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Why is the statement "the bayes estimator is bayes optimal" profound?

I'm trying to understand why people make a big deal about the optimality of a Bayes estimator. Certainly, if I have a Bayes estimator, then my expected loss is minimized, almost by definition. So, $$ \...
Y. S.'s user avatar
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Minimax Lower Bound on Mean Squared Error for average treatment effect in a Bernoulli randomized design?

Does anyone know of any references that show a proof of Minimax Lower Bound on Mean Squared Error for the average treatment effect in a Bernoulli randomized design? Take a population of $n$ units for ...
alene73's user avatar
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2 votes
1 answer
130 views

Counterexamples: Minimax but not efficient, efficient but not minimax

Are there examples of estimators that are minimax, but not efficient? Perhaps, to be more concrete, any of the following: (strong) Estimator sequence for each $n$ that is minimax but does not match ...
JohnA's user avatar
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1 vote
1 answer
81 views

Does large mutual information (between observations and parameter) imply the existence of a good estimator?

This question concerns the standard setting for applying Fano's inequality to derive minimax bounds for a parameter estimation problem. The goal is to estimate a parameter described by a random ...
forky40's user avatar
  • 163
9 votes
1 answer
416 views

Minimax estimator for geometric distribution

I'm trying to solve this problem: Let $X$ be a single sample from Geo($p$) where $p ∈ (0, 1)$. Find a minimax estimator for $p$ under the loss $L(p, δ(x)) = (p−δ(x))^2/ p(1−p)$ . I'm trying to put ...
user115608's user avatar
8 votes
1 answer
172 views

What do the terms "nearly-optimal rate", "near-minimax rate", "minimax optimal rate" and "minimax rate" mean in the context of posterior consistency?

Definition: A sequence $\epsilon_n$ is a posterior contraction rate at the parameter $θ_0$ if $$\Pi_n(θ: d(θ, θ_0) ≥ M_n \epsilon_n| X^{(n)}) → 0$$ in $P^{(n)}_{θ_0}$-probability, for every $M_n → ∞$. ...
user3911153's user avatar
2 votes
1 answer
187 views

Reference request for contemporary monographs on minimax theory

Are there any contemporary monographs on minimax theory$^1$, and if so, what are they? So far, I have seen minimax theory given some treatment in Theory of Point Estimation by Lehmann and Casella (...
microhaus's user avatar
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1 vote
1 answer
267 views

How to find the Bayesian equivalent of of $\bar{X}-\bar{Y}$

Let $X_1, \dots, X_n$ i.i.d. from $N(\mu, \sigma^2)$; $Y_1, \dots, Y_m$ i.i.d. from $N(\eta, \tau^2)$. $X_1, \dots, X_n$ are independent of $Y_1, \dots, Y_m$. And $\tau^2$ and $\sigma^2$ are ...
anonyx2's user avatar
  • 33
4 votes
1 answer
448 views

Showing $X\sim \operatorname{Poi}(\lambda)$ is minimax

Assume that $X$ has $\operatorname{Poisson} (\lambda)$ distribution and the loss function is $\ell(\lambda,a)=\frac{(\lambda-a)^2}{\lambda}$. Now, I want to show that $X$ is minimax. A hint that is ...
statwoman's user avatar
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5 votes
1 answer
173 views

Mid-range via minimax

Warning: crossposted at Mathematics SE. Given vector ${\rm a} \in \Bbb R^n$, $$\begin{array}{ll} \displaystyle\arg\min_{x \in {\Bbb R}} & \left\| x {\Bbb 1}_n - {\rm a} \right\|_2^2\end{array} = \...
Rodrigo de Azevedo's user avatar
1 vote
0 answers
109 views

Minimax estimators in Bayesian analysis

I got recently introduced to minimax methods in statistical decision theory. Is there an analogue in Bayesian analysis and some resources related to this?
Dion's user avatar
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4 votes
0 answers
388 views

The difference of normal means is also minimax?

Let $X_i \sim N(\xi, \sigma^2)$ and $Y_i \sim N(\eta, \tau^2)$ for known $\sigma^2$ and $\tau^2$. I know that $\bar{X}$ and $\bar{Y}$ are minimax under squared error loss since their variance is ...
Xiaomi's user avatar
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0 answers
34 views

Minimax equivalence of Matrix Norm

In a proof of random matrix theory, the author makes use of the following equivalence: \begin{equation} \inf_{v\in V(r)}\lVert Xv \rVert_2 = \inf_{v\in V(r)} \sup_{u \in S^{n-1}}u^TXv \end{equation} ...
Akshay Bansal's user avatar
1 vote
1 answer
645 views

Bayes Minimax Estimation

Let $S\sim B(n,\theta), l(\theta, a) = (\theta - a)^2,\delta=\bar{X} = S/n,$ and $$\delta^*(S)=\left(S+\frac{1}{2}\sqrt{n}\right)/\left(n+\sqrt{n}\right)$$ where $B(n,\theta)$ is Bernoulli ...
John Ling's user avatar
3 votes
1 answer
270 views

Why is the maximum risk of an estimator independent of a prior distribution over the parameter?

One way of choosing an estimator $\delta(x)$ for data $X$ distributed as $P_{\theta}(X)$, where $\theta \in \Theta$ is: $$minimize \sup_{\theta \in \Theta} Risk(\delta(x), \theta)$$ In this case why ...
wabbit's user avatar
  • 370
2 votes
1 answer
1k views

Is it possible to use alpha-beta pruning for non-zero-sum games with more than two players?

I have read somewhere that the minimax algorithm can be generalized for more than two players. Imagine that we have 3 players that each of them want to maximize its own answer. Is it possible to use ...
Amir Hooshang's user avatar
5 votes
0 answers
195 views

Do shrinkage estimators solve the Neyman-Scott paradox?

