By Rabi Bhattacharya, Lizhen Lin, Victor Patrangenaru

This graduate-level textbook is essentially aimed toward graduate scholars of records, arithmetic, technological know-how, and engineering who've had an undergraduate path in records, an top department path in research, and a few acquaintance with degree theoretic chance. It offers a rigorous presentation of the center of mathematical statistics.

Part I of this e-book constitutes a one-semester path on uncomplicated parametric mathematical records. half II offers with the big pattern thought of information - parametric and nonparametric, and its contents might be coated in a single semester besides. half III presents short bills of a couple of subject matters of present curiosity for practitioners and different disciplines whose paintings consists of statistical methods.

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**Additional resources for A Course in Mathematical Statistics and Large Sample Theory**

**Example text**

In other words, d(T is an estimator. 10) holds whether T is suﬃcient or not. 1. Let c(θ) be a real-valued parametric function of θ, and d(X) an estimator of c(θ), with Eθ d2 (X) ﬁnite for all θ. Let T be a suﬃcient statistic for θ. 11) ˆ )) = 1. , Eθ d(X) = c(θ) ∀ θ), then so ˆ ). is d(T Proof. 2, by letting A = R and noting that a → L(θ, a) ≡ (c(θ) − a)2 is strictly convex. Alternatively, one has ˆ ) + d(T ˆ ) − c(θ))2 Eθ (d(X) − c(θ))2 = Eθ (d(X) − d(T ˆ ) − c(θ))2 + Eθ (d(X) − d(T ˆ ))2 , = Eθ (d(T since ˆ ))(d(T ˆ ) − c(θ))] = Eθ [(d(T ˆ ) − c(θ)) · Eθ (d(X) − d(T ˆ ) | T )] = 0.

For the purpose of unbiased estimation the following generally stronger property than minimality is very useful. 3. t. Pθ ∀ θ ∈ Θ, Eθ g(T ) = 0 ∀ θ ∈ Θ =⇒ Pθ (g(T ) = 0) = 1 ∀ θ ∈ Θ. 12) holds for all bounded measurable functions g(T ) of T . A complete suﬃcient statistic is obviously boundedly complete. 1. Suppose a minimal suﬃcient statistic S exists. Then if a suﬃcient statistic T exists which is boundedly complete then T is minimal. 1 Suﬃcient Statistics and Unbiased Estimation 45 Proof. Suppose S is minimal suﬃcient and T is boundedly complete suﬃcient.

Xn be independent normal N (θ, σ 2 ) random variables (each with mean θ and variance σ 2 ). Assume σ 2 is known. Then with S = R, v(dx) = dx, the common pdf of the Xi ’s is 48 4 Suﬃcient Statistics, Exponential Families, and Estimation 1 p(x | θ) = (2πσ 2 )− 2 exp − (x − θ)2 2σ 2 −θ2 2σ 2 c(θ) = (2πσ 2 )−1/2 exp θ x σ2 −x2 h(x) = exp 2σ 2 (θ ∈ Θ = R), = c(θ)h(x)exp , . 21) Hence the natural parameter is π = θ/σ 2 and t(x) = x. The natural parameter space is = R. 22) where c(θ) = (2πnσ 2 )−1/2 exp{−nθ2 /2σ 2 }, h(t) = exp{−t2 /2nσ 2 }, and π = θ/σ 2 .