There are many situations in which researchers want to consider a set of response variables simultaneously rather than just one response variable. For instance, a possible example is when a researcher wishes to determine the effects of an exercise and diet program on both the cholesterol levels and the weights of obese subjects. Dietz (1989) proposed two multivariate generalizations of the Jonckheere test for ordered alternatives. In this study, we propose k-sample tests for nondecreasing ordered alternatives for bivariate normal data and compare their powers with Dietz's sum statistic. The proposed k-sample tests are based on transformations of bivariate data to univariate data. The transformations considered are the sum, maximum and minimum functions. The ideas for these transformations come from the Leconte, Moreau, and Lellouch (1994). After the underlying bivariate normal data are reduced to univariate data, the Jonckheere-Terpstra (JT) test (Terpstra, 1952 and Jonckheere, 1954) and the Modified Jonckheere-Terpstra (MJT) test (Tryon and Hettmansperger, 1973) are applied to the univariate data. A simulation study is conducted to compare the proposed tests with Dietz's test for k bivariate normal populations (k=3, 4, 5). A variety of sample sizes and various location shifts are considered in this study. Two different correlations are used for the bivariate normal distributions. The simulation results show that generally the Dietz test performs the best for the situations considered with the underlying bivariate normal distribution. The estimated powers of MJT sum and JT sum are often close with the MJT sum generally having a little higher power. The sum transformation was the best of the three transformations to use for bivariate normal data.

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The problem considered in this dissertation is the following: let π₁ and π₂ be two bivariate populations having unknown cumulative distribution functions F₁(x₁, x₂) and F₂(x₁, x₂), respectively. Assume that F₁ and F₂ are continuous and identical except possibly in location parameters. It is desired to test the null hypothesis H₀: F₁(x₁, x₂) ≡ F₂(x₁, x₂) against the alternative H₀: F₁(x₁, x₂) ≠ F₂(x₁, x₂) It cannot be assumed that the variables x₁ and x₂ are statistically independent. Suppose there are n₁pairs of observations (x₁₁, x₂₁),..., (x_1n₁, x_2n₁) from the population π₁ and n₂ pairs of observations (x_ln+1. X_2n₁+1),..., (x_1N, x_2N) from population π₂, where N = n₁ + n₂. The x₁ᵢ (i = 1,2,..., N) are ranked according to magnitude, the largest being assigned rank 1 and the smallest assigned rank N. In a similar manner, ranks are assigned to the observations x₂ᵢ (i = 1, 2, …, N). It is assumed that there are no ties in ranks. Let u₁ᵢ and u₂ᵢ denote the ranks assigned to x₁ᵢ and x₂ᵢ if these observations belong to population π₁, and let u’₁ᵢ and u’₂ᵢ denote the ranks of the same observations if they belong to population π₂. Since the sum of the first N integers is (N(N+1))/2, it follows that Σ_k=1^n₁ u_ik + Σ_{k=n₁ + 1}^N u_ik’ = (N(N+1))/2 If the N pairs of ranks are plotted on a plane, it is likely that the n₁ points from population π₁ and the n₂ points from population π₂ will be interspersed forming a circular or elliptical pattern under the assumption that F₁(x₁, x₂) and F₂(x₁, x₂) are identical. Under the alternative hypothesis, it is likely that there will be a segregation of the points into two groups. A test statistic, S₁² is constructed to measure the extent of this segregation . The S₁²-statistic proposed here, is based on the Euclidean distance between the centroids of the ranks belonging to π₁ and π₂, in particular S₁²= (ū₁-ū₁')² + (ū₂-ū₂')² where ūᵢ = n₁⁻¹ Σ_k=1^n₁ u_ik , uᵢ’ = n₂⁻¹ Σ_{k=n₁ + 1}^N u_ik’ The first two moments of S₁² are derived under the following conditional randomization procedures keeping the ranks paired as given in the sample, n₁ pairs are selected at random (with equal probabilities) from among the N = n₁ + n₂ pairs and assigned to population π₁; the remaining n₂ pairs are assigned to population π₂. It is shown that E(S₁²) = (N²(N+1))/6n₁n₂ and σ²_S₁² = a₀₀ + a₁₁A₁₁+ a₁₂A₁₂ + a₂₁A₂₁ + a₂₂A₂₂ + a₁₁,₁₁A²₁₁ Where A_rs = Σ_k=1^N₁ u_1k^ru_2k^s are parameters depending on the sample, and the coefficients a₀₀, a₁₁, a₁₂, a₂₁, a₂₂ and a₁₁,₁₁ have been tabulated for values of n₁ and n₂ up to 20. The exact sampling distribution of S₁² is unknown However, it is sho•Nn that the distribution of (kE(S₁²))/ σ²_S₁² is approximately χ² with (2[E(S₁²)]²/ σ²_S₁² degrees of freedom. A rank analogue of Wald’s modification of Hotelling's T² is given and the first two moments obtained. Also, a multivariate extension is considered and a statistic, S₁²(k,2), constructed. The expectation and variance of S₁²(k,2) are derived. A multi-populatiun extension for the case of bivariate populations is given and the expectation is derived for a statistic, S₁²(2,p). A statistic, S₁²(k,p) is constructed for the most general case and its expectation is given. An alternative approach to the problem, also investigated, is by means of discriminant analysis. In this case simplified formulas are given for the calculation of the components of a vector which provides optimum discrimination. It is shown that this method is not a fruitful one for the construction of tests of significance pertaining to the original null hypothesis. / Doctor of Philosophy