I read the following SE question: What problem do shrinkage methods solve? And I wondered if shrinkage estimators provide a consistent estimator of the sample variance in a "mixed-effects" model using ...
AdamO's user avatar
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3 votes
1 answer
182 views

Choose parameters ,such that MSE of an estimator is constant

I have an estimator : $X = (X_1,X_2,...,X_n)$ are iid and have distribution $B(1,\theta)$ $T(X) = X_1 + X_2 + ... + X_n$ I need to find such value of constants $\alpha$ and $\beta$ s.t MSE of ...
Daniil Yefimov's user avatar
2 votes
0 answers
31 views

Space-filling design algorithms from a discrete domain

Given $S = (X_1, \ldots, X_n)$, where each $X_i \in \mathbb{R}^2$, are there algorithms that produce $(X_{(1)}, \ldots, X_{(m)})$, where $m << n$ and the $X_{(i)}$ are sampled from $S$ to fill ...
Jimmy Risk's user avatar
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0 answers
271 views

Minimax decision function about poisson

Let $X_1,...,X_n$ be independent r.v.'s from the poisson $P(\theta)$ ,$\theta \in (0,\infty)$. and consider loss function $L(\theta ; \delta ) =\frac{[\theta - \delta(x_1,...,x_n)]^2}{\theta}$. then ...
Seongqjini's user avatar
1 vote
0 answers
113 views

Optimizing Hyperparameters with just a Score Function

I'm trying to create artificial intelligence for the popular game "2048" using this page as a reference: https://stackoverflow.com/questions/22342854/what-is-the-optimal-algorithm-for-the-game-2048/...
Connor Byron's user avatar
3 votes
1 answer
148 views

Minimax Test for Heterogeneous Gaussian Mean Shifting

For $i=1,…,N$, we have the data $X_i \sim \mathcal{N}( \delta \mu_i, \mu_i^2 )$, where $\mu_i>0$ and they are some unknown nuisance parameters. We want to test if there is a shift of the means, i.e....
Martin Zhang's user avatar
2 votes
1 answer
77 views

Auxiliary random experiment

This question is from Robert Hogg's Introduction to Mathematical Statistic 6th version exercise 8.5.2 page 461. The question is: Let $X_1, X_2,...,X_{10}$ be a random sample of size 10 from a Poisson ...
Deep North's user avatar
  • 4,746
4 votes
1 answer
1k views

Does the local triangle inequality holds for Kullback-Leibler divergence

Does the local triangle inequality holds for the Kullback-Leibler divergence? For the local triangle inequality, I mean the $$ d(\theta', \theta) + d(\theta'', \theta) \geq A d(\theta', \theta'') $$ ...
Steve's user avatar
  • 287
4 votes
1 answer
244 views

Simple question on graphical representation of minmax decision rule

In the picture below, I cannot understand why the minmax decision rule is on the line $R_1=R_2$. $R_i=R(\theta_i,d)$, where $\theta_i$ is the parameter and $d$ is the decision rule. $S$ is the risk ...
An old man in the sea.'s user avatar
6 votes
1 answer
753 views

Why are inf and sup used in the definition of minimax estimators?

An estimator $\hat{\delta}$ is minimax iff $$\sup_\theta R(\theta,\hat{\delta})=\inf_\delta\sup_\theta R(\theta,\delta)$$ or in english iff out of all estimators it has the least maximum risk. For ...
Julian Karch's user avatar
  • 1,960
4 votes
1 answer
171 views

Convergence rate: $E\|\hat f - f\|^2 = O(\psi_n)$ vs $\|\hat f - f\| = O_p(\psi_n^{1/2})$

I have seen two types of results on convergence rates for some estimator $\hat f$: $E\|\hat f - f\|^2 = O(\psi_n)$ and $\|\hat f - f\| = O_p(\sqrt{\psi_n})$. The first result seems to be stronger, ...
Lionville's user avatar
  • 395
3 votes
1 answer
907 views

minimax property of sample mean

Suppose $X_1,X_2,\ldots,X_n$ are iid $\mathcal{N}(\mu,\sigma^2)$, where $\sigma$ is known, but $\mu$ is not. We wish to construct a confidence interval of length $L$ (given) for $\mu$. Is it true that ...
Jeff's user avatar
  • 313
5 votes
1 answer
2k views

Bayesian Estimation: Bernoulli and Quadratic Loss Function

I am trying to understand a solution to this problem (I am a very beginner in Bayesian statistics) and I am terribly confused so I would appreciate it if someone could explain to me how exactly this ...
Jen's user avatar
  • 419
3 votes
0 answers
148 views

Finding the minimax estimator

Suppose $x$ comes from a Bernoulli distribution $f(x, \theta) = \theta^{x}(1-\theta)^{1-x}$ where $0 \leq \theta \leq 1$. Now suppose $y$ comes from the same distribution. An estimate for the ...
guestguy123's user avatar
4 votes
0 answers
224 views

Question about foundations of the uniform shrinkage prior

I am collecting papers about the uniform shrinkage prior for hierarchical Bayesian model. In "A prior for the variance in hierarchical models" of Michael J. Daniels it is stated at the end of page two ...
beuhbbb's user avatar
  • 5,063
4 votes
0 answers
204 views

Finding minimax estimator $\theta$

Suppose $X_1,\ldots,X_n$ is a random sample from $N(\theta,1)$ so that $\theta\geq a$ . How can I calculate minimax estimator $\theta$ under loss function $L(\delta,\theta)=(\delta-\theta)^2$
hadisanji's user avatar
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