## Department of Mathematical Statistics

#### Phd Students:

The Section of Mathematical Statistics was established on April 1, 2004, as a unification of Section of Mathematical Statistics and Its Applications and Section of Applied Probability. For brief history of the two last mentioned Sections see below.

The following researchers have been members of the Section of Mathematical Statistics: Tadeusz Bednarski, Przemys³aw Biecek, Katarzyna Danielak, Tadeusz Inglot, Teresa Ledwina, Ryszarda Rempa³a, Tomasz Rychlik, Grzegorz Wy³upek and Ryszard Zieliñski. The main scientific interests have been focused on the following topics:

• asymptotic optimality and efficiency of tests,
• characterizations by means of ordered statistical data,
• data driven tests,
• general methodology for robust estimation and testing,
• methods of model selection,
• multiple testing procedures,
• optimal bounds on statistical functionals,
• optimal nonparametric quantile estimation,
• sequential decision problems for controlling production, distribution and inventory processes.

#### Selected publications:

• T. Bednarski (2004). Robust estimation in the generalized Poisson model, Statistics 38, 149-159.
• M. Bogdan, F. Frommlet, P. Biecek, R. Cheng, J.K. Ghosh, R.W. Doerge (2008). Extending the modified Bayesian Information Criterion (mBIC) to dense markers and multiple interval mapping, Biometrics 64, 1162-1169.
• K. Danielak (2005). Distribution-free bounds for expectations of increments of records, J. Statist. Plann. Inf. 133, 239-247.
• T. Inglot, T. Ledwina (2004). On consistent minimax distinguishability and intermediate efficiency of Cramér-von Mises test, J. Statist. Plann. Inf. 124, 453-474.
• T. Inglot, T. Ledwina (2006). Data driven score tests for a homoscedastic linear regression model: asymptotic results, Probab. Math. Statist. 26, 41-61.
• J. Navarro, T. Rychlik (2007). Reliability and expectation bounds for coherent systems with exchangeable components, J. Multivar. Anal. 98, 102-113.
• W. Niemiro, R. Zieliñski (2007). Uniform asymptotic normality for the Bernoulli scheme, Appl. Math. 34, 215-221.
• P. Jaworski, T. Rychlik (2008). On distributions of order statistics for absolutely continuous copulas with applications to reliability problems, Kybernetika 44, 757--776.
• G. Wy³upek (2010). Data driven k sample tests, Technometrics 52, 107-123.
• T.Ledwina, J. Mielniczuk (2010). Variance function estimation via model selection, Appl. Math. 37, 387-411.
• J. Navarro, T. Rychlik (2010). Comparisons and bounds for expected lifetimes of reliability systems, Europ. J. Operational Res. 207, 309-317.
• K. Jasiñski, T. Rychlik (2012). Bounds on dispersion of order ststistics based on dependent symmetrically distributed random variables, J. Statist. Plann. Inf. 142, 2421-2429.
• T. Ledwina, G. Wy³upek (2012). Two-sample test against one-sided alternatives, Scand. J. Statist. 39, 358-381.

#### Seminars:

• Asymptotic Statistics (T. Inglot, T. Ledwina; Wroc³aw),
• Mathematical Statistics and other Probabilistic Applications (M. Mêczarski, J. Mielniczuk - until June 2010, P. Jaworski - since October 2010, T. Rychlik; Warszawa),
• Mathematical Statistics (W. Niemiro, T. Rychlik; Toruñ, since 2008),
• Statistics in Medicine (P. Biecek; Wroc³aw, 2007-2008).

### The Section of Mathematical Statistics and Its Applications

The Section of Mathematical Statistics and Its Applications stemmed from the Department of Applied Mathematics of the State Institute of Mathematics, which was established in November 20, 1948. In 1949 the Department consisted of six sections. Two of them: the General Section of Applications, headed by Hugo Steinhaus, and the Section of Technical Applications, directed by Jan Mikusiñski, were located in Wroc³aw. These two sections formed later on the Section of Applications to Biology, Economics and Technology, directed by Hugo Steinhaus until his retirement in 1960. Julian Perkal (1960-1965) and Stefan Zubrzycki (1965-1968) were later directors. The Section has played a very important role in the Wroc³aw scientific community, in particular through many collaborations with various professionals and educational work in applied probability and statistics. The journal Zastosowania Matematyki (the subtitle Applicationes Mathematicae added in 1965), founded by Hugo Steinhaus in 1953, with help of Jan Oderfeld, was a significant platform for the activity of the group. This activity was further supported by Julian Perkal, who founded the Listy Biometryczne journal in 1964. The journal is currently published as Biometrical Letters.

