This dissertation is a study of error in long haul optical fiber systems and how to coupe with them. First we characterize error events occurring during transmission, then we determine lower bounds on information capacity (achievable information rates) and at the end we propose coding schemes for these systems.Existing approaches for obtaining probability density functions (PDFs) for pulse energy in long-haul optical fiber transmission systems rely on numerical simulations or analytical approximations. Numerical simulations make far tails of the PDFs difficult to obtain, while existing analytic approximations are often inaccurate, as they neglect nonlinear interaction between pulses and noise.Our approach combines the instanton method from statistical mechanics to model far tails of the PDFs, with numerical simulations to refine the middle part of the PDFs. We combine the two methods by using an orthogonal polynomial expansion constructed specifically for this problem. We demonstrate the approach on an example of a specific submarine transmission system.Once the channel is characterized estimating achievable information rates is done by a modification of a method originally proposed by Arnold and Pfitser. We give numerical results for the same optical transmission system (submarine system at transmission rate 40Gb/s).The achievable information rate varies with noise and length of the bit patterns considered (among other parameters). We report achievable numerical rates for systems with different noise levels, propagation distances and length of the bit patterns considered.We also propose two iterative decoding schemes suitable for high-speed long-haul optical transmission. One scheme is a modification of a method, originally proposed in the context of magnetic media, which incorporates the BCJR algorithm (to overcomeintersymbol interference) and Low-Density Parity-Check (LDPC) codes for additional error resilience. This is a ``soft decision scheme" -meaning that the decoding algorithm operates with probabilities(instead of binary values). The second scheme is ``hard decision" -it operates with binary values. This scheme is based on the maximum likelihood sequence detection-Viterbi algorithm and a hard decision"Gallager B" decoding algorithm for LDPC codes.

Doctor of Philosophy / Department of Statistics / Haiyan Wang / Advances in technology easily collect a large amount of data in scientific research such as agricultural screening and micro-array experiments. We are particularly interested in data from one-way and crossed two-way designs that have a large number of treatment combinations but small replications with heteroscedastic variances. In this framework, several test statistics have been proposed in the literature. Even though the form of these proposed test statistics may be different, they all use limiting normal or chi-square distribution to conduct their tests. Such approximation approaches the true distribution very slowly when the sample size ni is small while the number of levels of treatments a gets large. A strategy to obtain better accuracy in the classical large sample size setting is to use the bootstrap procedure with studentized statistic. Unfortunately, the available bootstrap method fails when the number of treatment level combinations is large while the number of replications is small. The Fisher and Hall (1990) asymptotic pivotal statistic under large sample size setting is no longer pivotal under small sample size setting with large number of treatment levels. In the first part of this dissertation, we start with describing suitable bootstrap statistics and procedures for hypothesis tests in one- and two-way ANOVA with a large number of levels and small sample sizes. We prove that the theoretical type I error-rate of Akritas and Papadatos (2004) and Wang and Akritas (2006) test statistics and their corresponding bootstrap versions have accuracy of order O(1/√a). We then modify their statistics to obtain asymptotically pivotal statistics in our current framework. We prove that the theoretical type I error-rate of the bootstrap version of the pivotal statistics is accurate up to order O(1/√a). In the second part of the dissertation, we propose a new test statistic in one-way ANOVA which is asymptotically pivotal in the current setting. We improve the accuracy of approximation of the distribution of the test statistic by deriving asymptotic expansion of the statistic under the current framework and define a new test rejection region through Cornish-Fisher expansion of quantiles. The type I error-rate of the new test has a faster convergence rate and is accurate up to order O(1/a). Simulation studies show that our tests performs better in terms of type I error-rate but comparable power with that of Akritas and Papadatos (2004) in the large a small ni setting. The connection between our asymptotic expansions and bootstrap distribution in the large a small ni setting is discussed. Our proposed test based on asymptotic expansion and Cornish-Fisher expansion of quantiles have both the advantage of higher accuracy and computational efficiency due to no resampling is needed.

<p>We investigate mathematical tools, Edgeworth series expansion and the saddle point method, which are approximation techniques that help us to estimate the distribution function for the standardized mean of independent identical distributed random variables where we will take into consideration the lattice case. Later on we will describe one important application for these mathematical tools where game developing companies can use them to reduce the amount of time needed to satisfy their standard requests before they approve any game</p>

This thesis is focused around Edgeworths expansion for aproximation of distribution for parameter estimation. Aim of the thesis is to introduce term Edgeworths expansion, its assumptions and terminology associeted with it. Afterwords demonstrate process of deducting first term of Edgeworths expansion. In the end demonstrate this deduction on examples and compare it with different approximations (mainly central limit theorem), and show strong and weak points of Edgeworths expansion.

