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Estimation of asymptotic variance of stochastic processes
Štěrbová, Martina ; Prokešová, Michaela (referee) ; Pawlas, Zbyněk (advisor)
In this thesis we consider methods for estimating asymptotic variance of the sample mean for stationary stochastic sequences. We compare methods based on spectral density estimator at the origin with subsampling methods. These methods are parametrized by batch size. We present methods for estimating optimal batch size which minimizes the mean squared error. Special estimators are introduced for the case of Markov chains. We also mention recursive estimation which is suitable when the observations come consecutively. The the last but one section contains experiments of all methods considered in this thesis, for simulated data we compare individual methods by their relative mean squared error. In the last section we apply selected methods on real data.
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Distance between randomly chosen points
Bartoníčková, Ilona ; Pawlas, Zbyněk (referee) ; Anděl, Jiří (advisor)
In the present work we study expected value of distance between two points, which are chosen randomly and independently in given sets. This problem is often associated with travel distance between two cities of the fixed shape. Cities are mostly considered as circles or rectangles for simplification. The work deals with two separate problems. The first of them is introduced in chapter 2. Points are chosen randomly in two concentric circles. The described method uses the definition of geometric probability. Chapter 3 describes the same problem for two disjoint rectangles. The solution is based on transformation of variables. The limit case in one dimension is then obtained in chapter 4. The work is supplemented by numerous simulations.
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Statistical applications of urn models
Navrátil, Radim ; Omelka, Marek (referee) ; Pawlas, Zbyněk (advisor)
This work shows various applications of urn models in practice. First, basic properties of the occupancy distribution are derived together with its asymptotic approximation. This model is applied and generalized in the theory of database systems for records search from a given database. An application to random texts is mentioned, namely the computation of the expected number of missing and common words in random texts. There are presented exact formulas, their asymptotic approximations and the approximations via occupancy distribution. Then, some urn models, which are used in the randomized response theory for finding out respondents' answers to sensitive questions, are described. These models are compared according to their accuracy and respondents' goodwill to answer. Finally, two non-parametric tests of empty boxes are derived, one for the hypothesis whether a random sample comes from a given population and the second for the hypothesis whether two independent random samples come from the same population. The powers of these tests are compared with commonly used tests for these hypotheses.
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Segment point processes
Honzl, Ondřej ; Beneš, Viktor (referee) ; Pawlas, Zbyněk (advisor)
Naz-ev prace: Bodove proeesy usecek Aut.or: Ondrej Ilonzl Katedra: Katedra. pravdepodobnosti a matematicke statist,iky Vodonci bakalafske praee: RNDr. Zbynek Pawlas, Ph.D. e-mail vedouciho: zbynek.pawlas'O'mfl'.cmn.cz Abstrakt: Prace obsahnje strucny uvod do teoric bodovych procesii na nplnem se- parabilnim lokalne koutpakt iiiiu metrickem prost.oru. Hamcove je ziuinen si>ccii'ilni I>fipad st,acionariiilio ])roc:csu kouipaktiiich ninozin. Dale sc jiran1 vice1 /aincfujo na, Poissonuv prort\ usccek sc /nainyni ro/dcMrnim typickrho /rnu. V roviniirin pfi])aclo pak ukazujc n'i/nr odliady dolkovr int.i'ii/ily I'oissonova jiroccsu usccck, kU're jsou drfinovany na /aklade udaju /iskanych v okuc poxorovani. Hlavnim zajinoin prace st1 stava porovnavani tcclito odhadu die jujich rozpt.ylu. Cilem to- hoto sroviiavani ina byi stanovoni niezc \vlikoHti okna, klcra. fika, dokud jc Icpsi pou/it slozitojsi odhad a odkdy je naopak ro/ninno pouzit odliad. jclioz vvpocot jo snazsi, ale kl.cry pft'ilpoklada., zc inatiie vice iniormaci u po/orovanein JJI'OCCHU. Klicova slova: I'oissonuv i>roces. hodovy proces usccek, odhad delkove intonzit.y Title: Segment point Autlior: Ondrej Ilonzl Deijartuient: DepartiiieiiL ol'Prol)a.l>ilit.y and Mathcinalical Statistics Supervisor: KNDr. Zbyuek Pawlas, Ph.I). Supervisor's (^niail address:...