From 1968 to 1991 the Section was headed by Witold Klonecki and was renamed to the Section of Mathematical Statistics and Its Applications in 1973. At that time, the main activity was focused on PhD studies in modern statistical methodology, initially with the kind aid of colleagues from Wroc³aw University. Members of the group participated very actively in the organization of a series of conferences, beginning with one in Wis³a in 1973. Many of these conferences were international. Symposium to honour Jerzy Neyman (1974), the European Meeting of Statisticians (1981) and Banach Center Semester on Nonparametric and Robust Methods (1984) are prominent examples. Witold Klonecki was also one of the founders of the Probability and Mathematical Statistics journal (1980).

Since 1991 until 2004 the Section was headed by Tadeusz Bednarski. Apart from research in mathematical statistics, much attention was focused on direct applied work.

#### Selected publications:

1. K. Florek, J. £ukaszewicz, J. Perkal, H. Steinhaus, S. Zubrzycki (1951). Sur la liaison et la division des points d'un ensemble fini, Colloq. Math. 2, 282-285.
2. H. Steinhaus (1953/1954). On establishing paternity [in Polish], Zastosow. Mat. I, 67-82.
3. A. Ziêba (1953/1954). Elementary theory of pursuit [in Polish], Zastosow. Mat. I, 273-298.
4. S. Drobot, M. Warmus (1954). Dimensional analysis in sampling inspection of merchandise, Dissertationes Math. [Rozprawy Mat.] V.
5. S. G³adysz, A. Rybarski (1954/1956). On modelling three-dimensional fields by a plane field of current [in Polish], Zastosow. Mat. II, 150-160.
6. S. Zubrzycki (1954/1956). On the optimal method of water meter acceptance [in Polish], Zastosow. Mat. II, 199-209.
7. A. Huskowska (1954/1956). On the accuracy of some natural science measurements, [in Polish], Zastosow. Mat. II, 426-430.
8. J. Battek, J. Perkal (1956/1958). Quality and shape of forest stands [in Polish], Zastosow. Mat. III, 285-306.
9. J. £ukaszewicz, M. Warmus (1956). Numerical and Graphical Methods [in Polish], PWN, Warszawa.
10. S. Paszkowski i M. Warmus (1956). On some mathematical method in anthropology [in Polish], Przegl±d Antropologiczny XXII, 627-650.
11. H. Steinhaus (1957). The problem of estimation, Ann. Math. Statist. 28, 633-648.
12. S. Zubrzycki (1957). On estimating gangue parameters [in Polish], Zastosow. Mat. III, 105-153.
13. E. Marczewski, H. Steinhaus (1958/1959). On the systematic distance of biotopes, [in Polish], Zastosow. Mat. IV, 195-203.
14. H. Steinhaus, S. Trybu³a (1958/1959). Measurement by successive comparison [in Polish], Zastosow. Mat. IV, 204-212.
15. H. Steinhaus, K. Urbanik (1959). Poissonsche Folgen (Leon Lichtenstein zum Gedáchtnis), Math. Zeitsch. 72, 127-145.
16. T. Dalenius, J. Hájek, S. Zubrzycki (1961). On plane sampling and related geometrical problems, in : Proc. Fourth Berkeley Symp. Math. Statist. Probab. 1, 125-130.
17. S. Trybu³a (1961). On the paradox of three random variables, Zastosow. Mat. V, 321-332.
18. A. Bartkowiakowa, B. Gleichgewicht (1962). On the syllable characteristic of the Polish language [in Polish], Zastosow. Mat. VI, 309-319.
19. A. Krzywicki, A. Rybarski (1962). On a linearization of an equation of an elastic rod, Zastosow. Mat. VI, 321-332.
20. J. Perkal (1962). A design for a genetic test [in Polish], Zastosow. Mat. VI, 257-285.
21. H. Steinhaus (1963). Probability, credibility, possibility, Zastosow. Mat. VI, 341-361.
22. L. Zubrzycka (1963). On the adaptation of the typewriter keyboard to the structure of the language [in Polish], Zastosow. Mat. VI, 419-439.
23. K. Florek (1964). On a certain method of graphical integration and graphical harmonic analysis, Zastosow. Mat. VII, 353-370.
24. B. Kopociñski, L. Zubrzycka (1964). Remark on division of systems of features into harmonized subsystems [in Polish], Zastosow. Mat. VII, 317-321.
