The relationship between the norm square of the standardized cumulative
distribution and the chi-square statistic is examined using the form
of the covariance matrix as well as the projection perspective. This investigation
enables us to give uncorrelated components of the chi-square statistic
and to provide interpretation of these components as innovations standardizing
the cumulative distribution values. The norm square of the standardized
difference between empirical and theoretical cumulative distributions is also
examined as an objective function for parameter estimation. Its relationship
to the chi-square distance enables us to discuss the large sample properties
of these estimators and a difference in their properties in the cases that the
distribution is evaluated at fixed and random points.
In many applications we are faced with the problem of estimating object dimensions
from a noisy image. Some devices like a fluorescent microscope, X-ray
or ultrasound machines, etc., produce imperfect images. Image noise comes from a
variety of sources. It can be produced by the physical processes of imaging, or may
be caused by the presence of some unwanted structures (e.g. soft tissue captured
in images of bones ). In the proposed models we suppose that the data are drawn
from uniform distribution on the object of interest, but contaminated by an additive
error. Here we use two one-dimensional parametric models to construct confidence
intervals and statistical tests pertaining to the object size and suggest the possibility
of application in two-dimensional problems. Normal and Laplace distributions
are used as error distributions. Finally, we illustrate ability of the R-programs we
created for these problems on a real-world example.
I. Kosović, M. Benšić, Đ. Ačkar, A. Jozinović, Ž. Ugarčić, J. Babić, B. Miličević, D. Šubarić, Microstructure and cooking quality of barley-enriched pasta produced at different process parameters, Foods and Raw materials (2018), prihvaćen za objavljivanje
This paper describes an R package LeArEst that can be used for estimating object dimensions
from a noisy image. The package is based on the simple parametric model for data that are drawn
from uniform distribution contaminated by an additive error. Our package is able to estimate the
length of the object of interest on a given straight line that intersects it as well as to estimate the object
area if it is elliptically shaped. The input data may be a numerical vector as well as an image in JPEG
format. In this paper, background statistical models and methods for the package are summarized,
and algorithms and key functions implemented are described. Also, examples that demonstrate its
usage are provided.
A one-dimensional problem of a uniform distribution width estimation from
data observed with a Laplace additive error is analyzed. The error variance is
considered as a nuisance parameter and it is supposed to be known or consistently
estimated before. It is proved that the maximum likelihood estimator
in the described model is consistent and asymptotically efficient and sufficient
conditions for its existence are given. The method of moment estimator
is also analyzed in this model and compared with the maximum likelihood
estimator theoretically and in simulations. Finally, one real-world example
illustrates the possibility for applications in two-dimensional problems.
We introduce and analyze a class of estimators for distribution parameters based on the relationship between the distribution function and the empirical distribution function. This class includes the nonlinear least squares estimator and the weighted nonlinear least squares estimator which has been used in parameter estimation for lifetime data as well as the generalized nonlinear least squares estimator proposed in Bensic. Sufficient conditions for consistency and asymptotic normality are given. Capability and limitations are illustrated by simulations.
The primary concern is to introduce and illustrate the way of using a generalized nonlinear regression method for the purpose of parameter estimation in the classical parametric independent and identically distributed sample model. It is shown by simulation that the presented estimator has a root mean square error comparable to the maximum likelihood estimator in the model for which it is known that maximum likelihood has excellent properties. As this estimator is based on an empirical distribution function, it is also compared to the maximum goodness-of-fit estimators that minimize Cramer-von Mises and Anderson- Darling empirical distribution statistics and it is shown that it outperforms them in most cases.
The problem of estimating the width of a symmetric uniform distribution on the line together with the error variance, when data are measured with normal additive error, is considered. The main purpose is to analyze the maximum likelihood estimator and to compare it with the moment method estimator. It is shown that this two-parameter model is regular so that the maximum likelihood estimator is asymptotically e± ; cient. Necessary and su± ; cient conditions are given for the existence of the maximum likeli- hood estimator. As numerical problems are known to frequently occur while computing the maximum likelihood estimator in this model, useful suggestions for computing the maximum likelihood estimator are also given.