Let Y₁, Y₂... be a sequence of random variables obeying the law Y_i = β’_i + ε_i, where β₁, β₂, ... is a sequence of unknown k-dimensional regression vectors; x₁, x₂, ... is a sequence of known k-dimensional regressor vectors; and ε₁ , ε₂, ... is a sequence of independent and identically distributed random variables. Assume that β₁ = ... = β_m = β, m ≥ k, and that β̂₀ is an asymptotically normal estimate of β based on Y₁ , ..., Y_m. This study develops nonparametric procedures for testing H₀: = β = β_m+1 = β_m+2 = …. The proposed tests involve sequences of truncated sequential tests. That is, a function of the residuals Y_m+1 - β̂’₀ x_m+1, …, Y_m+N - β̂’₀ x_m+N is examined for a shift in the model. If no shift is indicated all m+N observations are pooled and a new estimate of β, β̂₁, is formed. The next N residuals are then examined for a shift. The procedure continues until a.shift is indicated. Brownian motion results are used to obtain approximate critical values when the function of the residuals is: the cumulative sum of the signs of the residuals; the sequential Wilcoxon scores; the ordinary cumulative sums of residuals. Exact results are obtained for the cumulative sum of signs procedure when testing for a shift in median. Asymptotic relative efficiency results are also obtained. / Ph. D.

This thesis seeks to identify factors affecting the probability of selection of a Surface Warfare Officer (SWO) to Executive Officer (XO) in the U.S. Navy. Selections to XO are made by a board that meets annually. Because a candidate is considered for selection in up to three consecutive boards, the possible outcomes in this process are selection to XO in one of three annual boards, failure to be selected to XO in the third board, or attrition from the process between boards. Using data on the board's selections over a three-year period (2002-2004) a hazards-based logistic regression model is developed to estimate the probabilities associated with a candidate's disposition based on his or her career profile. The model confirms that a candidate's recent fitness and evaluation report (FITREP) is the single-most-important factor affecting selection. Additionally, officers who have completed a tour in Washington D.C. or at the Bureau of Naval Personnel have higher probabilities of selection than do those who have completed other shore tours. But when an officer receives a poor FITREP, the probability of selection is low, regardless of other factors. A nonparametric statistical analysis is used to confirm these findings.

