| 科目 | 參考書籍及範圍 |
| 分析通論 | 1. Real Analysis (Royden) Part I (or Measure and Integral (by Weeden and Zygmund) ch.1-ch.8)
2. Principles of Mathematical Analysis (by Rudin) ch.7 and ch.11
Topics:
(i) Lebesgue measure and Lebesgue integral
(ii) Lebesgue's differentiation
(iii) L^p-spaces
(iv) sequences and series of functions
|
| 代數通論 | Algebra (by Hungerford), ch.1-ch.6 |
| 機率論 | 1. A Course in Probability Theory (by Kai Lai Chung)
2. Probability Theory (by Yuan Shih Chow and Henry Teicher)
(i) distribution funcation
(ii) classes of sets, measure and probability spaces
(iii) random variable, expectation, independence
(iv) convergence concepts
(v) law of large numbers, random series
(vi) characteristic function
(vii) conditional expectation, conditional independence, introduction to martingales |
| 數理統計 | 1. Theory of Point Estimation (by Lehmenn)
2. Testing Statistical Hupotheses (by Lehmann)
Topics:
(i) group families, exponential families, sufficient statistics, completeness
(ii) UMVU estimators, performance of the estimators, the information inequality
(iii) location-scale families, the principle of equivariance
(iv) Bayes estimation, minimax estimation, minimaxity and admissibility
(v) convergence in probability and in law, large-sample comparisons of estimators, the median as an estimator of location, trimmed mean
(vi) asymptotic efficiency, efficient likelihood estimations
(vii) the Neyman-Pearson fundamental lemmas, distributions with monotones likelihood ratio
(viii) unbiasedness for hypothesis testing, UMP unbiased test
(ix) confidence sets, unbiased confidence sets, Bayes confidence sets
(x) symmetry and invariance, maximal invariants, most powerful invariant test |
| 數值分析 | Numerical Analysis (by Burden and Faires) ch.1-ch.9 |