The purpose of the paper was to extract important features of children’s mathematical gift by using neural networks and logistic regression, in order to create a model that will assist teachers in elementary schools to recognize mathematically gifted children in an early stage, therefore enabling further development and realization of that gift. The initial model was created on the basis of a theoretical background and heuristical knowledge on giftedness in mathematics, including five components: (1) mathematical competencies, (2) cognitive components of gift, (3) personal components that contribute gift development, (4) environmental factors, and (5) efficiency of active learning and exercising methods, as well as grades and out-of-school activities of pupils in the fourth year of elementary school. The three neural network classification algorithms were tested in order to extract the important variables for detecting mathematically gifted children. The best neural network model was selected on the basis of a 10-fold cross-validation procedure. The model was also investigated by the logistic regression. Important predictors detected by two methods were compared and analyzed. The results show that both methods extract similar set of variables as the most important, including grades in mathematics, mathematical competencies of a child regarding numbers and calculating, but also grades in the literature, and environmental factors.
The problem of estimating the boundary of a uniform distribution on a disc is considered when data are measured with normally distributed additive random error. The problem is solved in two steps. In the first step the domain is subdivided into thin slices and the endpoints of slices are obtained within the framework of a corresponding one-dimensional problem. For the estimations implemented in that step the moment method and the maximum likelihood method are used. As there are numerical problems with calculating the variance of the estimator in the maximum likelihood approach, its good approximation is also given. In the second step the obtained endpoints are used to estimate the boundary using the total least-squares curve fitting procedure. A necessary and sufficient condition for the existence of the total least-squares solution is also given. Finally, simulation results are presented.
The paper deals with the problem of predicting the time to default in credit behavioural scoring. This area opens a possibility of including a dynamic component in behavioural scoring modelling which enables making decisions related to limit, collection and recovery strategies, retention and attrition, as well as providing an insight into the profitability, pricing or term structure of the loan. In this paper we compare survival analysis and neural networks in terms of modelling and results. The neural network architecture is designed such that its output is comparable to the survival analysis output. Six neural network models were created, one for each period of default. A radial basis neural network algorithm was used to test all six models. The survival model used a Cox modelling procedure. Further, different performance measures of all models were discussed since even in highly accurate scoring models, misclassification patterns appear. A systematic comparison ‘ 3+2+2’ procedure is suggested to find the most effective model for a bank. Additionally, the survival analysis model is compared to neural network models according to the relative importance of different variables in predicting the time to default. Although different models can have very similar performance measures they may consist of different variables. The dataset used for the research was collected from a Croatian bank and credit customers were observed during a twelve-month period. The paper emphasizes the importance of conducting a detailed comparison procedure while selecting the best model that satisfies the users' interest.
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.
H. Lepeduš, H. Fulgosi, M. Benšić, V. Cesar, Efficiency of the photosynthetic apparatus in developing needles of Notway spruce (Picea abies L. Karst), Acta Biologica Hungarica 59/2 (2008), 217-232
The problem of nonlinear weighted least squares fitting of the three-parameter Weibull distribution to the given data (wi,ti,yi), i=1,…,n, is considered. The part wi>0 of the data stands for the data weights. It is shown that the best least squares estimate exists provided that the data satisfy just the following two natural conditions: (i) 0
M. Benšić, K. Sabo, Border Estimation of a Two-dimensional Uniform Distribution if Data are Measured with Additive Error, Statistics - a Journal of Theoretical and Applied Statistics 41 (2007), 311-319
The paper considers estimation of the boundary of an elliptical domain when the data without a measurement error are distributed uniformly on this domain but are superimposed by random errors. The problem is solved in two phases. In the first phase the domain is subdivided into thin slices and the endpoints of these slices are estimated within the framework of a corresponding one-dimensional problem. In the second phase the estimated endpoints are used to estimate the boundary using the total least squares curve fitting procedure
M. Benšić, K. Sabo, Estimating the width of a uniform distribution when data are measured with additive normal errors with known variance, Computational Statistics & Data Analysis 51 (2007), 4731-4741
The problem of estimating the width of the symmetric uniform distribution on the line when data are measured with normal additive error is considered. The main purpose is to discuss the efficiency of the maximum likelihood estimator and the moment method estimator. It is shown that the model is regular and that the maximum likelihood estimator is more efficient than the moment method estimator. A sufficient condition is also given for the existence of both estimators.