A research report submitted to the Faculty of Science, University of the Witwatersrand, for the degree of Master of Science by Coursework and Research Report. / Researchers and data analysts often encounter a problem when analysing data with missing values. Methods for imputing continuous data are well developed in the literature. However, methods for imputing categorical data are not well established. This research report focuses on categorical data imputation using non-parametric and semi-parametric methods. The aims of the study are to compare different imputation methods for categorical data and to assess the quality of the imputation. Three imputation methods are compared namely; multiple imputation, hot deck imputation and random forest imputation. Missing data are created on a complete data set using the missing completely at random mechanism. The imputed data sets are compared with the original complete data set, and the imputed values which are the same as the values in the original data set are counted. The analysis revealed that the hot deck imputation method is more precise, compared to random forest and multiple imputation methods. Logistic regression is fitted on the imputed data sets and the original data set and the resulting models are compared. The analysis shows that the multiple imputation method affects the model fit of the logistic regression negatively.

The main goal of this thesis is to investigate the frequentist asymptotic properties of nonparametric Bayesian procedures in inverse problems and the Gaussian white noise model. In the first part, we study the frequentist posterior contraction rate of nonparametric Bayesian procedures in linear inverse problems in both the mildly and severely ill-posed cases. This rate provides a quantitative measure of the quality of statistical estimation of the procedure. A theorem is proved in a general Hilbert space setting under approximation-theoretic assumptions on the prior. The result is applied to non-conjugate priors, notably sieve and wavelet series priors, as well as in the conjugate setting. In the mildly ill-posed setting, minimax optimal rates are obtained, with sieve priors being rate adaptive over Sobolev classes. In the severely ill-posed setting, oversmoothing the prior yields minimax rates. Previously established results in the conjugate setting are obtained using this method. Examples of applications include deconvolution, recovering the initial condition in the heat equation and the Radon transform. In the second part of this thesis, we investigate Bernstein--von Mises type results for adaptive nonparametric Bayesian procedures in both the Gaussian white noise model and the mildly ill-posed inverse setting. The Bernstein--von Mises theorem details the asymptotic behaviour of the posterior distribution and provides a frequentist justification for the Bayesian approach to uncertainty quantification. We establish weak Bernstein--von Mises theorems in both a Hilbert space and multiscale setting, which have applications in $L^2$ and $L^\infty$ respectively. This provides a theoretical justification for plug-in procedures, for example the use of certain credible sets for sufficiently smooth linear functionals. We use this general approach to construct optimal frequentist confidence sets using a Bayesian approach. We also provide simulations to numerically illustrate our approach and obtain a visual representation of the different geometries involved.

