Šiame darbe yra nagrinejama baigtinė populiacija iš kurios išrinkta sluoksninė imtis su paprastąja atsitiktine imtimi sluoksniuose. Nagrinėjamas populiacijos parametras yra populiacijos dispersija. Pateikti trys jos įvertiniai ir parodyta, kad du iš jų yra nepaslinktieji, o vienas turi poslinkį ir šis poslinkis yra apskaičiuotas. Taip pat nagrinėjami penki savirankos metodai skirti dispersijos dispersijai vertinti: tradicinė saviranka, mastelio pakeitimo saviranka, veidrodinio sutapimo saviranka, negrąžintinė saviranka ir grąžintinė saviranka. Palyginimui yra naudojama apytikslės dispersijos įvertinio dispersijos formulė specialiai išvesta sluoksninėms imtims. Šio darbo tikslas palyginti savirankos metodais gautas dispersijos įvertinio dispersijos įvertinio reikšmes tarpusavyje ir su reikšmėmis gautomis skaičiuojant pagal apytikslę formulę. / Consider a finite population from which the stratified sample with simple random sample without replacement in each strata is drawn. The finite population parameter of interest is variance. There are viewed three estimators of variance and are shown that two estimators are unbiased and one estimator has a bias which is estimated. We studied five bootstrap methods, the naive bootstrap, the rescaling bootstrap, the mirror-match bootstrap, the without-replacement bootstrap and the with-replacement bootstrap for variance estimation of estimator of variance. In comparison, there is used approximation variance formula of variance which is derived special for stratified samples. Our purpose was to compare the result of the bootstrap methods with each other and with the result from the approximation variance formula.

Many interesting behaviours in the animal and human world involve cooperation among individuals. Yet, cooperating individuals are often susceptible to exploitation by cheaters. Because cheaters do better than the cooperators they exploit, the evolution and persistence of cooperation has been a challenging topic of study in biology, sociology and economics. Studies often abstract from real cooperative interactions, and construct simple games in which players can choose either cooperation with other players, or defection, e.g., the well known prisoner’s dilemma and the snowdrift game. In these games and other social dilemmas, mutual cooperation yields greater payoffs than mutual defection, but individuals are still tempted to defect (because of the possibility that if they cooperate, the other player will defect). Similar dilemmas also arise in situations where multiple individuals may be affected by the actions of one (such as volunteering for community service or evading taxes), and the main theme of this thesis is cooperation in groups. In chapter 2, we analyze pre-emptive vaccination for an outbreak of smallpox (following a bioterrorist attack or accidental release), from the public health (i.e., group) and individual perspectives. Chapters 3 and 4 deal with an extension of the snowdrift game to n interacting players and continuous strategy sets (where individuals decide on their degree of cooperation): in chapter 3, we analyze global evolutionary stability of cooperative strategies in a large class of n-player snowdrift games in infinite populations; chapter 4 analyzes general continuous n-person snowdrift games in finite populations, and compares the evolutionary dynamics with their infinite population analogues. In chapter 5, we present a general framework to model selection processes in finite populations, necessary for the analysis in chapter 4. / Thesis / Doctor of Philosophy (PhD)