N. Šarlija, M. Benšić, M. Zekić-Sušac, Logistic refression, survival analysis and neural networks in modeling customer credit scoring, WSEAS Transactions on Business and Economics 3/3 (2006), 64-70
A. Hunjet, Đ. Parac-Osterman, M. Benšić, Utjecaj boje okoline na doživljaj žutog i plavog tona, Tekstil 55/3 (2006), 121-126
M. Benšić, N. Šarlija, M. Zekić-Sušac, Modeling Small Business Credit Scoring Using Logistic Regression, Neural Networks, and Decision Trees, Intelligent Systems in Accounting, Finance and Management 13/3 (2005), 133-150
M. Benšić, M. Pavleković, Matematički časopis kao učiteljev izvor ideja za suvremenijom organizacijom nastave matematike, Život i škola 9/1 (2003), 39-49
S. Mihaljević, I. Karner, B. Dmitrović, M. Benšić, N. Mičunović, A. Včev, B. Škurla, M. Katičić, Hormonska regulacija želučane sekrecije i helicobacter pylori, Liječnički vjesnik 124/1 (2002), 13-16
M. Benšić, N. Šarlija, Influence of entrepreneur's educational background and experience on number of employees, Mathematical Communications - Supplement 1/1 (2001), 121-126
N. Leonenko, M. Benšić, On estimation of regression coefficients of long memory random fields observed on the arrays, Random operators and stochastics equations 5/3 (1997), 237-252
We study some problems of the parameter inference which are in connection with long memory homogeneous and isotropic random fields. We present the asymptotic behavior for the correlation matrix and the limit distributions of the LSE for regression coefficients in some regression models with long memory Gaussian and non-Gaussian errors.
N. Leonenko, M. Šilac-Benšić, On the asymptotic distributions of least square estimations in a regression model with singular erors, Dopovidi NAS of Ukraine 7 (1997), 26-31
N. Leonenko, M. Benšić, Asymptotic properties of the LSE in a regression model with long-memory Gaussian and non-Gaussian stationary errors, Random operators and stochastics equations 4/1 (1996), 17-32
We study some problems of the parameter inference which a in connection with long-memory covariance stationary processes. We present the asymptotic behavior for the variance and the limit distributions of the LSE for the regression coefficients in some cases of long-memory, stationary, Gaussian and non-Gaussian errors.
N. Leonenko, M. Šilac-Benšić, On estimation of regression coefficients in the case of long-memory noise, Theory of Stochastic Processes 18/3-4 (1996), 108-119
In this paper we consider some asymptotic properties of the LSE of continuous time and non-Gaussian long-memory errors. The precise description of the model we have been working with is given through the second section.
B. Dukić, D. Francišković, D. Jukić, R. Scitovski, M. Benšić, Strategije otplate zajma, Financijska teorija i praksa (1994), 15-26
M. Šilac, Otplata zajma varijabilnim anuitetima, Economic analysis and worker's management 23/2 (1989), 185-197
Refereed Proceedings
P. Taler, S. Hamedović, M. Benšić, E.K. Nyarko, LeArEst - The Software for Border and Area Estimation of Data Measured with Additive Error, 59th International Symposium ELMAR-2017, Zadar, 2017, 259-263
This paper proposes a solution to the problem of estimating object dimensions from a noisy image. Image noise can be produced by the physical processes of imaging, or can be caused by the presence of some unwanted structures (e.g. soft tissue captured in X-ray images of bones). We suppose that the data are drawn from uniform distribution on the object of interest, but contaminated by an additive error having normal or Laplace distribution. The software for border estimation of an object registered in such manner has been developed, and brief description of its key functions is given. The software is able to estimate the borders of an object on a given line that intersects it, as well as to estimate its area. Its input data may be numerical data, as well as images in JPEG format.
M. Zekić-Sušac, N. Šarlija, M. Benšić, Insolvency prediction by neural networks, 12th International Conference on Operational Research, Pula, Croatia, 2009, 175-188
M. Benšić, D. Jankov Maširević, Parameter estimation for a three-parameter Weibull distribution - a comparative study, 12th International Conference on Operational Research, Pula, Croatia, 2008, 159-164
N. Šarlija, M. Benšić, M. Zekić-Sušac, A neural network clasification of credit applicants in consumer credit scoring, IASTED International Conference on Artificial Intelligence and Applications, part of the 24th Multi-Conference on Applied Informatics, 2006, Innsbruck, 2006, 205-210
N. Šarlija, M. Benšić, M. Zekić-Sušac, Modeling customer revolving credit scoring using logistic regression, survival analysis and neural networks, 7th WSEAS International Conference on Neural Networks, Cavtat, 2006, 164-169
The aim of the paper is to discuss credit scoring modeling of a customer revolving credit depending on customer application data and transaction behavior data. Logistic regression, survival analysis, and neural network credit scoring models were developed in order to assess relative importance of different variables in predicting the default of a customer. Three neural network algorithms were tested: multilayer perceptron, radial basis and probabilistic. The radial basis function network model produced the highest average hit rate. The overall results show that the best NN model outperforms the LR model and the survival model. All three models extracted similar sets of variables as important. Working status and client's delinquency history are the most important features for customer revolving credit scoring on the observed dataset.
