Thesis (Ph. D.)--University of Chicago, Dept. of Statistics, June 2002. / Includes bibliographical references (p. 123-132) Also available on the Internet.

Thesis--Wisconsin. / Vita. Includes bibliographical references (leaves 68-69).

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D.Phil. (Mathematical Statistics) / Fractional Brownian motion and its increment process, fractional Gaussian noise, are syn- onymous with the concept of long range dependence. A strictly stationary time series is said to exhibit long range dependence or long memory if its autocorrelations decrease to zero as a power of the lag, but their sum over all lags is not absolutely convergent. This phenomenon has been observed in numerous scientific areas such as hydrology, ethernet traffic data, stock returns and exchange rates, to name just a few. The extent of long memory dependence is characterized by the value of the so called Hurst exponent or Hurst coefficient H. Approximate normality and unbiasedness of the maximum likelihood estimate of H hold reasonably well for sample sizes as small as 20 if the mean and scale parameters are known. We show in a Monte Carlo study that if the latter two parameters are unknown, the bias and variance of the maximum likelihood estimate of H both increase substantially. We also show that the bias can be reduced by using a jackknife or parametric bootstrap proce- dure. However, in very large samples, maximum likelihood estimation becomes problematic because of the large dimension of the covariance matrix that must be inverted. We consider an approach for estimating the Hurst exponent by taking first order differ- ences of fractional Gaussian noise. We find that this differenced process has short memory and that, consequently, we may assume approximate independence between the estimates of the Hurst exponents in disjoint blocks of data. We split the data into a number of con- tiguous blocks, each containing a relatively small number of observations. Computation of the likelihood function in a block then presents no computational problem. We form a pseudo likelihood function consisting of the product of the likelihood functions in each of the blocks and provide a formula for the standard error of the resulting estimator of H. This formula is shown in a Monte Carlo study to provide a good approximation to the true standard error. Application of the methodology is illustrated in two data sets. The long memory property of a time series is primarily characterized by H. In general, such series are exceptionally long, therefore it is natural to enquire whether or not H remains constant over the full extent of the time series. We propose a number of tests for the hypothesis that H remains constant, against an alternative of a change in one or more values of H. Formulas are given to enable calculation of asymptotic p-values. We also propose a permutational procedure for evaluating exact p-values. The proposed tests are applied to three sets of data.

Phase Retrieval is an inference problem where one seeks to recover an unknown complex-valued 𝓃-dimensional signal vector from the magnitudes of 𝓶 linear measurements. The linear measurements are specified using a 𝓶 × 𝓃 sensing matrix. This problem is a mathematical model for imaging systems arising in X-ray crystallography and other applications where it is infeasible to acquire the phase of the measurements. This dissertation presents some results regarding the analysis of this problem in the high-dimensional asymptotic regime where the number of measurements and the signal dimension diverge proportionally so that their ratio remains fixed. A limitation of existing high-dimensional analyses of this problem is that they model the sensing matrix as a random matrix with independent and identically (i.i.d.) distributed Gaussian entries. In practice, this matrix is highly structured with limited randomness. This work studies a correction to the i.i.d. Gaussian sensing model, known as the sub-sampled Haar sensing model which faithfully captures a crucial orthogonality property of realistic sensing matrices. The first result of this thesis provides a precise asymptotic characterization of the performance of commonly used spectral estimators for phase retrieval in the sub-sampled Haar sensing model. This result can be leveraged to tune certain parameters involved in the spectral estimator optimally. The second part of this dissertation studies the information-theoretic limits for better-than-random (or weak) recovery in the sub-sampled Haar sensing model. The main result in this part shows that appropriately tuned spectral methods achieve weak recovery with the information-theoretically optimal number of measurements. Simulations indicate that the performance curves derived for the sub-sampled Haar sensing model accurately describe the empirical performance curves for realistic sensing matrices such as randomly sub-sampled Fourier sensing matrices and Coded Diffraction Pattern (CDP) sensing matrices. The final part of this dissertation tries to provide a mathematical understanding of this empirical universality phenomenon: For the real-valued version of the phase retrieval problem, the main result of the final part proves that the dynamics of a class of iterative algorithms, called Linearized Approximate Message Passing schemes, are asymptotically identical in the sub-sampled Haar sensing model and a real-valued analog of the sub-sampled Fourier sensing model.