Disertacijoje nagrinėjamos papildomos informacijos panaudojimo galimybės konstruojant baigtinės populiacijos sumos, dispersijos ir kovariacijos įvertinius, bei sluoksniuojant baigtines populiacijas. Pirmiausia darbe sprendžiamas populiacijų sluoksniavimo uždavinys, kai tyrimo kintamojo skirstinys yra asimetrinis. Pasiūlomas naujas - pataisytasis geometrinis - sluoksniavimo metodas. Šis metodas modeliuojant lyginamas su trimis kitais žinomais metodais: kvadratinės šaknies iš skirstinio dažnio, geometriniu ir laipsninio sluoksniavimo metodu. Modeliavimo rezultatai rodo, kad vidutiniškai asimetrinėms populiacijoms geriausiai tinka laipsninio sluoksniavimo metodas, o ypač asimetrinėms populiacijoms geriausias yra pataisytasis geometrinis sluoksniavimas. Toliau nagrinėjami baigtinės populiacijos sumos kalibruotieji įvertiniai, sukonstruoti taikant skirtingas atstumo funkcijas. Modeliuojant tiriama šių įvertinių kokybė. Sukonstruoti nauji populiacijos kovariacijos kalibruotieji įvertiniai, naudojantys vieną, dvi ir tris svorių sistemas. Šie įvertiniai konstruojami pasirenkant skirtingas kalibravimo lygtis. Remiantis modeliais pagrįstų įvertinių teorija, čia taip pat sukonstruojamas pataisytasis tiesiniu regresiniu modeliu pagrįstas kalibruotasis populiacijos kovariacijos įvertinys. Modeliuojant įvertiniai lyginami tarpusavyje ir su standartiniais atitinkamų parametrų įvertiniais. Kalibruotieji įvertiniai yra kur kas tikslesni, jei tyrimo ir papildomų kintamųjų koreliacija yra... [toliau žr. visą tekstą] / The dissertation analyzes how to incorporate auxiliary information into the estimation of the finite population total, variance, covariance, and how to use it for the stratification of finite populations. First of all, the problem of efficient stratification in the case of skewed population is considered. A new adjusted geometric stratification method is introduced. This method is compared by simulation with the cumulative root frequency method, the geometric method, and the power method. The simulation results show that in most cases considered the power method is the most efficient one, but the adjusted geometric stratification method outperforms all the methods in the case of highly skewed populations. The calibrated estimators of finite population total, constructed using different distance functions, are considered. The quality of such estimators is analyzed by simulation. The new calibrated estimators of the finite population covariance (variance) are derived, using one or more weighting systems. Applying the model calibration theory, we construct here an adjusted linear regression model-assisted and calibrated estimator of the population covariance. The estimators derived are compared by simulation with the standard estimators of the respective parameters. The calibrated estimators of the population covariance are more efficient compared to the straight estimators provided the auxiliary variables are well correlated with the study variables. The problem of estimation... [to full text]

The dissertation analyzes how to incorporate auxiliary information into the estimation of the finite population total, variance, covariance, and how to use it for the stratification of finite populations. First of all, the problem of efficient stratification in the case of skewed population is considered. A new adjusted geometric stratification method is introduced. This method is compared by simulation with the cumulative root frequency method, the geometric method, and the power method. The simulation results show that in most cases considered the power method is the most efficient one, but the adjusted geometric stratification method outperforms all the methods in the case of highly skewed populations. The calibrated estimators of finite population total, constructed using different distance functions, are considered. The quality of such estimators is analyzed by simulation. The new calibrated estimators of the finite population covariance (variance) are derived, using one or more weighting systems. Applying the model calibration theory, we construct here an adjusted linear regression model-assisted and calibrated estimator of the population covariance. The estimators derived are compared by simulation with the standard estimators of the respective parameters. The calibrated estimators of the population covariance are more efficient compared to the straight estimators provided the auxiliary variables are well correlated with the study variables. The problem of estimation... [to full text] / Disertacijoje