25. S. Zubrzycki (1966). Explicit formulas for minimax admissible estimators in some cases of restrictions imposed on the parameter, Zastosow. Mat. IX, 31-52.
26. B. Bednarek-Kozek (1973). On estimation in the multidimensional Gaussian model, Zastosow. Mat. XIII, 511-520.
27. H. Drygas, J. Srzednicka (1976). A new result on Hsu's model on regression analysis, Bull. Acad. Polon. Sci., Sér. Sci. Math. Phys. Astronom. 24, 1133-1136.
28. S. Gnot (1976). The mean efficiency of block designs, Math. Operationsforsch. Statist. 7, 75-84.
29. W. Klonecki (1977). Optimal C($\alpha$) tests for homogeneity, in : Proceedings of the Symposium to honour Jerzy Neyman (Warsaw 1974), PWN, 161-175.
30. A.S. Kozek (1977). Efficiency and Cramér-Rao type inequalities for convex loss functions, J. Multivariate Anal. 7, 89-106.
31. M. Musiela (1977). Sequential estimation of parameters of a stochastic differential equation, Math. Operationsforsch. Statist. Ser. Statist. 8, 483-498.
32. T. Ledwina (1978). Admissible tests for exponential families with finite support, Math. Operationsforsch. Statist. Ser. Statist. 9, 105-118.
33. R. Zmy¶lony (1980). A characterization of best unbiased estimators in the general linear model, in : Mathematical Statistics and Probability Theory (Proc. Sixth International Conf. Wis³a 1978), Lecture Notes in Statist. 2, Springer-Verlag, 365-378.
34. W. Wawrzyniak (1981). A characterization of minimum variance unbiased estimators in the general linear model with restrictions on parameter space. Math. Operationsforsch. Statist. Ser. Statist. 12, 456-477.
35. T. Bednarski (1982). Binary experiments, minimax tests and 2-alternating capacities, Ann. Statist. 10, 226-232.
36. A.S. Kozek (1982). Towards a calculus for admisibility, Ann. Statist. 10, 825-837.
37. S. Gnot, J. Kleffe, R. Zmy¶lony (1985). Nonnegativity of admissible invariant quadratic estimates in mixed linear models with two variance components, J. Statist. Plann. Inference 12, 249-258.
38. J. Zabrzeski, R. Zmy¶lony (1987). Experimental description of the rate of platinum and rodium losses in the process of ammonia oxidation, Appl. Catalisis 35, 13-22.
39. R.H. Farrel, W. Klonecki, S. Zontek (1989). All admissible linear estimators of the vector of gamma scale parameters with application to random effects models, Ann. Statist. 17, 268-281.
40. W. Klonecki, S. Zontek (1992). Admissible estimators of variance components obtained via submodels, Ann. Statist. 20, 1454-1467.
41. T. Bednarski (1993). Robust estimation in the Cox regression model, Scand. J. Statist. 20, 213-225.
42. T. Bednarski, S. Zontek (1996). Robust estimation of parameters in mixed unbalanced models, Ann. Statist. 24, 1493-1510.
43. K. Drosik, A. Michalski, S. Sadowski, A. Tukiendorf, R. Zmy¶lony (1998). Neoplasmatic Disease Incidence in Opole Province, Wydawnictwo Uniw. Opolskiego.
44. T. Inglot, W.C.M. Kallenberg, T. Ledwina (1998). Vanishing shortcoming of data driven Neyman's tests, in : Asymptotic Methods in Probability and Statistics, A Volume in Honour of Miklós Csörgõ, North-Holland, 811-829.
45. T. Bednarski, W. Florczak (1999). On a local uniform bootstrap validity, Statist. Neerl. 53, 111-121.
46. W.C.M. Kallenberg, T. Ledwina (1999). Data driven rank tests for independence, J. Amer. Statist. Assoc. 94, 285-301.
47. A. Janic-Wróblewska, T. Ledwina (2000). Data driven rank test for two-sample problem. Scand. J. Statist. 27, 281-297.
48. T. Inglot, T. Ledwina (2001). Intermediate approach to comparison of some goodness-of-fit tests, Ann. Inst. Statist. Math. 53, 810-834.
49. G.R. Ducharme, T. Ledwina (2003). Efficient and adaptive nonparametric test for the two-sample problem, Ann. Statist. 31, 2036-2058.
50. T. Inglot, W.C.M. Kallenberg (2003). Moderate deviations of minimum contrast estimators under contamination, Ann. Statist. 31, 852-879.

