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Wang, Yuqian. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaves 36-39). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.ii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Properties of approximate solutions --- p.6 / Chapter 3 --- Liapunov-Schmidt Reduction --- p.17 / Chapter 4 --- Critical point of the reduced energy functional --- p.28 / Bibliography --- p.36

In this thesis, two topics in asymptotic statistics and spline functions are studied. / The first one is a study of testing the equality of Sharpe ratios. We compare two approaches to testing for the equality of many Sharpe ratios: the multivariate test of Wright et al. (2014) and Ledoit and Wolf's (2008) pairwise test. Firstly, Ledoit and Wolf's pairwise test is generalized to a multivariate one for direct comparison. We conclude by proposing a modified version that incorporates the Warp-Speed calibration method of Giacomini et al. (2013). The resulting procedure is much less computationally expensive but is comparable in its accuracy. / The second one is a study of the theoretical properties of an exponential weighting aggregated (EWA) penalized spline estimator, where the smoothing parameter is being averaged over an exponential reweighted posterior distribution. We show that the finite sample mean squared error of the EWA estimator is smaller than that of the smooth penalized spline estimator when the smoothing parameter is chosen to be a fixed value. Consistency and asymptotic normality of the EWA estimator are also developed under general situations. / 這篇論文研究了兩個關於漸近統計和樣條函數的話題。 / 其一是在假設檢驗夏普比率的相等性這個問題中的應用。我們首先比較了Wright et al.（2014）的多元檢驗方法和Ledoit & Wolf（2008）的二元檢驗方法。我們先將Ledoit & Wolf (2008)的二元檢驗法拓展到多元層面，以方便比較。最後，我們提出了採用Giacomini et al.（2013）的曲速法修改后的自助抽樣檢驗法。這種方法極大地降低了實際計算成本，同時保持了其準確性。 / 其二是對樣條估計的漸近表現的研究。我們研究了指數權重合計平滑樣條（EWA）估計，這種方法通過貝葉斯法給予平滑參數一個後驗分佈并將其加權合計。我們通過擴展一個oracle不等式驗證了EWA估計的方差比選定了一個平滑參數的一般平滑樣條估計的方差要小。此外，我們還驗證了EWA估計在一般情況下的一致性和漸近正態性。 / Huang, Wei. / Thesis M.Phil. Chinese University of Hong Kong 2014. / Includes bibliographical references (leaves 80-84). / Abstracts also in Chinese. / Title from PDF title page (viewed on 13, September, 2016). / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only.

by Choi Chun-Man. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 56-58). / Abstract also in Chinese. / Chapter 1 --- Radial symmetry for solutions of a semilinear el- liptic equation on a bounded domain --- p.6 / Chapter 2 --- Asymptotic symmetry of singular solutions to a semilinear elliptic equation --- p.12 / Chapter 2.1 --- Introduction --- p.12 / Chapter 2.2 --- Some preliminary analysis --- p.14 / Chapter 2.3 --- Proof of Theorem 2.1 --- p.20 / Chapter 3 --- Classification of non-negative solutions to Yamabe type equations --- p.32 / Chapter 3.1 --- Introduction --- p.32 / Chapter 3.2 --- The Proof of Theorem 3.1 for k > 0 --- p.38 / Chapter 3.3 --- Case k <0 --- p.48 / Bibliography

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D. Phil. / There have been two rather distinct approaches to the analysis of time series: the time domain approach and frequency domain approach. The former is exemplified by the work of Quenouille (1957), Durbin (1960), Box and Jenkins (1970) and Ljung and Box (1979). The principal names associated with the development of the latter approach are Slutsky (1929, 1934), Wiener (1930, 1949), Whittle (1953), Grenander (1951), Bartlett (1948, 1966) and Grenander and Rosenblatt (1957). The difference between these two methods is discussed in Wold (1963). In this thesis, we are concerned with a frequency domain approach. Consider a model of the "signal plus noise" form yt = g (2t — 1 2n ) + 77t t= 1,2,—. ,n (1.1) where g is a function on (0, 1) and Ti t is a white noise process. Our interest is primarily in testing the hypothesis that g is constant, that is, that it does not change over time. There is a vast literature related to this problem in the special case where g is a step function. In that case (1.1) specifies an abrupt change model. Such abrupt change models are treated extensively by Csorgo and Horvath (1997), where an exhaustive bibliography can also be found. The methods associated with the traditional abrupt change models are, almost without exception, time domain methods. The abrupt change model is in many respects too restrictive since it confines attention to signals g that are simple step functions. In practical applications the need has arisen for tests of constancy of the mean against a less precisely specified alternative. For instance, in the study of variables stars in astronomy (Lombard (1998a)) the appropriate alternative says something like: "g is non-constant but slowly varying and of unspecified functional form". To accommodate such alternatives within a time domain approach seems to very difficult, if at all possible. They can, however, be accommodated within a frequency domain approach quite easily, as shown by, for example, Lombard (1998a and 1998b). Tests of the constancy of g using the frequency domain characteristics of the observations have been investigated by a number of authors. Lombard (1988) proposed a test based on the maximum of squared Fourier cosine coefficients at the lowest frequency oscillations. Eubank and Hart (1992) proposed a test which is based on the maximum the averages of Fourier cosine coefficients. The essential idea underlying these tests is that regular variation in the time domain manifests itself entirely at low frequencies in the frequency domain. Consequently, when g is "high frequency" , that is consists entirely of oscillations at high frequencies, the tests of Lombard (1988) and of Eubank and Hart (1992) lose most of their power. The fundamental tool used in frequency domain analysis is the periodogram; see Chapter 2 below for the definition and basic properties of the latter. A new class of tests was suggested by Lombard (1998b) based on the weighted averages of periodogram ordinates. When 7i t in model (1.1) are i.i.d. random variables with zero mean and variance cr-2 , one form of the test statistic is T1r, = Etvk fiy (A0/0-2 - (1.2) k=1 where wk is a sequence of constants that decrease as k increases and m = [i]. The rationale for such tests is discussed in detail in Lombard (1998a and 1998b). The greater part of the present Thesis consists of an investigation of the asymptotic null distributions, and power, of such tests. It is also shown that such tests can be applied directly to other, seemingly unrelated problems. Three instances of the latter type of application that are investigated in detail are (i) frequency domain competitors of Bartlett's test for white noise, (ii) frequency domain-based tests of goodness-of-fit and (iii) frequency domain-based tests of heteroscedasticity in linear or non-linear regression. regression. The application of frequency domain methods to these problems are, to the best of our knowledge, new. Until now, most research has been restricted to the case where m in (1.1) are i.i.d. random variables. As far as the correlated data are concerned, the changepoint problem was investigated by, for instance, Picard (1985), Lombard and Hart (1994) and Bai (1994) using time domain methods. Kim and Hart (1998) proposed two test statistics derived from frequency domain considerations and that are modeled along the lines of the statistics considered by Eubank and Hart (1992) in the white noise case. An analogue of the type of test statistic given in (1.2) for use with correlated data was proposed, and used, by Lombard (1998a). The latter author does not, however, provide statements or proofs regarding the asymptotic properties of the proposed test.

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