nagrinėjamos papildomos informacijos panaudojimo galimybės konstruojant baigtinės populiacijos sumos, dispersijos ir kovariacijos įvertinius, bei sluoksniuojant baigtines populiacijas. Pirmiausia darbe sprendžiamas populiacijų sluoksniavimo uždavinys, kai tyrimo kintamojo skirstinys yra asimetrinis. Pasiūlomas naujas - pataisytasis geometrinis - sluoksniavimo metodas. Šis metodas modeliuojant lyginamas su trimis kitais žinomais metodais: kvadratinės šaknies iš skirstinio dažnio, geometriniu ir laipsninio sluoksniavimo metodu. Modeliavimo rezultatai rodo, kad vidutiniškai asimetrinėms populiacijoms geriausiai tinka laipsninio sluoksniavimo metodas, o ypač asimetrinėms populiacijoms geriausias yra pataisytasis geometrinis sluoksniavimas. Toliau nagrinėjami baigtinės populiacijos sumos kalibruotieji įvertiniai, sukonstruoti taikant skirtingas atstumo funkcijas. Modeliuojant tiriama šių įvertinių kokybė. Sukonstruoti nauji populiacijos kovariacijos kalibruotieji įvertiniai, naudojantys vieną, dvi ir tris svorių sistemas. Šie įvertiniai konstruojami pasirenkant skirtingas kalibravimo lygtis. Remiantis modeliais pagrįstų įvertinių teorija, čia taip pat sukonstruojamas pataisytasis tiesiniu regresiniu modeliu pagrįstas kalibruotasis populiacijos kovariacijos įvertinys. Modeliuojant įvertiniai lyginami tarpusavyje ir su standartiniais atitinkamų parametrų įvertiniais. Kalibruotieji įvertiniai yra kur kas tikslesni, jei tyrimo ir papildomų kintamųjų koreliacija yra... [toliau žr. visą tekstą]

Consideramos a predição ótima de valores latentes com base em dados sujeitos a erros de medida endógenos e exógenos, obtidos a partir de uma amostra aleatória de uma população finita. Consideramos o modelo misto para populações finitas (MMPF) com erros de medida exógenos e endógenos usando o enfoque proposto por Stanek et al. (2004) e Stanek & Singer (2004), e calculamos o melhor preditor linear não enviesado (BLUP) do valor latente da i-ésima unidade selecionada na amostra. Quando as variâncias endógenas são heterocedásticas, o preditor obtido sob o MMPF é diferente do preditor obtido sob o modelo misto usual, pois a constante de encolhimento depende da média das variâncias individuais. Utilizamos simulação para comparar o preditor obtido sob o modelo misto usual (utilizado conforme a interpretação usual) com o preditor obtido sob o MMPF, mostrando que apesar do primeiro ser enviesado, ele geralmente apresenta erro quadrático médio (EQM) menor (ou ligeiramente maior) do que aquele obtido sob o MMPF. Adicionalmente, mostramos como utilizar dois pacotes de \\emph estatístico (Proc MIXED do SAS e lme(nlme) do R), construídos sob o modelo misto usual, para ajustar corretamente modelos em situações com erros exógenos e endógenos, heterocedásticos ou homocedásticos. / We consider optimal estimation and prediction of latent values based on data subject to endogenous and exogenous measurement errors, obtained via simple random sample from a finite population. We consider a finite population mixed model (FPMM) with endogenous and exogenous measurement errors proposed by Stanek III et al. (2004) and Stanek III & Singer (2004) and obtained the best linear unbiased predictor (BLUP) of the latent value of the i-th unit selected in the sample. When the endogenous variances are heteroscedastic, the predictor obtained under the FPMM is different than the predictor obtained with the usual mixed model, because the shrinkage constant depends on the average of the individual variances. We consider simulation studies to compare the predictor obtained under the usual mixed model (used according to the usual interpretation) with the predictor obtained under the FPMM, and show that the former is biased, but usually presents smaller (or slightly larger) mean squared error (MSE) than the predictor obtained under the FPMM. Additionally, we indicate how two commonly used statistical software packages (SAS\'s Proc MIXED and R\'s lme(nlme) ) may be employed to fit mixed models in situations with heteroscedastic or homoscedastic exogenous and endogenous errors.

