Questions tagged [minimax]

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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 ...
5
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0answers
71 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 ...
4
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1answer
1k 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 ...
4
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1answer
844 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 $$ d(\theta', \theta) + d(\theta'', \theta) \geq A d(\theta', \theta'') $$ ...
4
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0answers
112 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 ...
3
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1answer
104 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, ...
3
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1answer
116 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....
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0answers
86 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 ...
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190 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 ...
2
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1answer
447 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 ...
2
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1answer
60 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 ...
2
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1answer
80 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 ...
2
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0answers
20 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 ...
2
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1answer
35 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 ...
1
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1answer
81 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 ...
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0answers
64 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?
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1answer
219 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 ...
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0answers
82 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/...
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11 views

minimax estimator in applications

In the applications, does it really matter if the estimator is minimax or not if, say, we are interested in forecasting and with current non-minimax estimator we have better out-of-sample scores?
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12 views

Detection Theory minimax with non differentiable interior

The minimax is used in detection theory and decision theory for minimizing the overall average risk for the worst case prior. $$ \min_{\delta} \max_{\pi_0} r(\pi_0,\delta) = \max_{\pi_0} \min_{\delta}...
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6 views

Finding and presenting solution for optimized supplier portfolio

I am trying to find the best solution to create a "model" and presentation of the following challenge: I have several products which shall be bought from as few suppliers as possible. For each ...
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21 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} ...
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127 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 ...