S. Pfeifer, M. Benšić, N. Šarlija, An insight into using entrepreneurship intentions as predictor of entrepreneurial behavior, The Entrepreneurship - Innovation - Marketing Interface, Karlsruhe, 2005, 89-108
Entrepreneurial behavior is vital for economic recovery and growth. Potential for business start-ups or potential for wealth creating behavior is a prerequisite of any economic development. Permanent supply of the new venture creators or entrepreneurship career seekers can be supported through identification of what and how many influences are critical for the emergence of the entrepreneurial intentions. The research examines well documented and theoretically consistent intent approach, while methodologically it tests the robustness of the structural equation model. This model enables measurement of direct and indirect influences on entrepreneurial intentions and behavior. A sample of Croatian final year students of economics is used to investigate intentions of the future labor force for choosing the entrepreneurial self-owned business start-up as a career option. Consistent with other studies examining the influence of the intentions on behavior, it was hypothesized that the more favorable the attitude and perceived norms about entrepreneurial behavior are and the greater the perceived feasibility (control) of the entrepreneurial act is, the stronger the intention to perform the behavior will be. Furthermore, the greater the degree to which the behavior can be controled, the greater the influence of the knowledge about the intentions to perform the defined behavior. The research provides several benefits. In particular, recognition of the critical antecedents of the entrepreneurial behavior and the empirical testing of the participant prior to her/his choice enables policy makers to determine propensity of entrepreneurial potential transition into an entrepreneurial performance. The findings of the study provide evidence that the university environment could play a more important role in entrepreneurship encouragement and support. University programs designers would get a clear message about the perception and expectations that future labor force has about entrepreneurship ; therefore, more efficient program curricula could be implemented in the university education. Empirical testing of the participant prior to his or her career choice enables one to determine whether perception of the effects of government entrepreneurship reforms are ecouraging or not, and whether the overall community is fostering or hindering entrepreneurism. Another benefit of this study would be to ensure that supportive measures hit the right targets in the process of using the entrepreneurial vehicle as the crucial one for economic development.
The aim of this paper is to discuss credit scoring modeling of a customer revolving
credit depending on customer application data and transaction behavior data
influenced by specific economic conditions that exist in Croatia. Since Croatia is a
country in transition with war consequences, changes in political, institutional and
social systems and, above all, with specific economic conditions characterized by
slow economy, a high unemployment rate and a relatively low personal income, it is
assumed that this influences credit behavior of customers and, as a consequence, a
composition of credit scoring models. For instance, it has been shown that a small
business credit scoring model developed in Croatia is influenced by economic
conditions. The data set for our research consisted of 50,000 customer accounts
(application data and transaction data) in Croatia over the period of 12 months. We
have developed a logistic credit scoring model and a survival-based credit scoring
model in order to assess the relative importance of different variables in predicting
default as well as profitability of a customer. The paper analyzes influences of
economic conditions on credit scoring modeling.
N. Šarlija, M. Benšić, Z. Bohaček, Multinomial model in consumer credit scoring, 10th International Conference on Operational Research KOI 2004, Trogir, 2004, 193-202
Credit scoring is a process of determing how likely applicants are to default withe their repayments. It has been used in consumer lending for more then three decades. Data sample of consumer credits used in this research is divided into three groups - good, poor and bad. The aim of this paper is to explore the possibility of using poor applicants in developing a credit scoring model. In ordet to accomplish this, multinomial models have been developed. Also, for the purpose of obtaining results as good as possible, three binomial models have been built. These models are compared according to significant variables and classification accuracy after which the best model is extracted.
M. Zekić-Sušac, N. Šarlija, M. Benšić, Small business credit scoring: A comparison of logistic regression, neural network and decision tree models, 26th International Conference on Information Tehnology Interfaces, Cavtat, Dubrovnik, 2004, 265-270
The paper compares the models for small business credit scoring developed by logistic regression, neural networks, and CART decision trees on a Croatian bank dataset. The models obtained by all three methodologies were estimated ; then validated on the same hold-out sample, and their performance is compared. There is an evident significant difference among the best neural network model, decision tree model, and logistic regression model. The most successful neural network model was obtained by the probabilistic algorithm. The best model extracted the most important features for small business credit scoring from the observed data.
