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Neste trabalho analisamos, de maneira descritiva utilizando simulações, o desempenho um conjunto de estimadores para os parâmetros funcionais f, g, h e k de uma equação diferencial parcial. Para as simulações, selecionamos doze EDPs nas quais aplicaremos os estimadores as soluções dessas EDPs. Iremos medir o desempenho dos estimadores utilizando o erro quadrático entre as estimativas obtidas e os coeficientes conhecidos das EDPs. As estimativas dos coeficientes serão obtidas para soluções com e sem ruídos. Os estimadores propostos serão comparados com um grupo de estimadores diretos. Os programas para as simulações foram desenvolvidos com programas no repositório R, versão 3.2.3. Veremos que os estimadores propostos apresentaram um desempenho superior aos dos estimadores diretos, para soluções com e sem ruídos. Nas soluções com ruídos os estimadores propostos tiveram um desempenho mais significativo que para soluções sem ruídos. / In this paper we shall analyze, on a descriptive way using simulations, the performance of a set of estimators for the coefficients f, g, h and k using the following. For the simulations, we select twelve EDPs in which we will apply the estimators from the solutions from the mentioned EDPs. We will measure the performance using the quadratic error between the estimates obtained and the known coefficients from the EDPs. The estimates will be obtained for solutions with and without noises. The proposed estimators will be compared with a group of direct estimators. The programs used for the simulations were developed with the programs in the repository R, version 3.2.3. We shall see that the proposed estimators present a superior performance compared to the direct estimators, for solutions with or without noises. The proposed estimators have had a better performance in the solutions with noises than in the solutions without a noise.

Neste trabalho analisamos, de maneira descritiva utilizando simulações, o desempenho um conjunto de estimadores para os parâmetros funcionais f, g, h e k de uma equação diferencial parcial. Para as simulações, selecionamos doze EDPs nas quais aplicaremos os estimadores as soluções dessas EDPs. Iremos medir o desempenho dos estimadores utilizando o erro quadrático entre as estimativas obtidas e os coeficientes conhecidos das EDPs. As estimativas dos coeficientes serão obtidas para soluções com e sem ruídos. Os estimadores propostos serão comparados com um grupo de estimadores diretos. Os programas para as simulações foram desenvolvidos com programas no repositório R, versão 3.2.3. Veremos que os estimadores propostos apresentaram um desempenho superior aos dos estimadores diretos, para soluções com e sem ruídos. Nas soluções com ruídos os estimadores propostos tiveram um desempenho mais significativo que para soluções sem ruídos. / In this paper we shall analyze, on a descriptive way using simulations, the performance of a set of estimators for the coefficients f, g, h and k using the following. For the simulations, we select twelve EDPs in which we will apply the estimators from the solutions from the mentioned EDPs. We will measure the performance using the quadratic error between the estimates obtained and the known coefficients from the EDPs. The estimates will be obtained for solutions with and without noises. The proposed estimators will be compared with a group of direct estimators. The programs used for the simulations were developed with the programs in the repository R, version 3.2.3. We shall see that the proposed estimators present a superior performance compared to the direct estimators, for solutions with or without noises. The proposed estimators have had a better performance in the solutions with noises than in the solutions without a noise.

Electricity prices in competitive markets are extremely volatile with salient features such as mean-reversion and jumps and spikes. Modeling electricity spot prices is essential for asset and project valuation as well as risk management. I introduce the mean-reversion feature into a classical variance gamma model to model the electricity price dynamics as a mean-reverting variance gamma (MRVG) process. Derivative pricing formulae are derived through transform analysis and model parameters are estimated by the generalized method of moments and the Markov Chain Monte Carlo method. A real option approach is proposed to value a tolling contract incorporating operational characteristics of the generation asset and contractual constraints. Two simulation-based methods are proposed to solve the valuation problem. The effects of different electricity price assumptions on the valuation of tolling contracts are examined. Based on the valuation model, I also propose a heuristic scheme for hedging tolling contracts and demonstrate the validity of the hedging scheme through numerical examples. Autoregressive Conditional Heteroscedasticity (ARCH) and Generalized ARCH (GARCH) models are widely used to model price volatility in financial markets. Considering a GARCH model with heavy-tailed innovations for electricity price, I characterize the limiting distribution of a Value-at-Risk (VaR) estimator of the conditional electricity price distribution, which corresponds to the extremal quantile of the conditional distribution of the GARCH price process. I propose two methods, the normal approximation method and the data tilting method, for constructing confidence intervals for the conditional VaR estimator and assess their accuracies by simulation studies. The proposed approach is applied to electricity spot price data taken from the Pennsylvania-New Jersey-Maryland market to obtain confidence intervals of the empirically estimated Value-at-Risk of electricity prices. Several directions that deserve further investigation are pointed out for future research.

