# Questions tagged [minimax]

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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 (...
1answer
68 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 ...
1answer
120 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 ...
0answers
19 views

### Giving a job lecture on “minimax statistical estimation”. What kind of questions to expect?

As part of a job application process, I am giving a short "mock lecture" for students on minimax estimation in statistics. I will introduce the basic concept and give an example. I am able ...
1answer
93 views

1answer
110 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, ...
1answer
536 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 ...
1answer
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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 ...
0answers
100 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 ...
0answers
194 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 ...