### Section of Applied Probability

The Section of Applied Probability originated in a long process rooted in the Department of Applied Mathematics of the State Mathematical Institute, founded in 1948. Two Warsaw's sections of this department were forerunners of it. Namely, the Section of Statistical Quality Control and the Actuarial Section. In accordance with the evolving interests of the members, the Section of Statistical Quality Control was renamed to the Industrial Applications Section in 1962. In the period from 1952 to 1970 these sections were directed by Jan Oderfeld, with short break in 1959 when Czes³aw Rajski was in charge of it. At that time, the research was mainly focused on statistical quality control, operations research and reliability theory. The Actuarial Section evolved into the Section of Mathematical Statistics. In 1951, Oskar Lange was put in charge of the section and he did it until 1958. In 1958-1960 Marek Fisz was the leader of it while later on the section was headed by Wies³aw Sadowski. Interests of the group were focused on statistics, decision making and econometry.

These two evolving groups were the predecessors the Applied Probability Section, formally founded in 1972.

The section was headed by Robert Bartoszyñski, Ryszard Zieliñski and Tomasz Rychlik in the periods 1972-1985, 1985-2002 and 2002-2004, respectively. In the early seventies, the scientific interests of the group switched to probabilistic modelling (mostly, but not only, of biological phenomena), controlled Markov chains, stochastic approximation and random search. In recent years, various issues of mathematical statistics, including robust statistics, fixed precision estimation and moment bounds have become major focuses of the section's research.

The section was active in the organization of courses in applied mathematics, enhancing education in a wide spectrum of applied probability. Also, starting from 1971, Eugeniusz Fidelis annually organized the very popular National Conferences on Applications of Mathematics. Members of the group organized many international conferences in statistics, including Banach Center Semesters on Mathematical Statistics (1976) and Sequential Methods in Statistics (1981), among others.

