\mathcal{F}_t)_{t\geq 0}, \mathbb{P})\) be given filtered probability space which satisfies usual conditions and let \(W = (W_t, t \geq 0)\) be an one-dimensional standard Brownian motion defined on that space. Let \(X = (X_t, t \geq 0)\) be an one-dimensional diffusion which satisfies Ito's stochastic differential equation (SDE) of the form \(dX_t = \mu(X_t, \theta)dt + \sigma_0\nu(X_t)dW_t, X(0) = x_0, t \geq 0\), where \(\nu\) and \(\mu\) are real functions and \(x_0\) is a given deterministic initial value of \(X, \sigma_0 > 0\) is a given parameter, and \(\theta_0\) is true parameter value. Let \(X\) be a solution of given SDE for true parameter value \(\theta_0\). We assume that \(\theta\) belongs to the parameter space \(\Theta\) which is an open, relatively compact, convex set in the Euclidean space \(\mathbb{R}^d\). Let \(T > 0\) be a fixed real number and \(0 =: t_0 < t_1 < \cdots < t_n := T, n \in \mathbb{N}\) be deterministic subdivision of segment \([0, T]\). Given a discrete observation \((X_{t_i}, 0 \leq i \leq n)\) of the trajectory \((X_t, t \in [0, T])\), we estimate the unknown drift parameter \(\theta\) of \(X\), and we get estimator \(\Bar {\theta}_n\), which we call AMLE of the parameters \(\theta\). Using continuous observations \((X_t, t \in [0, T])\), we can get estimator for \(\theta\) which we call MLE of the parameter \(\theta\). Let \(\Delta_n := max_{i=1,\dots, n}(t_i-t_{i-1})\). For each \(\theta \in \Theta\) let \(\sum(\theta)\) be \(d \times d\) random matrix which \(j, k\) component is defined by \(\sum(\theta)^{jk} = \frac{1}{2} \int_{0}^{T}\nu^4(X_s)\frac{\partial}{\partial x} \frac{\frac{\partial}{\partial \theta_j}\mu(X_s, \theta)}{\nu^2(X_s)} \frac{\partial}{\partial x} \frac{\frac{\partial}{\partial \theta_k}\mu (X_s, \theta)}{\nu^2(X_s)}ds.\) (0.1) We will say that a random vector \(Y\) has mixed normal distribution with covariance \(\mathcal{F}_T\) -measurable random matrix \(C\), and we write \(Y \sim MN(0, C)\) if \(Y \overset{d}{=} \sqrt{C} Z\), where \(\sqrt{C}\) is square symmetric root of \(C\) and \(Z \sim N(0,I)\) is standard normal random vector independent of \(\mathcal{F}_T\). If \(Y \sim MN(0, C)\), then \(\mathbb{E} [e^{i \langle t,Y \rangle} | \mathcal{F}_T]=e^{- \frac{1}{2} \sum_{j,k=1,\dots ,d^{t_j t_k C^{jk}}}}\). If we denote by \(\overset{st}{\Rightarrow}\) stable convergence in law, then, we got new results, that, under some assumptions on our model, we have \(\frac{1}{\sqrt{\Delta_n}}(\Bar {\theta}_n-\hat{\theta}_T) \overset{st}{\Rightarrow} MN(0, (D^2 L_T (\hat{\theta}_T))^{-1} \sum (\hat{\theta}_T)(D^2 L_T(\hat{\theta}_T))^{-1})\), and \((\sqrt{\sum_{n}(\Bar {\theta}_n)})^{-1} D^2 L_n (\Bar {\theta}_n) \frac{1}{\sqrt{\Delta_n}} (\Bar {\theta}_n-\hat{\theta}_T) 1_{\{\sum_{n}(\Bar{\theta}_n) \mbox{is regular matrix}\}} \overset{st}{\Rightarrow} N(0,I)\). where \(D^2 L_T\) and \(D^2 L_n\) are matrices of derivatives of second order for the functions \(L_T (\theta)= \int_{0}^{T} \frac{\mu(X_s, \theta)}{\sigma_{0}^{2}\nu^2(X_s)}dX_s-\frac{1}{2} \int_{0}^{T} \frac{\mu^2(X_s, \theta)}{\sigma_{0}^{2}\nu^2(X_s)} ds\), and \(L_n(\theta)=-\frac{n}{2} ln(\sigma_{0}^{2})-\frac{1}{2} \sum_{i=1}^{n} \frac{(X_{t_i}-X_{t_{i-1}}-\mu(X_{t_{i-1}}, \theta)(t_i-t_{i-1}))^{2}}{\sigma_{0}^{2}\nu^2(X_{t_{i-1}})(t_i-t_{i-1})}\) , respectively. |
