Ce mémoire propose une adaptation lisse de méthodes bootstrap par pseudo-population aux fins d'estimation de la variance et de formation d'intervalles de confiance pour des quantiles de population finie. Dans le cas de données i.i.d., Hall et al. (1989) ont montré que l'ordre de convergence de l'erreur relative de l’estimateur bootstrap de la variance d’un quantile échantillonnal connaît un gain lorsque l'on rééchantillonne à partir d’une estimation lisse de la fonction de répartition plutôt que de la fonction de répartition expérimentale. Dans cet ouvrage, nous étendons le principe du bootstrap lisse au contexte de population finie en le mettant en œuvre au sein des méthodes bootstrap par pseudo-population. Étant donné un noyau et un paramètre de lissage, cela consiste à lisser la pseudo-population dont sont issus les échantillons bootstrap selon le plan de sondage initial. Deux plans sont abordés, soit l'échantillonnage aléatoire simple sans remise et l'échantillonnage de Poisson. Comme l'utilisation des algorithmes proposés nécessite la spécification du paramètre de lissage, nous décrivons une méthode de sélection par injection et des méthodes de sélection par la minimisation d'estimés bootstrap de critères d'ajustement sur une grille de valeurs du paramètre de lissage. Nous présentons des résultats d'une étude par simulation permettant de montrer empiriquement l'efficacité de l'approche lisse par rapport à l'approche standard pour ce qui est de l'estimation de la variance d'un estimateur de quantile et des résultats plus mitigés en ce qui concerne les intervalles de confiance. / This thesis introduces smoothed pseudo-population bootstrap methods for the purposes of variance estimation and the construction of confidence intervals for finite population quantiles. In an i.i.d. context, Hall et al. (1989) have shown that resampling from a smoothed estimate of the distribution function instead of the usual empirical distribution function can improve the convergence rate of the bootstrap variance estimator of a sample quantile. We extend the smoothed bootstrap to the survey sampling framework by implementing it in pseudo-population bootstrap methods. Given a kernel function and a bandwidth, it consists of smoothing the pseudo-population from which bootstrap samples are drawn using the original sampling design. Two designs are discussed, namely simple random sampling and Poisson sampling. The implementation of the proposed algorithms requires the specification of the bandwidth. To do so, we develop a plug-in selection method along with grid search selection methods based on bootstrap estimates of two performance metrics. We present the results of a simulation study which provide empirical evidence that the smoothed approach is more efficient than the standard approach for estimating the variance of a quantile estimator together with mixed results regarding confidence intervals.

This paper discusses results of a computer simulation to investigate the effect of jittering to simulate measurement error. In addition, the classical F ratio, the bootstrap F and the F for ranked data are compared. Empirical powers and p-values suggest the bootstrap is a good and robust procedure and the rank procedure seems to be too liberal when compared to the classical F ratio.

We explore the application of the bootstrap unit root test to time series with heavy-tailed errors. The size and power of the tests are estimated for two different autoregressive models (AR(1)) using computer simulated data. Real-data examples are also presented. Two different bootstrap methods and the subsampling approach are compared. Conclusions on the optimal bootstrap parameters, the range of applicability, and the performance of the tests are made.

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In this thesis, I am interested in estimating the difference between dependent proportions from a 2 × 2 contingency table when there are missing data. The Expectation-Maximization (EM) algorithm is used to obtain an estimate for the difference between correlated proportions. To obtain the standard error of this difference I employ a resampling technique known as bootstrapping. The performance of the bootstrap standard error is evaluated for different sample sizes and different fractions of missing information. Finally, a 100(1-α)% bootstrap confidence interval is proposed and its coverage is evaluated through simulation.

For any manufacturing process, one of the key challenges after a solid foundation has been built is how improvements can be made. Management has to consider how possible changes will affect both the process as a whole in addition to every individual part before implementation. The groundwork for this is to have a clear overview of every part and the possibility to investigate effects of changes. This thesis thus aims to provide a clear overview of the complex painting process at Volvo GTO in Umeå and a template for investigating how differently implemented changes will affect the process. The means for doing this is to use statistics, modeling and discrete event simulation. Modeling shall provide an approximate recreation of reality and the subsequent analysis shall take into account similarities and differences to estimate the effects of changes. Recreation of real-world data and variability is based on bootstrap resampling for multiple independent weeks of observations. Results obtained from simulation are compared to observed data in order to validate the model and investigate discrepancies. Given the results of model validation, modifications are implemented and information obtained from model validation is used to evaluate the results of the modifications. Further, strengths and weaknesses of the thesis are presented and a recommendation of altering the stance on process improvements is provided to Volvo GTO.

