 # Darija Marković Associate Professor
Department of Mathematics
Josip Juraj Strossmayer University of Osijek
Trg Ljudevita Gaja 6
Osijek, HR-31000, Croatia¸
 phone: +385-31-224-823 fax: +385-31-224-801 email: darija @ mathos.hr office: 20 (first floor)

## Research Interests

Applied Mathematics
Numerical Mathematics
Parameter Estimation

## Degrees

PhD in Numerical Mathematics, Faculty of Natural Science, Department of Mathematics, University of Zagreb, 2009.
MSc in Mathematics, Faculty of Natural Science, Department of Mathematics, University of Zagreb, 2005.
BSc in Mathematics and Computer Science, Department of Mathematics, University of Osijek, Croatia, 2000.

## Publications

Journal Publications

1. D. Jukić, D. Marković, Nonlinear least squares estimation of the shifted Gompertz distribution, European Journal of Pure and Applied Mathematics 10/2 (2017), 157-166
The focus of this paper is the existence of the best nonlinear least squares estimate for the shifted Gompertz distribution. As a main result, two theorems on the existence of the least squares estimate are obtained, as well as their generalization in the lp norm (1 ≤ p < ∞ ).
2. D. Marković, Preponderantly increasing/decreasing data in regression analysis, Croatian Operational Research Review 7/2 (2016), 269-276
For the given data (wi,xi,yi), i=1, ..., n, and the given model function f(x;θ), where θ is a vector of unknown parameters, the goal of regression analysis is to obtain estimator θ∗ of the unknown parameters θ such that the vector of residuals is minimized in some sense. The common approach to this problem of minimization is the least-squares method, that is minimizing the L2 norm of the vector of residuals. For nonlinear model functions, what is necessary is finding at least the sufficient conditions on the data that will guarantee the existence of the best least-squares estimator. In this paper we will describe and examine in detail the property of preponderant increase/decrease of the data, which ensures the existence of the best estimator for certain important nonlinear model functions.
3. D. Marković, L. Borozan, On Parameter Estimation by Nonlinear Least Squares in Some Special Two-Parameter Exponential Type Models, Applied Mathematics & Information Sciences 9/6 (2015), 2925-2931
Two-parameter growth models of exponential type f(t;a,b) = g(t)exp(a+bh(t)), where a and b are unknown parameters and g and h are some known functions, are frequently employed in many different areas such as biology, finance, statistic, medicine, ect. The unknown parameters must be estimated from the data (w_i, t_i, y_i), i = 1,...,n, where t_i denote the values of the independent variable, y_i are respective estimates of regression function f and w_i > 0 are some data weights. A very popular and widely used method for parameter estimation is the method of least squares. In practice, to avoid using nonlinear regression, this kind of problems are commonly transformed to linear, which is not statistically justified. In this paper we show that for strictly positive g and strictly monotone h original nonlinear problem has a solution. Generalization in the lp norm (1 ≤ p < ∞) and some illustrative examples are also given.
4. D. Marković, D. Jukić, Total least squares fitting the three-parameter inverse Weibull density, European Journal of Pure and Applied Mathematics 7/3 (2014), 230-245
The focus of this paper is on a nonlinear weighted total least squares fitting problem for the three-parameter inverse Weibull density which is frequently employed as a model in reliability and lifetime studies. As a main result, a theorem on the existence of the total least squares estimator is obtained, as well as its generalization in the l_q norm (1≤q<∞).
5. D. Marković, D. Jukić, On parameter estimation in the bass model by nonlinear least squares fitting the adoption curve, International Journal of Applied Mathematics and Computer Science 23/1 (2013), 145-155
The Bass model is one of the most well-known and widely used first-purchase diffusion models in marketing research. Estimation of its parameters has been approached in the literature by various techniques. In this paper, we consider the parameter estimation approach for the Bass model based on nonlinear weighted least squares fitting of its derivative known as the adoption curve. We show that it is possible that the least squares estimate does not exist. As a main result, two theorems on the existence of the least squares estimate are obtained, as well as their generalization in the ls norm (1 ≤ s < ∞). One of them gives necessary and sufficient conditions which guarantee the existence of the least squares estimate. Several illustrative numerical examples are given to support the theoretical work.
6. M. Marušić, D. Marković, D. Jukić, Least squares fitting the three-parameter inverse Weibull density, Mathematical Communications 15/2 (2010), 539-553
The inverse Weibull model was developed by Erto . In practice, the unknown parameters of the ppropriate inverse Weibull density are not known and must be estimated from a random sample. Estimation of its parameters has been approached in the literature by various techniques, because a standard maximum likelihood estimate does not exist. To estimate the unknown parameters of the three-parameter inverse Weibull density we will use a combination of onparametric and parametric methods. The idea consists of using two steps: in the first step we calculate an initial nonparametric density estimate which needs to be as good as possible, and in the second step we apply the nonlinear least squares method to estimate the unknown parameters. As a main result, a theorem on the existence of the least squares estimate is obtained, as well as its generalization in the l_p norm (1 p < 1). Some simulations are given to show that our approach is satisfactory if the initial density is of good enough quality.
7. D. Jukić, D. Marković, On nonlinear weighted errors-in-variables parameter estimation problem in the three-parameter Weibull model, Applied mathematics and computation 215/10 (2010), 3599-3609
This paper is concerned with the three-parameter Weibull distribution which is widely used as a model in reliability and lifetime studies. In practice, the Weibull model parameters are not known in advance and must be estimated from a random sample. Difficulties in applying the method of maximum likelihood to three-parameter Weibull models have led to a variety of alternative approaches in the literature. In this paper we consider the nonlinear weighted errors-in-variables (EIV) fitting approach. As a main result, two theorems on the existence of the EIV estimate are obtained. An illustrative example is also included.
8. D. Jukić, D. Marković, On nonlinear weighted least squares fitting of the three-parameter inverse Weibull distribution, Mathematical Communications 15/1 (2010), 13-24
In this paper we consider nonlinear least squares fitting of the three-parameter inverse Weibull distribution to the given data (wi; ti; yi), i = 1,...,n, n>3. As the main result, we show that the least squares estimate exists provided that the data satisfy just the following two natural conditions: (i) 0
9. D. Marković, D. Jukić, On nonlinear weighted total least squares parameter estimation problem for the three-parameter Weibull density, Applied Mathematical Modelling 34/7 (2010), 1839-1848
The three-parameter Weibull density function is widely employed as a model in reliability and lifetime studies. Estimation of its parameters has been approached in the literature by various techniques, because a standard maximum likelihood estimate does not exist. In this paper we consider the nonlinear weighted total least squares fitting approach. As a main result, a theorem on the existence of the total least squares estimate is obtained, as well as its generalization in the total l_q norm (\$qgeq 1\$). Some numerical simulations to support the theoretical work are given.
10. D. Marković, D. Jukić, M. Benšić, Nonlinear weighted least squares estimation of a three-parameter Weibull density with a nonparametric start, Journal of Computational and Applied Mathematics, 228/1 (2009), 304-312
This paper is concerned with the parameter estimation problem for the three-parameter Weibull density which is widely employed as a model in reliability and lifetime studies. Our approach is a combination of nonparametric and parametric methods. The basic idea is to start with an initial nonparametric density estimate which needs to be as good as possible, and then apply the nonlinear least squares method to estimate the unknown parameters. As a main result, a theorem on the existence of the least squares estimate is obtained. Some simulations are given to show that our approach is satisfactory if the initial density is of good enough quality.

