# Local asymptotic properties of approximative maximum likelihood estimator of drift parameters in diffusion model

Lubura, Snježana (2015) Local asymptotic properties of approximative maximum likelihood estimator of drift parameters in diffusion model. Doctoral thesis, Faculty of Science > Department of Mathematics.

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## Abstract

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 \textit{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.

Item Type: Thesis (Doctoral thesis) AMLE, diffusion, drift parameters, likelihood function, mixed normal distribution, stable convergence in law Huzak, Miljenko 2015 76 NATURAL SCIENCES > MathematicsNATURAL SCIENCES > Mathematics > Probability Theory and Statistics Faculty of Science > Department of Mathematics Iva Prah 09 Nov 2015 09:54 09 Nov 2015 09:54 http://digre.pmf.unizg.hr/id/eprint/4345