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We observe a stochastic process 𝑌 on [0,1]^𝑑 (𝑑 ≥ 1) satisfying 𝑑𝑌(𝑡)=𝑛¹/²𝑓(𝑡)𝑑𝑡 + 𝑑𝑊(𝑡), 𝑡 ∈ [0,1]^𝑑, where 𝑛 ≥ 1 is a given scale parameter (`sample size'), 𝑊 is the standard Brownian sheet on [0,1]^𝑑 and 𝑓 ∈ L₁([0,1]^𝑑) is the unknown function of interest. We propose a multivariate multiscale statistic in this setting and prove that the statistic attains a subexponential tail bound; this extends the work of 'Dumbgen and Spokoiny (2001)' who proposed the analogous statistic for 𝑑=1. In the process, we generalize Theorem 6.1 of 'Dumbgen and Spokoiny (2001)' about stochastic processes with sub-Gaussian increments on a pseudometric space, which is of independent interest. We use the proposed multiscale statistic to construct optimal tests (in an asymptotic minimax sense) for testing 𝑓 = 0 versus (i) appropriate Hölder classes of functions, and (ii) alternatives of the form 𝑓 = 𝜇_𝑛𝕀_{𝐵_𝑛}$, where 𝐵_𝑛 is an axis-aligned hyperrectangle in [0,1]^𝑑 and 𝜇_𝑛 ∈ ℝ; 𝜇_𝑛 and 𝐵_𝑛 unknown. In Chapter 3 we use this proposed multiscale statistics to construct honest confidence bands for multivariate shape-restricted regression including monotone and convex functions.

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In the first chapter of this dissertation, I explore how forecaster attention, or the degree to which new information is incorporated into forecasts, is reflected at the lower-dimensional factor representation of multivariate forecast data. When information is costly to acquire, forecasters may pay more attention to some sources of information and ignore others. How much attention they pay will determine the strength of the forecast correlation (factor) structure. Using a factor model representation, I show that a forecast made by a rationally inattentive agent will include an extra shrinkage and thresholding "attention matrix" relative to a full information benchmark, and propose an econometric procedure to estimate it. Differences in the degree of forecaster attentiveness can explain observed differences in empirical shrinkage in professional macroeconomic forecasts relative to a consensus benchmark. Forecasters share the same reduced-form model, but differ in their measured attention. Better-performing forecasters have higher measured attention (lower shrinkage) than their poorly-performing peers. Measured forecaster attention to multiple dimensions of the information space can largely be captured by a single scalar cost parameter. I propose a new class of information cost functions for the classic multivariate linear-quadratic Gaussian tracking problem called separable spectral cost functions. The proposed measure of attention and mapping from theoretical model of attention allocation to factor structure in the first chapter is valid for this set of cost functions. These functions are defined over the eigenvalues of prior and posterior variance matrices. Separable spectral cost functions both nest known cost functions and are consistent with the definition of Uniformly Posterior Separable cost functions, which have desirable theoretical properties. The third chapter is coauthored work with Professor Serena Ng. We estimate higher frequency values of monthly macroeconomic data using different factor based imputation methods. Monthly and weekly economic indicators are often taken to be the largest common factor estimated from high and low frequency data, either separately or jointly. To incorporate mixed frequency information without directly modeling them, we target a low frequency diffusion index that is already available, and treat high frequency values as missing. We impute these values using multiple factors estimated from the high frequency data. In the empirical examples considered, static matrix completion that does not account for serial correlation in the idiosyncratic errors yields imprecise estimates of the missing values irrespective of how the factors are estimated. Single equation and systems-based dynamic procedures that account for serial correlation yield imputed values that are closer to the observed low frequency ones. This is the case in the counterfactual exercise that imputes the monthly values of consumer sentiment series before 1978 when the data was released only on a quarterly basis. This is also the case for a weekly version of the CFNAI index of economic activity that is imputed using seasonally unadjusted data. The imputed series reveals episodes of increased variability of weekly economic information that are masked by the monthly data, notably around the 2014-15 collapse in oil prices.

In this thesis we demonstrate a technique for robust controller design for an autonomous underwater vehicle (AUV) that explicitly handles the trade-off between reference tracking, agility, and energy efficient performance. AUVs have many sources of modeling uncertainty that impact the uncertainty in maneuvering performance. A robust control design process is proposed to handle these uncertainties while meeting control system performance objectives. We investigate the relationships between linear system design parameters and the control performance of our vehicle in order to inform an H∞ controller synthesis problem with the objective of balancing these tradeoffs. We evaluate the controller based on its reference tracking performance, agility and energy efficiency, and show the efficacy of our control design strategy. / Master of Science / In this thesis we demonstrate a technique for autopilot design for an autonomous underwater vehicle (AUV) that explicitly handles the trade-off between three performance metrics. Mathematical models of AUVs are often unable to fully describe their many physical properties. The discrepancies between the mathematical model and reality impact how certain we can be about an AUV's behavior. Robust controllers are a class of controller that are designed to handle uncertainty. A robust control design process is proposed to handle these uncertainties while meeting vehicle performance objectives. We investigate the relationships between design parameters and the performance of our vehicle. We then use this relationship to inform the design of a controller. We evaluate this controller based on its energy efficiency, agility and ability to stay on course, and thus show the effectiveness of our control design strategy.