Neste trabalho, consideramos uma população finita composta por N elementos, sendo que para cada unidade está associado um número (ou vetor) de tal forma que temos para a população o vetor de valores X = (X1, ... ,XN), onde Xi denota a característica de interesse do i-ésimo indivíduo da população, que suporemos desconhecida. Aqui assumimos que a distribuição do vetor X é permutável e que existe disponível uma amostra composta por n < N elementos. Os objetivos são a construção de testes de hipóteses para os parâmetros operacionais, através das distribuições a posteriori obtidas sob a abordagem preditivista para populações finitas e a comparação com os resultados obtidos a partir dos modelos Bayesianos de superpopulação. Nas análises consideramos os modelos Bernoulli, Poisson, Uniforme Discreto e Multinomial. A partir dos resultados obtidos, conseguimos ilustrar situações nas quais as abordagens produzem resultados diferentes, como prioris influenciam os resultados e quando o modelo de populações finitas apresenta melhores resultados que o modelo de superpopulação. / We consider a finite population consisting of N units and to each unit there is a number (or vector) associated such that we have for the population the vector of values X = (X1, ..., XN), where Xi denotes the characteristic of interest of the i-th individual in the population, which we will suppose unknown. Here we assume that the distribution of the vector X is exchangeable and that there is available a sample of size n < N from this population. The goals are to derive tests of hipotheses for the operational parameters through the corresponding posterior distributions obtained under the predictivistic approach for finite populations and to compare them with the results obtained from the usual Bayesian procedures of superpopulation models. In the analysis, the following models are considered: Bernoulli, Poisson, Discrete Uniform and Multinomial. From the results, we can identify situations in which the approaches yield dierent results, how priors influence the results of hipothesis testing and when the finite population model performs better than the superpopulation one.

Consideramos a predição ótima de valores latentes com base em dados sujeitos a erros de medida endógenos e exógenos, obtidos a partir de uma amostra aleatória de uma população finita. Consideramos o modelo misto para populações finitas (MMPF) com erros de medida exógenos e endógenos usando o enfoque proposto por Stanek et al. (2004) e Stanek & Singer (2004), e calculamos o melhor preditor linear não enviesado (BLUP) do valor latente da i-ésima unidade selecionada na amostra. Quando as variâncias endógenas são heterocedásticas, o preditor obtido sob o MMPF é diferente do preditor obtido sob o modelo misto usual, pois a constante de encolhimento depende da média das variâncias individuais. Utilizamos simulação para comparar o preditor obtido sob o modelo misto usual (utilizado conforme a interpretação usual) com o preditor obtido sob o MMPF, mostrando que apesar do primeiro ser enviesado, ele geralmente apresenta erro quadrático médio (EQM) menor (ou ligeiramente maior) do que aquele obtido sob o MMPF. Adicionalmente, mostramos como utilizar dois pacotes de \\emph estatístico (Proc MIXED do SAS e lme(nlme) do R), construídos sob o modelo misto usual, para ajustar corretamente modelos em situações com erros exógenos e endógenos, heterocedásticos ou homocedásticos. / We consider optimal estimation and prediction of latent values based on data subject to endogenous and exogenous measurement errors, obtained via simple random sample from a finite population. We consider a finite population mixed model (FPMM) with endogenous and exogenous measurement errors proposed by Stanek III et al. (2004) and Stanek III & Singer (2004) and obtained the best linear unbiased predictor (BLUP) of the latent value of the i-th unit selected in the sample. When the endogenous variances are heteroscedastic, the predictor obtained under the FPMM is different than the predictor obtained with the usual mixed model, because the shrinkage constant depends on the average of the individual variances. We consider simulation studies to compare the predictor obtained under the usual mixed model (used according to the usual interpretation) with the predictor obtained under the FPMM, and show that the former is biased, but usually presents smaller (or slightly larger) mean squared error (MSE) than the predictor obtained under the FPMM. Additionally, we indicate how two commonly used statistical software packages (SAS\'s Proc MIXED and R\'s lme(nlme) ) may be employed to fit mixed models in situations with heteroscedastic or homoscedastic exogenous and endogenous errors.