S. Mihaljević, M. Katičić, B. Dmitrović, I. Karner, A. Včev, V. Borzan, S. Canecki, Ž. Vranješ, V. Prus, J. Barbić, M. Benšić, The Influence of Stomach Mucosa Morphological Changes on Somatostatin Cell Number in Atrum Mucosa, 11th International Conference on Ulcer Research, Dubrovnik, 2003, 307-312
M. Benšić, Z. Bohaček, N. Šarlija, M. Zekić-Sušac, Neural Network-and-Logit-Based Modeling Strategy for Small Business Credit Scoring, International Conference Forecasting Financial Markets : Advances for Exchange Rates, Interest Rates and Asset Management, London, 2002
Credit scoring has been so far investigated using both logistic regression and neural networks mostly for the purpose of comparing the accuracy of two methods (Desai et al, 1997 ; West, 2000), using commonly recognized credit scoring models. However, due to specific characteristics of small business loans, the importance of selecting different variables from other company loans is emphasized by practitioners and researchers (Feldman, 1997). Specific economic conditions, especially in transitional countries, that also influence model effectiveness, emphasize a close relationship between methodology accuracy and variable selection. This paper investigates such relationship by performing a neural network forward cross-validation modeling strategy based on hit rates, logit univariate and logit forward selection analysis. Comparing the accuracy of both methods, the system is able to extract the best model for the given data. Tested on a Croatian small business loans dataset, it proposes the set of important features for credit scoring in that specific economic environment.
M. Benšić, Đ. Borozan, Nonlinearity in the foreign-exchange market: the application of close returns analysis, 7th International Conference on Operational Research KOI 1998, Rovinj, 1999, 311-316
M. Benšić, R. Scitovski, Određivanje intervala povjerenja za neke specijalne nelinearne regresije, 6th International Conference on Operational Research KOI 1996, Rovinj, 1996, 87-92
U radu se analiziraju rezultati primjene metode linearizacije na određivanje intervala i područja povjerenja za nepoznate parametre logističke i 3-parametarske eksponencijalne regresije.
Others
M. Benšić, G. Benšić, Kamatni račun, Osječki matematički list 11/2 (2011), 113-126
U radu je prezentiran jednostavan metodološki pristup u obradi kamatnog računa temeljen samo na dvjema ključnim formulama te razumijevanju i primjeni principa jednostavnog i složenog ukamaćivanja. Pokazano je da izvedene formule za jednostavno i složeno ukamaćivanje određuju funkciju rasta kapitala po razlicitim modelima kapitalizacije što ukazuje na potrebu strogog poštovanja odabranog principa ukamaćivanja u svakom pojedinom problemu. Navedeni su neki primjeri iz hrvatskih udžbenika matematike u kojima se o tome ne vodi računa te se u jednom zadatku koriste oba principa. Ovakvi i slični zadaci mogu biti uzrok zbrci koju u glavi imaju učenici kada pokušavaju naučiti kamatni račun.
M. Benšić, Jedna igra na sreću, Osječki matematički list 1 (2001), 5-8
M. Benšić, M. Crnjac, Optimalan Lp procjenitelj nepoznatih parametara regresijskog modela, Ekonomski vjesnik 10/1 (1998), 71-74
M. Benšić, Asymptotic distributions of least square estimations in a regression model with singular errors, Mathematical Communications 1/2 (1996), 33-38
M. Šilac-Benšić, Problemi optimalnog izbora, Ekonomski vjesnik 2/5 (1992), 327-328
R. Scitovski, M. Šilac, D. Francišković, Problemi i nesporazumi u primjeni financijske matematike, Privreda 33 (1989), 243-257
Package provides methods for estimating borders of uniform distribution on the interval (one-dimensional) and on the elliptical domain (two-dimensional) under measurement errors. For one-dimensional case, it also estimates the length of underlying uniform domain and tests the hypothesized length against two-sided or one-sided alternatives. For two-dimensional case, it estimates the area of underlying uniform domain. It works with numerical inputs as well as with pictures in JPG format.
Standardizing the empirical distribution function yields a statistic with norm
square that matches the chi-square test statistic. To show this one may use the
covariance matrix of the empirical distribution which, at any finite set of points, is
shown to have an inverse which is tridiagonal. Moreover, a representation of the
inverse is given which is a product of bidiagonal matrices corresponding to a representation
of the standardization of the empirical distribution via a linear combination
of values at two consecutive points. These properties are discussed also in the context
of minimum distance estimation
Projects
Leader of the following projects
University J.J. Strossmayer in Osijek:
Exploration of optimization and estimation properties of Generalized method of moments and Nonlinear least squares (2016)