Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [minimax]

24 questions
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Mid-range via minimax

Warning: crossposted at Mathematics SE. Given vector ${\rm a} \in \Bbb R^n$, $$\begin{array}{ll} \displaystyle\arg\min_{x \in {\Bbb R}} & \left\| x {\Bbb 1}_n - {\rm a} \right\|_2^2\end{array} = \frac1n {\Bbb 1}_n^\top {\rm a} \tag{mean}$$ is the…
Rodrigo de Azevedo
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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 fixed effects for cluster adjustment. This is…
AdamO
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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 details see e.g.…
Julian Karch
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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 risk function was obtained. I would also…
Jen
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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 fixed, and a sequence of Bayes estimators can be…
Xiaomi
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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 $$ d(\theta', \theta) + d(\theta'', \theta) \geq A d(\theta', \theta'') $$ for some $A \in (0,1]$. Also what about the…
Steve
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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, because of Markov's inequality. However, it is the…
Lionville
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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 that: An alternative proper prior, shown to…
peuhp
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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 given is to consider the gamma $\Gamma(k-1,1)$…
statwoman
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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 is this risk independent of a prior over…
wabbit
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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. $\delta = 0$ v.s. $\delta \neq 0$. Note that in…
Martin Zhang
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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 predictor distribution is $p_{\delta}(y) =…
guestguy123
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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 (1998), and a more extensive treatment in…
microhaus
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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 estimator is constant. $MSE(\hat \theta) = Var(\hat…
Daniel Yefimov
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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 space (e.g. maximize distances)? My problem arises from…
Jimmy Risk
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