This dissertation contains three essays. They are related to the exponential series estimation of copulas and the application of parametric copulas in financial econometrics. Chapter II proposes a multivariate exponential series estimator (ESE) to estimate copula density nonparametrically. The ESE attains the optimal rate of convergence for nonparametric density. More importantly, it overcomes the boundary bias of copula estimation. Extensive Monte Carlo studies show the proposed estimator outperforms kernel and log-spline estimators in copula estimation. Discussion is provided regarding application of the ESE copula to Asian stock returns during the Asian financial crisis. The ESE copula complements the existing nonparametric copula studies by providing an alternative dedicated to the tail dependence measure. Chapter III proposes a likelihood ratio statistic using a nonparametric exponential series approach. The order of the series is selected by Bayesian Information Criterion (BIC). I propose three further modifications on my test statistic: 1) instead of putting equal weight on the individual term of the exponential series, I consider geometric and exponential BIC average weights; 2) rather than using a nested sequence, I consider all subsets to select the optimal terms in the exponential series; 3) I estimate the likelihood ratio statistic using the likelihood cross-validation. The extensive Monte Carlo simulations show that the proposed tests enjoy good finite sample performances compared to the traditional methods such as the Anderson-Darling test. In addition, this data-driven method improves upon Neyman’s score test. I conclude that the exponential series likelihood ratio test can complement the Neyman’s score test. Chapter IV models and forecasts S&P500 index returns using the Copula-VAR approach. I compare the forecast performance of the Copula-VAR model with a classical VAR model and a univariate time series model. I use this approach to forecast S&P500 index returns. I apply a modified Diebold-Mariano test to test the equality of mean squared forecast errors and utilize a forecast encompassing test to evaluate forecasts. The findings suggest that allowing a more flexible specification in the error terms using copula tends improve the forecast accuracy. I also demonstrate combined forecasts improved forecasts accuracy over individual models.

Based on Lyapunov stability theorem, a design methodology of nth order adaptive integral variable structure derivative estimator (AIVSDE) is proposed in this thesis. The proposed derivative estimator not only is an improved version of the existing AIVSDE, but also can be used to estimate the nth differentiation of a smooth signal which has continuous and bounded derivatives up to n+1. A low pass filter is cascaded with AIVSDE so that the effects of noise can be alleviated by adjusting the designing parameters of filter and AIVSDE. The adaptive algorithm is incorporated in the control scheme for removing the a priori knowledge of the upper bound of the observed signal. The stability of the proposed derivative estimator is guaranteed, and the comparison of upper bound of derivative estimation error between recently proposed nonlinear adaptive variable structure derivative estimator (NAVSDE) and AIVSDE is also demonstrated. An example is given for showing the applicability of the proposed AIVSDE.

Abstract Under known signals environments, the problem of frequency estimation can be regarded as that of sinusoidal frequency estimation. Therefore, the frequency estimation of a single complex sinusoid signal in a white Gaussian noise channel is an important problem in the field of signal processing. Some of the applications include array signal processing, spectral estimation, carrier and clock synchronization for digital communications, Doppler rate estimation, and many others in radar and sonar systems. Frequency estimations based on the information of phase have threshold effects. While the length of the observation data is fixed, the performance of the estimator will be degraded and the variance will not achieve Cramer-Rao lower bound under the condition that signal-to-noise ratio (SNR) is below a certain threshold. In this thesis, two modified frequency estimation methods are proposed in additive white Gaussian noise channels. These two methods, estimating the frequency value by linearly combining the phase difference of correlated data, are basically extended from Kim¡¦s method. These estimators have lower complexity than optimal maximum likelihood estimator and attain as good performance at moderately high SNR¡¦s. These two methods, at high frequency values, yield a considerably lower variance threshold than Kay¡¦s method and Kim¡¦s method and remain unbiased.

This thesis investigates improved estimators of the parameters of the linear regression models with normal errors, under sample and non-sample prior information about the value of the parameters. The estimators considered are the unrestricted estimator (UE), restricted estimator (RE), shrinkage restricted estimator (SRE), preliminary test estimator (PTE), shrinkage preliminary test estimator (SPTE), and shrinkage estimator (SE). The performances of the estimators are investigated with respect to bias, squared error and linex loss. For the analyses of the risk functions of the estimators, analytical, graphical and numerical procedures are adopted. In Part I the SRE, SPTE and SE of the slope and intercept parameters of the simple linear regression model are considered. The performances of the estimators are investigated with respect to their biases and mean square errors. The efficiencies of the SRE, SPTE and SE relative to the UE are obtained. It is revealed that under certain conditions, SE outperforms the other estimators considered in this thesis. In Part II in addition to the likelihood ratio (LR) test, the Wald (W) and Lagrange multiplier (LM) tests are used to define the SPTE and SE of the parameter vector of the multiple linear regression model with normal errors. Moreover, the modified and size-corrected W, LR and LM tests are used in the definition of SPTE. It is revealed that a great deal of conflict exists among the quadratic biases (QB) and quadratic risks (QR) of the SPTEs under the three original tests. The use of the modified tests reduces the conflict among the QRs, but not among the QBs. However, the use of the size-corrected tests in the definition of the SPTE almost eliminates the conflict among both QBs and QRs. It is also revealed that there is a great deal of conflict among the performances of the SEs when the three original tests are used as the preliminary test statistics. With respect to quadratic bias, the W test statistic based SE outperforms that based on the LR and LM test statistics. However, with respect to the QR criterion, the LM test statistic based SE outperforms the W and LM test statistics based SEs, under certain conditions. In Part III the performance of the PTE of the slope parameter of the simple linear regression model is investigated under the linex loss function. This is motivated by increasing criticism of the squared error loss function for its inappropriateness in many real life situations where underestimation of a parameter is more serious than its overestimation or vice-versa. It is revealed that under the linex loss function the PTE outperforms the UE if the nonsample prior information about the value of the parameter is not too far from its true value. Like the linex loss function, the risk function of the PTE is also asymmetric. However, if the magnitude of the scale parameter of the linex loss is very small, the risk of the PTE is nearly symmetric.

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