All Questions
7 questions
2
votes
1
answer
676
views
Distribution of ratio between complex Gaussian and Chi-square R.V.s
What would be the distribution (p.d.f.) of the following ratio?
$$z = \frac{x_{1}}{|x_{1}|^2 + |x_{2}|^2 + ... + |x_{M}|^2}$$
where $x_{i} \sim \mathcal{CN}(0,a), \forall i$ and $a > 1$. As can ...
2
votes
0
answers
87
views
A complex problem involving densities (likelihood functions) and optimization
Consider the following autoregressive process with normal errors:
\begin{equation}\label{7YlUV4i8nuO}\tag{I}
y_t = \phi y_{t-1}+ u_t, \quad u_t \overset{iid}{\sim} N(0,\sigma^2)
\end{equation}
We ...
- 13
2
votes
0
answers
78
views
Distribution of unbiased estimator of covariance matrix with missing values
Initial setup
Assuming $X_1, ..., X_n \in \mathbb{R}^m$ are iid, sampled from $\mathcal{N}(\mu, V)$, one can define the estimators for the sample mean $\hat{\mu} = \frac{1}{n} := X^T 1_n$, and sample ...
- 255
2
votes
0
answers
56
views
Rate of $L_1$ loss in estmating density on $[0,1]$
Let $f$ be a density on $[0,1]$ and let $X_1,X_2,\ldots$ be $\textit{iid}$ $f$-distributed. Also, let $f_n$ denote the kernel density estimator, i.e.
$$f_n(x) = \frac{1}{nh_n} \sum_{i=1}^n K\left(\...
- 121
1
vote
1
answer
341
views
Form of minimax estimator
Let $\Delta$ be the set of all probability distributions over $\mathbb{N}=\{1,2,\ldots\}$ and fix some $\mathcal{P}\subseteq\Delta$.
Suppose additionally that $\Delta$ is endowed with some norm $||\...
- 6,541
1
vote
0
answers
148
views
conjecture for general form of minimax estimator
I had previously posed an overly ambitious version of this conjecture here,
Form of minimax estimator,
which was quickly shot down by Václav Voráček (on twitter) and Iosif Pinelis (MO answer in the ...
- 6,541
1
vote
0
answers
108
views
Bootstrap-$t$ confidence intervals
I'm writing a dissertation about bootstrap methods and the main book I'm using is Efron, B., & Tibshirani, R.J. (1994), An Introduction to the Bootstrap (1st ed.), Chapman and Hall/CRC. Now I need ...