This thesis is focused around Edgeworth's expansion for approximation of distribution for parameter estimation. Aim of the thesis is to introduce term Edgeworth's expansion, its assumptions and terminology associated with it. Afterwards demonstrate process of deducting first term of Edgeworth's expansion. In the end demonstrate this deduction on examples and compare it with different approximations (mainly central limit theorem), and show strong and weak points of Edgeworth's expansion.

We investigate mathematical tools, Edgeworth series expansion and the saddle point method, which are approximation techniques that help us to estimate the distribution function for the standardized mean of independent identical distributed random variables where we will take into consideration the lattice case. Later on we will describe one important application for these mathematical tools where game developing companies can use them to reduce the amount of time needed to satisfy their standard requests before they approve any game

被廣泛應用在選擇權定價的Black-Scholes 模型[3] 時常在深價內與深價外 的選擇權價格有錯價的現象，也就是理論價格估計實際市場價格的偏差。藉由 Black-Scholes 評價公式所反推出的隱含波動度往往不像我們所期待的在不同履約價格具有一致性，這種現象被稱為波動度的微笑曲線。在這份論文裡，我們參考Jarrow and Rudd [13] 提出的方法，將Edgeworth展開式套用在Black-Scholes模型作延伸應用，進而推導出偏態峰態修正後的的評價公式，再利用台指選擇權的市場資料作實證分析並與Filho and Rosenfeld [1] 的研究作比較。我們發現從台指選擇權的實證結果得到非常態分配的隱含偏態和隱含峰態。此外，理論價格的估計偏誤比例顯著的被新的模型改善且隱含波動度的微笑曲線也變的較為平坦，這個方法提供我們一個有效的方法，利用標的資產的偏態峰態得到該資產的近似分配。 / The Black-Scholes [3] option pricing model widely applied in option contracts frequently misprices deep-in-the-money and deep-out-of-the-money options. The implied volatilities computed by the Black-Scholes formula are not identical on each strike price as we expect. This phenomenon is called the volatility smile or skew. In this thesis, we derived a skewness- and kurtosis-adjusted option pricing model using an Edgeworth expansion constructed by Jarrow and Rudd [13] to an investigation of TAIEX option prices and compare the results with those in Filho and Rosenfeld [1]. We found that non-normal skewness and kurtosis are implied by TAIEX option returns. Moreover, the magnitude of price deviations were signicantly corrected and the volatility skew is attened. This approach provides an useful way to derive an approximate distribution of a underlying security with its skewness and kurtosis.

Doctor of Philosophy / Department of Statistics / Haiyan Wang / Various tests have been created to compare the means of two populations in many scenarios and applications. The two-sample t-test, Wilcoxon Rank-Sum Test and bootstrap-t test are commonly used methods. However, methods for skewed two-sample data set are not well studied. In this dissertation, several existing two sample tests were evaluated and four new tests were proposed to improve the test accuracy under moderate sample size and high population skewness. The proposed work starts with derivation of a first order Edgeworth expansion for the test statistic of the two sample t-test. Using this result, new two-sample tests based on Cornish Fisher expansion (TCF tests) were created for both cases of common variance and unequal variances. These tests can account for population skewness and give more accurate test results. We also developed three new tests based on three transformations (T[subscript i] test, i = 1; 2; 3) for the pooled case, which can be used to eliminate the skewness of the studentized statistic. In this dissertation, some theoretical properties of the newly proposed tests are presented. In particular, we derived the order of type I error rate accuracy of the pooled two-sample t-test based on normal approximation (TN test), the TCF and T[subscript i] tests. We proved that these tests give the same theoretical type I error rate under skewness. In addition, we derived the power function of the TCF and TN tests as a function of the population parameters. We also provided the detailed conditions under which the theoretical power of the two-sample TCF test is higher than the two-sample TN test. Results from extensive simulation studies and real data analysis were also presented in this dissertation. The empirical results further confirm our theoretical results. Comparing with commonly used two-sample parametric and nonparametric tests, our new tests (TCF and Ti) provide the same empirical type I error rate but higher power.