#### Selected publications:

• J. Oderfeld (1951). On the dual aspects of sampling plans. Colloq. Math. 2, 89-97.
• M. Fisz (1954). Probability Theory and Mathematical Statistics (first edition; in Polish), PWN.
• T. Czechowski, W. Sadowski, W. Zasêpa (1954/1956). On determining the safety factor [in Polish], Zastosow. Mat. 2, 190-198.
• W. Sadowski (1954/1956). On a non-parametric test of comparing dispersions [in Polish], Zastosow. Mat., 2, 161-171.
• C. Rajski (1954/1956). On the verification of hypotheses concerning two populations consisting of items marked by attributes [in Polish], Zastosow. Mat., 2, 179-189.
• M. Fisz, K. Urbanik (1956). Analytical characterization of the composed non-homogeneous Poisson process, Stud. Math. 15, 328-336.
• M. Fisz (1958). A limit theorem for empirical distribution functions, Stud. Math. 17, 71-77.
• O. Lange (1959). Introduction to Econometrics (first edition; in Polish), PWN.
• J. £ukaszewicz, Sadowski, W. (1960/1961). On comparing several populations with a control population . Zastosow. Mat. 5, 309-320.
• R. Bartoszyñski (1961). A characterization of the weak convergence of measures. Ann. Math. Statist. 32, 561-576.
• J. Oderfeld, E. Pleszczyñska (1961). A linear estimate of the mean deviation in normal population [in Polish], Zastosow. Mat. VI, 111-117.
• E. Fidelis, J. Oderfeld (1962). Two-step control taking into account the measurement errors [in Polish], Zastosow. Mat. VI, 249-256.
• W. Rudzki (1962). Estimation of mean properties of shapeless products [in Polish], Zastosow. Mat. VI, 235-248.
• R. Bartoszyñski (1967). Branching processes and the theory of epidemics, Proc. Fifth Berkeley Symp. on Math. Statist. Probab. IV, 615-618.
• E. Pleszczyñska (1973). Trend estimation problems in time-series analysis. Dissertationes Math. (Rozprawy Mat.) 104.
• R. Zieliñski (1973). A new class of estimators with an application to statistical quality control. Zastosow. Mat. XIII, 279-300.
• R. Bartoszyñski (1974). On a metric structure derived from subjective judgements: scaling under perfect and imperfect discrimination, Ecomometrica 42, 55-71.
• R. Bartoszyñski (1975). On risk of rabies, Math. Biosci. 24, 357-377.
• R. Zieliñski (1977). Global stochastic approximation. Dissertationes Math. (Rozprawy Mat.) 142.
• R. Bartoszyñski, W.J. Bühler (1978). On survival in hostile environment, Math. Biosci. 38, 293-301.
• R. Bartoszyñski, B.W. Brown, C.M. McBridge, J.R. Thompson (1981). Some nonparametric techniques for estimating the intensity function of a cancer related to nonstationary Poisson process. Ann. Statist. 9, 150-160.
• R. Zieliñski (1983). Robust statistical procedures: a general approach, in: Stability Problems for Stochastic Models'' (V.V. Kalashnikov and V.M. Zolotarev, eds.), Lecture Notes in Mathematics 982, Springer-Verlag, Berlin, 283-295.
• B. Go³dys, M. Mêczarski, R. Zieliñski (1986). An asymptotic fixed-precision confidence interval for the minimum of a quadratic regression function. Probab. Math. Statist. 7, 7-11.
• J. Koronacki, W. Wertz (1988). A global stopping rule for recursive density estimators. J. Statist. Plann. Inference 20, 23-39.
• J. Koronacki (1989). Stochastic Approximation. Optimization Methods under Random Conditions [in Polish]. WNT, Warsaw.
• B. Go³dys (1990). Regularity properties of solutions to stochastic evolution equations, Colloq. Math. 58, 327-338.
• R. Rempa³a (1991). Forecast horizon in a dynamic family of one-dimensional control problems, Dissertationes Math. (Rozprawy Mat.) 315.
• R. Zieliñski (1991). Fixed precision estimation in the Blum-Rosenblatt time series. Amer. J. Management Sci. 11, 233-239.
• T. Rychlik (1993). Bounds for expectation of L-estimates for dependent samples, Statistics 24, 1-7.
• L. Gajek, D. Zagrodny (1995). Geometric mean value theorems for the Dini derivative. J. Math. Anal. Appl. 191, 56-76.
• W. Niemiro (1995). Estimation of nuisance parameters for inference based on least absolute deviations. Zastosow. Mat. 22, 515-529.
• W. Niemiro, P. Pokarowski (1995). Tail events of some nonhomogeneous Markov chains, Ann. Appl. Probab. 5, 261-293.
• L. Gajek, T. Rychlik (1996). Projection method for moment bounds on order statistics from restricted families. I. Dependent case. J. Multivar. Anal. 57, 156-174.
• L. Gajek, T. Rychlik (1998). Projection method for moment bounds on order statistics from restricted families. II. Independent case. J. Multivar. Anal. 64, 156-182.
• R. Zieliñski (1998). Uniform strong consistency of sample quantiles. Statist. Probab. Lett. 37, 115-119.
• T. Rychlik (2001). Projecting Statistical Functionals. Lecture Notes in Statistics 160, Springer-Verlag, New York.
• S. Bylka, R. Rempa³a (2003). Selected Problems in the Mathematical Inventory Theory [in Polish], Akademicka Oficyna Wydawnicza Exit, Warsaw.