# Questions tagged [minimax]

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### 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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### 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 ...
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### 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} ...
64 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 ...
51 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 ...
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### 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 ...
58 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 ...
61 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 ...
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### 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 ...
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### 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 ...
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### 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/...
114 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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### 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 ...
730 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 $$d(\theta', \theta) + d(\theta'', \theta) \geq A d(\theta', \theta'')$$ ...
429 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 ...
100 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, ...
401 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 ...
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### 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 ...
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 ...