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Structural properties of graphs---probabilistic and deterministic point of view
Hladký, Jan ; Pawlas, Zbyněk (referee) ; Šámal, Robert (advisor)
We study bipartite subgraphs of a random cubic graph in the thesis. We show, that an edge-maximum bipartite subgraph of a random cubic graph on n vertices has asymptotically almost surely less then 3 2 · 0.9351n edges. We also show that the number of vertices of a vertex-maximum induced bipartite subgraph of a random cubic graph lies within interval [0.75n; 0.9082n]. To obtain the lower bound we design a randomized algorithm for finding a large induced bipartite subgraph of a random cubic graph. We discuss consequences of the results for graph homomorphisms, namely for Pentagon Conjecture posed by Nešetřil.
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Firm Valuation
Baran, Jaroslav ; Hurt, Jan (referee) ; Pawlas, Zbyněk (advisor)
Xazev prace: Ocenovani pndnikii Aulor: Jaroslav Ha ran Katedra (iistav): Katedra pravdepodobnost i a mateniat icke statisliky Vedouci bakalarske pra.ce: HNDr. Zbynek Pawlas. Ph.D. e-mail vedoucfho: pawlas'.^'karlin.mtf.cnni.c/, Abslrakt: C'ileni prace je seznamif c.tenafe sc zfddadnhni a nejpon/iVanejsimi nastroji pro ohodnoceni akcii akciovycli spolecnosti. Druha cast. pra.ce so zabyva nitr'todaini odvoxt'iii a odliadu pai'ainetru pou/itych v modnlccli ororiovani. Jsiiu pr(i/('iit,ovaiiy zakladnf niodcly otvnnvani jako in^lnda sourasnr hodnoly budouci'cli prijniii a rclalivtii Dci-fiuvani, ktcrr \wr v uvahu tr/ui ccny srovnatelnych firciti. N'a konci pnicc jc jirakticky pri'klad occnrni spulrrnusti Nukia. Prace ina ypis in- Ibriuativiii chara.kU'i' a, ji-.jnn cilnn uoni prusadil konkirl.ni morlcl. ale srznainit, ctcnafc sc '/iikladiiiini prinripy ixTfiovani a poukaxat na vyhucly a ncvyhody uvc- drnych inodcln. Investor se nakoiiec nefidi ponze ]>odle niodehi (K-in'iovani. ale /a- iijine poatoj i diky oritatiiini /drojiiin inlorniaci. jako jsou media, vyrorni zpravy spcileeiiostf. /.pravy analytiku. vylih'dky trim atd. Klirova slova: ocenovani. soneasna liodnota, rust, urokova infra Title: Irinu valuation Author: .laroslav I'ia.ran De]jarlment: Depa,rlinent ol Proba.bility and Ma.thcnia,! ical Staiistics...
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Spatial modelling
Voldán, Adam ; Pawlas, Zbyněk (referee) ; Beneš, Viktor (advisor)
Na/,ev praoe: Prost.orovc modolovam Antor: Adam Voldan Katedra: Katodra pravdepodobnosti ;v matematicke stal.istiky Vodouci bakalafske pracc: Proi. RNDr. Viktor Benes DrSc. e-mail vedouciho: Viktor.Bones'ohnff. emii.cz Abstrakt: V pfedlozene praci je stndovan nahodnv bodovy procos, konkretne permanent procos. Podrobne je probrana tcoric nahodnych bodovych pro- ce.su danych husLotou vzhlodeiu k Pois.sonove procesn. Sainotny proces je si- nuilovan inctodou Markov chain Mont.o Carlo, poinoci Mrtropolis-Hastingso- va algoritinu pro pcvuy pocct boili'i. Tento al^oi'itiuub ju naprogramovau v jazyce Pascal a vystupy toholo tnodoln json flalc vyhodnocovany v prograinn R-Spatstat, st.udovaiKj bylo pfodcvsiiu prostorove roznn'wteni bodii poinoci indexn dispor/c Kh'coxra .slova: Mc:!ro])olis-Ha,sting,siiv algoritnin.s. Poissc^nnv bodovy procos. pcniument procea Title: Spatial modelling Avithor: Adam VokUin De]jartment: Department of Probability and Mathoniatica.l Statistics Supervisor: Prof. RNDr. Viktor Bonos DrSc. Supervisor's e-mail address: Viktor.Bene.s^Cnifr.euni.e/, Abstract: In the present work we study stochastic point processes, especi- ally the permanent process. We introduce the theory of the stochastic point processes given by the density with respect, to the Poisson process. The per- manent process is...
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