Mathematical Statistics

By Motoya Machida. 2006

This material is designed to introduces basic concepts and fundamental theory of mathematical statistics. A review of basic concepts will include likelihood functions, sufficient statistics, and exponential family of distributions. Then point estimation will be discussed, including minimum variance unbiased estimates, Cramér-Rao inequality, maximum likelihood estimates and asymptotic theory. Topics in general theory of statistical tests include Neyman–Pearson theorem, uniformly most powerful tests, and likelihood ratio tests.

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Citations in the corpus

Learning Distributions from their Samples under Communication Constraints (2019, background): “These two conditions justify interchanging differentiation and integration as in for each j , and they also ensure that p ( m | θ ) is continuously differentiable with respect to each θ j (see Lemma 1, Section 26 in [14]).”
The structure of optimal private tests for simple hypotheses (2018, methodology): “…estimation—the well-known characterization of parametric estimation in terms of Fisher information is obtained by showing that the Fisher information measures variability in the Hellinger distance and then employing the Hellinger-based characterization of nonprivate simple tests (e.g. [Bor99, Chap.”
Fisher Information Under Local Differential Privacy (2020, background): “…inequality due to Gill and Levit [11], which bounds the average ℓ 2 risk by R where λ ( θ ) = Q =1 λ j ( θ j ) is a prior for the parameter θ and J ( λ ) is the Fisher information associated with the prior λ : Assuming that Θ = [ B, B ] d , the prior λ can be chosen to minimize J ( λ ) [8], [22].”
Non-commercial Research and Educational Use including without Limitation Use in Instruction at Your Institution, Sending It to Specific Colleagues That You Know, and Providing a Copy to Your Institution's Administrator. All Other Uses, Reproduction and Distribution, including without Limitation Comm (2005, unknown): “This number is small for molecules with a limited number of spins (66) and high symmetry.”
The epic story of maximum likelihood (2007, unknown): “Hodges (see Figure 1 above) showed you could improve locally on maximum likelihood, basically by shrinking the estimate toward zero, and as such it might also be viewed as an early hint of the 1955 shrinkage estimates of Charles Stein that in multiparameter problems can improve uniformly on maximum ”
Modelling the Supply of Ecosystem Services from Agriculture: A Minimum-Data Approach (2006, unknown): “Using the spatial distribution of opportunity cost defined above, ϕ (ω), we can make a change of variable and define the spatial distribution of opportunity cost per unit of ecosystem service as φ (ω/e) = ϕ (ω)e, as illustrated in the left side of Figure 1 (Freund 1962, p. 120).”
Consistency of trace norm minimization (2007, methodology): “The following proposition—which is a consequence of standard results in M-estimation (Shao, 2003, Van der Vaart, 1998)—considers the case where n 1 / 2 λ n is tending to zero, where we obtain that ˆ W is asymptotically normal with mean W and covariance matrix n − 1 σ 2 Σ − 1 mm , i.e., for fast…”
Lower Bounds for Learning Distributions under Communication Constraints via Fisher Information (2019, methodology): “These two conditions justify interchanging diﬀerentiation and integration as in and they also ensure that p ( m | θ ) is continuously diﬀerentiable with respect to θ coordinate-wise (Borovkov, 1998, Section 26, Lemma 1).”
Loss Functions for Binary Class Probability Estimation and Classification: Structure and Applications (2005, background): “…in view of a simple asymptotic argument which is as follows: Standard formulas for asymptotic variances in estimating equations (for example, Shao 2003, p. 369) yield AVar( ˆ b) ≈ N E ∂ 2 L(b) −1 Var (∂L(b)) E ∂ 2 L(b) −1 From Section 9, Equations (30) and (31), we obtain Var ∂L(b) = 1 N 2…”
Face Recognition (2018, unknown): “When many samples from many classes are put together into a single set, The performance of the system for di(cid:11)erent numbers of MEF/MDF features.”
The History of the Cross Section of Stock Returns (2016, unknown): “, Bickel and Doksum (1977)). Multiple comparison methods exist to address this issue (e.g., White (2000)) but these methods rely on the reporting of all tests to adjust test sizes and p-values.”
Dynamic Pricing Without Knowing the Demand Function: Risk Bounds and Near-Optimal Algorithms (2009, unknown): “The following technical conditions articulate standard regularity assumptions in the context of Maximum Likelihood estimation (cf. Borovkov (1998)), and are used to define the admissible class of parametric demand functions.”
Dynamic Pricing and Learning: Historical Origins, Current Research, and New Directions (2013, unknown): “Similarly, results on the convergence rate of maximum-likelihood estimators, as in [308,309], are crucial in the analysis of pricing policies by Besbes and Zeevi [188], Broder and Rusmevichientong [184] and den Boer and Zwart [185].”