Refereed Proceedings

1. D. Marković, D. Jukić, A review of some existence results on parameter estimation problem in the three-parameter Weibull model, 12th International Conference on Operational Research, Pula, Croatia, 2008, 103-111
2. D. Marković, D. Dudaš, Bezierove krivulje i de Casteljauov algoritam, Programski sustav Mathematica u znanosti, tehnologiji i obrazovanju. PrimMath. , Zagreb, Hrvatska, 2003, 51-67
Predstavit ćemo neke osnovne ideje CAGD-a (Computer Aided Geometric Design). Ključno je da s geometrijskim objektima možemo brzo i učinkovito manipulirati na računalu, tj. da ulazni parametri imaju geometrijsko značenje, drugim riječima da imamo predodžbu kako će zadana krivulja izgledati. U tu svrhu definirat ćemo Bernsteinove polinome, predstaviti njihova osnovna svojstva, te uvesti pojam Bezierovih krivulja kao prikaz polinoma u Bernsteinovoj bazi. Također ćemo pokazati de Casteljauov algoritam. Svi programi bit će izradeni primjenom programskog sustava Mathematica. Pri tome koristit će se grafičke mogućnosti i animacija iterativnog procesa.
3. D. Jukić, D. Marković, M. Ribičić Penava, A. Krajina, On the choice of initial approximation of the least squares estimate in some growth models of exponential type, 9th International Conference on Operational Research KOI 2002, Trogir, 2002, 47-55
4. D. Marković, QR dekompozicija velikih vrpčastih matrica i primjena na konstrukciju aproksimirajućeg spline-a, Programski sustav Mathematica u znanosti, tehnologiji i obrazovanju. PrimMath, Zagreb, 2001, 215-227

