A New Algorithm for Global Minimization Based on the Combination of Adaptive Random Search and Simplex Algorithm of Nelder and Mead

We propose a new general-purpose algorithm for locating global minima of differentiable and nondifferentiable multivariable functions. The algorithm is based on combination of the adaptive random search approach and the Nelder-Mead simplex minimization. We show that the new hybrid algorithm satisfies the conditions of the theorem for convergence (in probability) to global minimum. By using test functions we demonstrate that the proposed algorithm is far more efficient than the pure adaptive random search algorithm, Some of the considered test functions are related to membership set estimation method for model parameter determination which was successfully applied to kinetic problems in chemistry and biology

A general theorem on approximate maximum likelihood estimation

In this paper a version of the general theorem on approximate maximum likelihood estimation is proved. We assume that there exists a log-likelihood function L(θ) and a sequence (Ln(θ)) of its estimates defined on some statistical structure parametrized by θ from an open set Θ ⊆ Rd, and dominated by a probability P. It is proved that if L(θ) and Ln(θ) are random functions of class C2(Θ) such that there exists a unique point θ ∈ Θ of the global maximum of L(θ) and the first and second derivatives of Ln(θ) with the respect to θ converge to the corresponding derivatives of L(θ) uniformly on compacts in Θ with the order OP(γn), limn γn = 0, then there exists a sequence of Θ-valued random variables θn which converges to θ with the order OP(γn), and such that θn is a stationary point of Ln(θ) in asymptotic sense. Moreover, we prove that under two more assumptions on L and Ln, such estimators could be chosen to be measurable with respect to the σ-algebra generated by Ln(θ)

Ruin probabilities and decompositions for general perturbed risk processes

We study a general perturbed risk process with cumulative claims modelled by a subordinator with finite expectation, with the perturbation being a spectrally negative Levy process with zero expectation. We derive a Pollaczek-Hinchin type formula for the survival probability of that risk process, and give an interpretation of the formula based on the decomposition of the dual risk process at modified ladder epochs

Local asymptotic mixed normality of approximate maximum likelihood estimator of drift parameters in diffusion model

We assume that the diffusion X satisfies a stochastic differential equation of the form: dXt=μ(Xt,θ)dt+σ0ν(Xt)dWt, with unknown drift parameter θ and known diffusion coefficient parameter σ0. We prove that approximate maximum likelihood estimator of drift parameter θ n obtained from discrete observations (XiΔn,0≤ i≤ n) along fixed time interval [0,T], and when Δn =T/n tends to zero, is locally asymptotic mixed normal, with covariance matrix which depends on MLE obtained from continuous observations (Xt,0≤ t≤ T) along fixed time interval [0,T], and on path (Xt,0≤ t≤ T)

Wiener-Hopfova faktorizacija

U ovom radu pomoću dualnih vremena zaustavljanja dokazana je Wiener- Hopfova faktorizacija slučajne šetnje na R te je primijenjena u dokazu Baxterovih jednakosti. Ovaj rad je posvećen uspomeni na docenta Antu Mimicu

Estimation of the killing rate parameter in a diffusion model

We consider a parameter estimation problem for a diffusion with killing, starting at a point in an open and bounded set. The infinitesimal killing rate function depends on a control variable and parameters. Values of the control variable are known while parameters have unknown values which have to be estimated from data. The minimum of three times: the maximum observation time, the first exit time from the open set, and the killing time, is observed. Instead of the maximum likelihood estimation method we propose and use the minimum $chi^2$-estimation method that is based on the conditional mean of the data observed before the maximum observation time is reached, and on the frequency of data that are equal to the maximum observation time. We prove that the estimator exists and is consistent and asymptotically normal. The method is illustrated by an example

An approximate maximum likelihood estimator of drift parameters in a multidimensional diffusion model

For a fixed $T$ and $k \geq 2$, a $k$-dimensional vector stochastic differential equation $dX_t=\mu(X_t, \theta)dt+\nu(X_t)dW_t,$ is studied over a time interval $[0,T]$. Vector of drift parameters $\theta$ is unknown. The dependence in $\theta$ is in general nonlinear. We prove that the difference between approximate maximum likelihood estimator of the drift parameter $\overline{\theta}_n\equiv \overline{\theta}_{n,T}$ obtained from discrete observations $(X_{i\Delta_n}, 0 \leq i \leq n)$ and maximum likelihood estimator $\hat{\theta}\equiv \hat{\theta}_T$ obtained from continuous observations $(X_t, 0\leq t\leq T)$, when $\Delta_n=T/n$ tends to zero, converges stably in law to the mixed normal random vector with covariance matrix that depends on $\hat{\theta}$ and on path $(X_t, 0 \leq t\leq T)$. The uniform ellipticity of diffusion matrix $S(x)=\nu(x)\nu(x)^T$ emerges as the main assumption on the diffusion coefficient function.Comment: 38 page

De Finettijev teorem

Statistički model i slučajni uzorak osnovni su pojmovi matematičke statistike. Za slučaj beskonačne populacije slučajni uzorak se najčešće definira kao niz nezavisnih i jednako distribuiranih slučajnih veličina u odnosu na svaku vjerojatnost iz pretpostavljenog statističkog modela. Budući da je slučajni uzorak model za niz opažanja određene veličine kao funkcije nekog slučajnog eksperimenta, postavlja se pitanje nije li pretpostavka o nezavisnosti i jednakoj distribuiranosti opažanih pokusa prejaka. Ako jest, koja pretpostavka je slabija od te, a da i dalje povlači poželjne rezultate inferencijalne statistike? Pokazuje se da je to pretpostavka izmjenjivosti