Disertacijoje tiriamos negrąžintinių imčių pozicinių statistikų tiesinių kombinacijų (L-statistikų) savybės. Pagrindinis disertacijos uždavinys yra L-statistikų skirstinių normaliosios aproksimacijos patikslinimas trumpaisiais Edgeworth'o skleidiniais. Šių aproksimacijų tikslumui įvertinti disertacijoje naudojamas baigtinių populiacijų simetrinių statistikų Hoeffding'o skleidinys. Pirmame disertacijos skyriuje gautos išreikštinės pirmųjų L-statistikos Hoeffding'o skleidinio narių ir skleidinio liekamųjų narių formulės. Jomis naudojantis, antrame disertacijos skyriuje išspręsti tokie uždaviniai: gautas optimalus imties ekstremaliųjų reikšmių dispersijų viršutinysis įvertis; nustatytos pakankamosios L-statistikų asimptotinio normalumo sąlygos; sukonstruotas trumpasis L-statistikos Edgeworth'o skleidinys ir nustatytos pakankamosios šios aproksimacijos sąlygos. Trečiame disertacijos skyriuje sukonstruoti L-statistikos dispersijos ir Edgeworth'o skleidinio parametrų įvertiniai. Ketvirtame disertacijos skyriuje sukonstruoti ir ištirti Stjudentizuotų ir kartotinių imčių L-statistikų trumpieji Edgeworth'o skleidiniai. / Properties of linear combinations of order statistics (L-statistics), where samples are drawn without replacement, are considered in the thesis. The main object of the thesis is an improvement of the normal approximation to distributions of L-statistics by one-term Edgeworth expansions. An accuracy of these approximations is estimated using the Hoeffding decomposition of finite population symmetric statistics. In the first chapter of the thesis, explicit expressions of the first terms and remainder terms of the Hoeffding decomposition of L-statistics are obtained. The main applications of the decomposition are given in the second chapter: the optimal upper bound for variances of the sample minimum and maximum is obtained; sufficient conditions for the asymptotic normality of L-statistics are established; the one-term Edgeworth expansion for L-statistics is constructed and sufficient conditions for the validity of this approximation are obtained. In the third chapter, estimators of the variance and parameters that define the Edgeworth expansion of an L-statistic are constructed. In the fourth chapter, a one-term Edgeworth expansion for a Studentized L-statistic and empirical Edgeworth expansions are constructed and analyzed.

本論文主要討論貝氏漸近的比較，推導出參數的聯合後驗分配與利用圖形來診斷馬可夫鏈蒙地卡羅的收斂。Johnson (1970)利用泰勒展開式得到個別後驗分配的展開式，此展開式是根據概似函數與先驗分配。 Weng (2010b) 和 Weng and Hsu (2011) 利用 Stein’s 等式且由概似函數與先驗分配估計後驗動差；將這些後驗動差代入Edgeworth 展開式得到近似後驗分配，此近似分配的誤差可精確到大O的負3/2次方與Johnson’s 相同。另外Weng and Hsu (2011)發現Weng (2010b) 和Johnson (1970)的近似展開式各別項誤差到大O的負1次方不一致，由模擬結果得到Weng’s 在此項表現比Johnson’s 好。另外由Weng (2010b)得到一維參數 的Edgeworth 近似後驗分配延伸到二維參數的聯合後驗分配；並應用二維參數的聯合後驗分配於多階段資料。本論文我們提出利用圖形來診斷馬可夫鏈蒙地卡羅收斂的方法，並且應用一般化線性模型與混合常態模型做為模擬。 關鍵字: Edgeworth 展開式；馬可夫鏈蒙地卡羅；個別後驗分配；Stein’s 等式 / Johnson (1970) obtained expansions for marginal posterior distributions through Taylor expansions. The expansion in Johnson (1970) is expressed in terms of the likelihood and the prior. Weng (2010b) and Weng and Hsu (2011) showed that by using Stein's identity we can approximate the posterior moments in terms of the likelihood and the prior; then substituting these approximations into an Edgeworth series one can obtain an expansion which is correct to O(t{-3/2}), similar to Johnson's. Weng and Hsu (2011) found that the O(t{-1}) terms in Weng (2010b) and Johnson (1970) do not agree and further compared these two expansions by simulation study. The simulations confirmed this finding and revealed that our O(t{-1}) term gives better performance than Johnson's. In addition to the comparison of Bayesian asymptotics, we try to extend Weng (2010a)'s Edgeworth series for the distribution of a single parameter to the joint distribution of all parameters. Since the calculation is quite complicated, we only derive expansions for the two-parameter case and apply it to the experiment of multi-stage data. Markov Chain Monte Carlo (MCMC) is a popular method for making Bayesian inference. However, convergence of the chain is always an issue. Most of convergence diagnosis in the literature is based solely on the simulation output. In this dissertation, we proposed a graphical method for convergence diagnosis of the MCMC sequence. We used some generalized linear models and mixture normal models for simulation study. In summary, the goals of this dissertation are threefold: to compare some results in Bayesian asymptotics, to study the expansion for the joint distribution of the parameters and its applications, and to propose a method for convergence diagnosis of the MCMC sequence. Key words: Edgeworth expansion; Markov Chain Monte Carlo; marginal posterior distribution; Stein's identity.