## Projects

• Problem procjene parametara u nekim dvoparametarskim monotonim matematičkim modelima (Parameter estimation problem in some two-parameter monotonic mathematical models)

Scientific project run in  2013/14; supported by the J. J. Strossmayer University of Osijek (principal investigator)

Participation (as researcher) in work of following projects funded by Croatian Ministry of Science:

• Scientific project "Passive control of mechanical models" (235-2352818-1042) within scientific program "Various aspects of parameter estimation problem in nonlinear mathematical models“ (2352818), since 2007;
• Scientific project "Parameter estimation in mathematical models“ (0235001), 2002 - 2006;

## Professional Activities

Refereeing/Reviewing

Conferences and Workshops

with talk:

• On the existence of the nonlinear weighted least squares estimate for some special exponential type models, 15th International Conference on Operational Research, Osijek, September 2014
• On parameter estimation by nonlinear least squares in some special two-parameter exponential type models, International Conference on Advances in Applied Mathematics and Mathematical Physics, Istanbul, Turkey, August 2014
• Parameter estimation problem for Weibull model, 14th Young Statisticians Meeting, Basovizza, October 2009
• Least squares fitting the three-parameter inverse Weibull density, Applied Mathematics and Scientific Computing, Zadar, September 2009
• On nonlinear weighted least squares fitting of the three-parameter inverse Weibull distribution, Applied Mathematics and Scientific Computing, Zadar, September 2009
• A review of some existence results on parameter estimation problem in the three-parameter Weibull model, 12th International Conference on Operational Research, Pula, October 2008
• On nonlinear weighted errors-in-variables parameter estimation problem in the three-parameter Weibull model, 4th Croatian Mathematical Congress, Osijek, June 2008
• Bézierove krivulje i de Casteljauov algoritam, Prim-Math, Zagreb, September 2003
• On the choice of initial approximation of the least squares estimate in some growth models of exponential type, 9th International Conference on Operational Research, Trogir, October 2002
• QR dekompozicija velikih vrpčastih matrica i primjena na konstrukciju aproksimirajućeg spline-a, PrimMath, Mathematica u znanosti, tehnologiji i obrazovanju, Zagreb, September 2001

without talk:

• 5th Croatian Mathematical Congress, Rijeka, June 2012
• 4th Conference on Applied Mathematics and Scienti c Computing, Brijuni, June 2005
• 3rd Conference on Applied Mathematics and Scienti c Computing, Brijuni, June 2003
• 8th International Conference on Operational Research, Rovinj, September 2000
• 2nd Croatian Mathematical Congress, Zagreb, June 2000

Invited Lectures

• Problem procjene parametara u 3-parametarskom Weibullovu modelu, Stručno-znanstveni kolokvij AMACIZ-a,Zagreb, March 2009
• Težinski spline-ovi, Matematički kolokvij, Osijek, January 2004
• Least squares spline, Matematički kolokvij, Osijek, March 2002

Study Visits Abroad and Professional Improvements

• Eidgenössische Technische Hochschule Zürich, Switzerland, February, 2008
• Technische Universität Berlin, Germany, June 2007
• Max Planck Institut für Informatik, Saarbrücken, Germany, August-September 2005

Professional Society Membership

• HMD - Croatian Mathematical Society, Department Osijek
• HDOI - Croatian Operational Research Society

## Teaching

Konzultacije (Office Hours): konzultacije se održavaju po dogovoru.

Prijedlog tema diplomskih radova na Odjelu za matematiku:

1. Polinomijalne matrice;
2. Parovi polinomijalnih matrica;
3. Osnovna svojstva racionalnih matrica;
4. Matrične grupe;
5. Egzistencija i konstrukcija generaliziranog inverza
6. Moguće su i druge teme u dogovoru sa zainteresiranim studentom

Nastavne aktivnosti u zimskom semestru akademske 2020./2021.

Nastavne aktivnosti u ljetnom semestru akademske 2020./2021.

Teaching experience

List of courses taught: Linear Algebra 1, Linear Algebra 2, Differential calculus, Integral calculus, Numerical mathematics, Ordinary Differential Equations, Vector Spaces, Algebra, Elementary Mathematics 2, Analytic Geometry, Didactics of Mathematics 2, Mathematics (Faculty of Economics), Applied Mathematics (Faculty of Food Technology), Engineering Mathematics (Faculty of Food Technology), Mathematics 1 (Department of Physics), Mathematics 2 (Department of Physics), Mathematics 3 (Faculty of Electrical Engineering)

## Personal

• Birthdate: July 7, 1976
• Birthplace: Osijek, Croatia
• Citizenship: Croatian