有各式各樣的參數與非參數期貨定價模型被廣泛應用於金融領域。其中一些模型的組合能顯著提升期貨定價的準確性。更具體的說，可以先通過參數模型擬合數據，再使用非參數模型學習並修正誤差估價誤差。本論文作為范和Mancini(2009) 結果的延伸，將市場交易的歐式期權價格作為輸入數據，運用「有參數模型指導的非參數定價方法」對障礙期權進行估價。「自動誤差修正估價法」運用非參數方法對由參數估價法產生的誤差進行修正，使得障礙期權的非參數定價模型可以被視為一系列的歐式期權定價的組合。在整個障礙期權的估價過程中，本論文同時提供了一種分數階快速傅裡葉變換的應用，可通過由非參數方法獲得的標的資產對數的存活函數計算標的資產對數最大值分佈的特徵函數。 / There are a variety of parametric and nonparametric option pricing models commonly used in Finance. A combination of them can enhance the pricing performance significantly. Specifically, one proposes to fit the data with a parametric method and then correct the pricing errors empirically with a nonparametric learning approach. This thesis extends Fan and Mancini's (2009) model-guided nonparametric method to barrier option pricing using market traded European option data. Adopting automatic correction of errors (ACE) method to estimate the risk neutral conditional survivor function, by which the pricing error of the initial parametric estimates is captured nonparametrically, enables the nonparametric pricing procedure to value a barrier option as a sum of sequence of European options. As a byproduct from the valuation process, this thesis also provides a modified fractional fast Fourier transform technique compute the characteristic function of the running maximum log-price of the underlying asset nonparametrically through the calibrated survivor functions. / Detailed summary in vernacular field only. / Chi, Chengzhan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 38-39). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Nonparametric Local Regression Modelling --- p.4 / Chapter 2.1 --- Function Estimation by Local Constant --- p.4 / Chapter 2.2 --- Function Estimation by Local Linear Regression --- p.5 / Chapter 3 --- Nonparametric ACE European Option Pricing --- p.7 / Chapter 3.1 --- European Option Prices and Risk Neutral Survivor Functions --- p.7 / Chapter 3.2 --- Estimation of Risk Neutral Survivor Functions --- p.10 / Chapter 3.2.1 --- Risk Neutral Survivor Functions and Traded Options --- p.10 / Chapter 3.2.2 --- Survivor Function Estimation with Nonparametric ACE Method --- p.11 / Chapter 3.3 --- Representation of European Option Prices at Log-asset Level and Numerical Example --- p.15 / Chapter 4 --- Nonparametric ACE Barrier Option Pricing Framework --- p.20 / Chapter 4.1 --- Continuous-time Barrier Option --- p.20 / Chapter 4.2 --- Discrete Approximation and Backward Induction --- p.21 / Chapter 4.3 --- Decomposed Problems --- p.25 / Chapter 5 --- Nonparametric Estimation of Cumulative Distribution Function of M{U+2C7C}(R{U+209C}) --- p.28 / Chapter 5.1 --- Survivor Functions and Maxima Probabilities --- p.28 / Chapter 5.2 --- Characteristic Functions of Maxima --- p.30 / Chapter 5.2.1 --- Algorithm --- p.30 / Chapter 5.2.2 --- Preparation --- p.31 / Chapter 5.2.3 --- Fast Fourier Transform (FFT) --- p.31 / Chapter 5.2.4 --- Fractional Fast Fourier Transform (FRFT) --- p.33 / Chapter 5.2.5 --- Derivation of ΦR{U+209C} --- p.34 / Chapter 5.3 --- Numerical Experiments --- p.35 / Chapter 6 --- Conclusion --- p.37 / Bibliography --- p.38

Poon, Ka Ho. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 62-63). / Abstracts in English and Chinese. / Abstract of the thesis entitled --- p.ii / 摘要 --- p.iii / Acknowledgements --- p.iv / Chapter Section 1. --- Introduction --- p.1 / Chapter Section 2. --- Review of Useful Concepts --- p.4 / Chapter 2.1 --- Terms and Methodologies - Pattern Recognition --- p.4 / Chapter 2.1.1 --- Rolling Windows --- p.4 / Chapter 2.1.2 --- Smoothing Function - Kernel Regression --- p.5 / Chapter 2.1.3 --- Filtering Function ´ؤ Search for Extrema --- p.6 / Chapter 2.1.4 --- Filtering Function - The Pattern Detection Algorithm --- p.7 / Chapter 2.1.5 --- Risk-adjustment Model --- p.10 / Chapter Section 3. --- Data and Methodology --- p.12 / Chapter 3.1 --- Data --- p.12 / Chapter 3.2 --- Methodology --- p.12 / Chapter Section 4. --- Results --- p.17 / Chapter Section 5. --- Further Extension --- p.21 / Chapter Section 6. --- Discussions and Conclusion --- p.22 / APPENDIX 1 --- p.23 / References --- p.62

Semiparametric maximum likelihood analysis allows inference in errors-invariables models with small loss of efficiency relative to full likelihood analysis but with significantly weakened assumptions. In addition, since no distributional assumptions are made for the nuisance parameters, the analysis more nearly parallels that for usual regression. These highly desirable features and the high degree of modelling flexibility permitted warrant the development of the approach for routine use. This thesis does so for the special cases of linear and nonlinear regression with measurement errors in one explanatory variable. A transparent and flexible computational approach is developed, the analysis is exhibited on some examples, and finite sample properties of estimates, approximate standard errors, and likelihood ratio inference are clarified with simulation. / Graduation date: 2001