Questions tagged [minimax]
The minimax tag has no usage guidance.
33
questions
1
vote
0
answers
16
views
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 ...
- 11
0
votes
0
answers
17
views
Showing that an expression is an unbiased estimator of the risk (Berger, Question 5.53a)
Suppose $X \sim Normal_p(\theta, \Sigma)$, $\Sigma$ known and that $L(\theta,\delta) = (\theta-\delta)^T Q (\theta - \delta)$, $Q$ a known $p \times p$ positive definite matrix. Let $\delta(x)$ = $x + ...
- 11
2
votes
1
answer
79
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 ...
- 692
1
vote
1
answer
45
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 ...
- 163
9
votes
1
answer
281
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 ...
- 231
0
votes
0
answers
43
views
Minimax hypothesis testing - prove equality of the empirical cost
For the general minimax test, when the Bayes decision rule is designed assuming the prior is $q$, prove the following equality of the empirical cost
$$
\left.\frac{\mathrm{d} \varphi\left(\hat{H}_p, p\...
- 1
6
votes
1
answer
98
views
What do the terms "nearly-optimal rate", "near-minimax rate", "minimax optimal rate" and "minimax rate" mean in the context of posterior consistency?
Definition: A sequence $\epsilon_n$ is a posterior contraction rate at the parameter $θ_0$ if $$\Pi_n(θ: d(θ, θ_0) ≥ M_n \epsilon_n| X^{(n)}) → 0$$ in $P^{(n)}_{θ_0}$-probability, for every $M_n → ∞$.
...
- 278
2
votes
1
answer
136
views
Reference request for contemporary monographs on minimax theory
Are there any contemporary monographs on minimax theory$^1$, and if so, what are they?
So far, I have seen minimax theory given some treatment in Theory of Point Estimation by Lehmann and Casella (...
- 2,358
1
vote
1
answer
213
views
How to find the Bayesian equivalent of of $\bar{X}-\bar{Y}$
Let $X_1, \dots, X_n$ i.i.d. from $N(\mu, \sigma^2)$; $Y_1, \dots, Y_m$ i.i.d. from $N(\eta, \tau^2)$. $X_1, \dots, X_n$ are independent of $Y_1, \dots, Y_m$. And $\tau^2$ and $\sigma^2$ are ...
- 23
3
votes
1
answer
353
views
Showing $X\sim \operatorname{Poi}(\lambda)$ is minimax
Assume that $X$ has $\operatorname{Poisson} (\lambda)$ distribution and the loss function is $\ell(\lambda,a)=\frac{(\lambda-a)^2}{\lambda}$. Now, I want to show that $X$ is minimax. A hint that is ...
- 561
5
votes
1
answer
161
views
Mid-range via minimax
Warning: crossposted at Mathematics SE.
Given vector ${\rm a} \in \Bbb R^n$,
$$\begin{array}{ll} \displaystyle\arg\min_{x \in {\Bbb R}} & \left\| x {\Bbb 1}_n - {\rm a} \right\|_2^2\end{array} = \...
1
vote
0
answers
96
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?
- 924
4
votes
0
answers
309
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 ...
- 2,346
0
votes
0
answers
30
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}
...
- 213
1
vote
1
answer
520
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 ...
- 13
3
votes
1
answer
223
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 ...
- 370
1
vote
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 ...
- 233
5
votes
0
answers
161
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 ...
- 57.8k
3
votes
1
answer
178
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 ...
- 1,390
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 ...
- 21
0
votes
0
answers
243
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 ...
- 101
1
vote
0
answers
102
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/...
- 11
3
votes
1
answer
141
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....
- 61
2
votes
1
answer
64
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 ...
- 4,638
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
$$
d(\theta', \theta) + d(\theta'', \theta) \geq A d(\theta', \theta'')
$$
...
- 287
4
votes
1
answer
188
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 ...
- 5,360
6
votes
1
answer
683
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 ...
- 1,810
4
votes
1
answer
155
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, ...
- 375
3
votes
1
answer
760
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 ...
- 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 ...
- 419
3
votes
0
answers
126
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 ...
- 31
3
votes
0
answers
213
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 ...
- 4,733
4
votes
0
answers
198
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 $L(\delta,\theta)=(\delta-\theta)^2$
- 865