Consider infecting a subset $A_0 \subseteq V(G)$ of the vertices of a graph $G$. Let an uninfected vertex $v \in V(G)$ become infected if $|N_G(v) \cap A_0| \geq r$, for some integer $r$. Define $A_t = A_{t-1} \cup \{v \in V(G) : |N_G(v) \cap A_{t-1}| \geq r \},$ and say that the set $A_0$ is \emph{lethal} under $r$-neighbor percolation if there exists a $t$ such that $A_t = V(G)$. For a graph $G$, let $m(G,r)$ be the size of the smallest lethal set in $G$ under $r$-neighbor percolation. The problem of determining $m(G,r)$ has been extensively studied for grids $G$ of various dimensions. We define $$m(a_1, \dots, a_d, r) = m\left (\prod_{i=1}^d [a_i], r\right )$$ for ease of notation. Famously, a lower bound of $m(a_1, \dots, a_d, d) \geq \frac{\sum_{j=1}^d \prod_{i \neq j} a_i}{d}$ is given by a beautiful argument regarding the high-dimensional ``surface area" of $G = [a_1] \times \dots \times [a_d]$. While exact values of $m(G,r)$ are known in some specific cases, general results are difficult to come by. In this thesis, we introduce a novel technique for viewing $3$-neighbor lethal sets on three-dimensional grids in terms of lethal sets in two dimensions. We also provide a strategy for recursively building up large lethal sets from existing small constructions. Using these techniques, we determine the exact size of all lethal sets under 3-neighbor percolation in three-dimensional grids $[a_1] \times [a_2] \times [a_3]$, for $a_1,a_2,a_3 \geq 11$. The problem of determining $m(n,n,3)$ is discussed by Benevides, Bermond, Lesfari and Nisse in \cite{benevides:2021}. The authors determine the exact value of $m(n,n,3)$ for even $n$, and show that, for odd $n$, $$\ceil*{\frac{n^2+2n}{3}} \leq m(n,n,3) \leq \ceil*{\frac{n^2+2n}{3}} + 1.$$ We prove that $m(n,n,3) = \ceil*{\frac{n^2+2n}{3}}$ if and only if $n = 2^k-1$, for some $k >0$. Finally, we provide a construction to prove that for $a_1,a_2,a_3 \geq 12$, bounds on the minimum lethal set on the the torus $G = C_{a_1} \square C_{a_2} \square C_{a_3}$ are given by $$2 \le m(G,3) - \frac{a_1a_2 + a_2a_3 + a_3a_1 -2(a_1+a_2+a_3)}{3} \le 3.$$ / Graduate

<p>While Reinforcement Learning (RL) has achieved phenomenal success in diverse fields in recent years, the statistical properties of the underlying algorithms are still not fully understood. One key aspect in this regard is the evaluation of the value associated with the RL agent. In this dissertation, we propose two statistically sound methods for policy evaluation and inference, and study their theoretical properties within the RL setting. </p> <p><br></p> <p>In the first work, we propose an online bootstrap method for statistical inference in policy evaluation. The bootstrap is a flexible and efficient approach for inference in online learning, but its efficacy in the RL setting has yet to be explored. Existing methods for online inference are restricted to settings involving independently sampled observations. In contrast, our method is shown to be distributionally consistent for statistical inference in policy evaluation under Markovian noise, which is a standard assumption in the RL setting. To demonstrate the effectiveness of our method in practical applications, we include several numerical simulations involving the temporal difference (TD) learning and Gradient TD (GTD) learning algorithms across a range of real RL environments. </p> <p><br></p> <p>In the second work, we propose a tensor Markov Decision Process framework for modeling the evolution of a sequential decision-making process when the state-action features are tensors. Under this framework, we develop a low-rank tensor estimation method for off-policy evaluation in batch RL. The proposed estimator approximates the Q-function using a tensor parameter embedded with low-rank structure. To overcome the challenge of nonconvexity, we introduce an efficient block coordinate descent approach with closed-form solutions to the alternating updates. Under standard assumptions from the tensor and RL literature, we establish an upper bound on the statistical error which guarantees a sub-linear rate of computational error. We provide numerical simulations to demonstrate that our method significantly outperforms standard batch off-policy evaluation algorithms when the true parameter has a low-rank tensor structure.</p>

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