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Neste trabalho, consideramos uma população finita composta por N elementos, sendo que para cada unidade está associado um número (ou vetor) de tal forma que temos para a população o vetor de valores X = (X1, ... ,XN), onde Xi denota a característica de interesse do i-ésimo indivíduo da população, que suporemos desconhecida. Aqui assumimos que a distribuição do vetor X é permutável e que existe disponível uma amostra composta por n < N elementos. Os objetivos são a construção de testes de hipóteses para os parâmetros operacionais, através das distribuições a posteriori obtidas sob a abordagem preditivista para populações finitas e a comparação com os resultados obtidos a partir dos modelos Bayesianos de superpopulação. Nas análises consideramos os modelos Bernoulli, Poisson, Uniforme Discreto e Multinomial. A partir dos resultados obtidos, conseguimos ilustrar situações nas quais as abordagens produzem resultados diferentes, como prioris influenciam os resultados e quando o modelo de populações finitas apresenta melhores resultados que o modelo de superpopulação. / We consider a finite population consisting of N units and to each unit there is a number (or vector) associated such that we have for the population the vector of values X = (X1, ..., XN), where Xi denotes the characteristic of interest of the i-th individual in the population, which we will suppose unknown. Here we assume that the distribution of the vector X is exchangeable and that there is available a sample of size n < N from this population. The goals are to derive tests of hipotheses for the operational parameters through the corresponding posterior distributions obtained under the predictivistic approach for finite populations and to compare them with the results obtained from the usual Bayesian procedures of superpopulation models. In the analysis, the following models are considered: Bernoulli, Poisson, Discrete Uniform and Multinomial. From the results, we can identify situations in which the approaches yield dierent results, how priors influence the results of hipothesis testing and when the finite population model performs better than the superpopulation one.

Šiame magistro darbe nagrinėjamos papildomos informacijos panaudojimo galimybės konstruojant baigtinės populiacijos koreliacijos koeficiento įvertinius. Taikant imties plano svorių kalibravimo metodus sukonstruojami aštuoni nauji koreliacijos koeficiento įvertiniai. Įvertiniams konstruoti panaudotos kelios kalibravimo lygčių ir atstumo funkcijų kombinacijos. Kalibruotieji svoriai išvesti pasitelkiant Lagranžo neapibrėžtųjų daugiklių metodą. Naudojant Teiloro ištiesinimo ir visrakčio metodus sukonstruojami koreliacijos koeficiento įvertinių dispersijos įvertiniai. Darbe taip pat atliekamas matematinis modeliavimas, kuriuo siekiama palyginti naujai sukonstruotus įvertinius tarpusavyje ir su standartiniu įvertiniu. Įvertinių tikslumą lemia pakankamai didelė koreliacija tarp tyrimo kintamųjų ir jų papildomų kintamųjų, o du įvertiniai yra tikslūs ir esant mažai koreliacijai. Tiriama, kaip įvertinių tikslumą įtakoja imties dydis bei koreliacija tarp tyrimo ir papildomų kintamųjų. Matematinio modeliavimo eksperimentai atlikti, naudojant darbo autorės sukurtas MATLAB programas. / This master’s thesis analyzes the opportunities of auxiliary information usage while constructing the correlation coefficient estimators of the finite population. By applying the methods of weight calibration of sample design the new eight correlation coefficient estimators are derived. Several combinations of calibration equations and distance measures are used to construct estimators. Calibration weights were derived through Lagrange multiplier method. Using Taylor linearization and jackknife methods variance estimators of the correlation coefficient estimators were constructed. The paper also contains a mathematical simulation aimed at comparing the new derived estimators in between and with a standard estimator. The accuracy of estimators is determined by rather substantial correlation between study and auxiliary variables, whereas two of the estimators are accurate even in the presence of bad correlation. It is tested how sample size and correlation between study and auxiliary variables influences the accuracy of estimators. All the mathemathical simulation tests were performed employing MATLAB programs that have been created by the author of this work.