All Questions
Tagged with estimation-theory pr.probability
14 questions with no upvoted or accepted answers
3
votes
0
answers
82
views
Uniform mean-square-error estimates
Consider a standard statistical estimation problem with iid real observations $\{X_i\}_{i=1}^N$. For a collection of real functions $\mathcal{F}$, I want to get an estimate of the uniform rate of ...
- 111
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
51
views
MLE and CRLB with mismatched likelihoods
Suppose that I can do a Karhunen-Loeve expansion of a log-likelihood function $p(\bf{x};\theta)$ into N terms and that these accounts for a fraction $1-\delta$ of the total energy. Now consider ...
- 21
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
81
views
Calculating the mean squared error for an estimate of a large sum
Consider the set of all Boolean function $f: \{0, 1\}^{n} \rightarrow \{-1, 1\}$. Now, let's pick a function uniformly at random from this set. Let $F$ be the random variable corresponding to the ...
1
vote
0
answers
75
views
Percentile interval Lemma
Let $\theta$ be a parameter and $\hat{\theta}$ the plug-in estimate, I need a proof of the following lemma, as given in [1], p. 173, in the form of a reference or of a direct argument:
Percentile ...
1
vote
0
answers
154
views
How to retrieve back the input using Bussgang theorem?
If we have a non-linear function $f$, that is applied to input $x$,
we have then the output $y=f(x)$
Using Bussgang decomposition we can linearize this nonlinearity and express $y$ as
$y=Bx+ η$,
...
- 39
1
vote
0
answers
62
views
Distances between up and down crosses in Gaussian Processes
Given a gaussian process $g := \mathcal{GP}\left(\mu, \Sigma \right)$,
where $\mu$ is the mean and $\Sigma$ is the covariance function, I am interested in estimating the mean value $L_m$ of the ...
- 91
1
vote
0
answers
49
views
A question about the prediction error
I am reading about the prediction error estimation and I found the following:
Suppose we have ${\mathbf{Y}}=\mathbf{x}_0+ \epsilon$, where, $\epsilon$ is normally distributed as $\sim \mathcal{N}(0, \...
- 11
1
vote
0
answers
46
views
Equivalence of Graphical model selection algorithms
Suppose, a jointly Gaussian random vector is denoted by $X \in \mathbb{R}^{p}$ and $X$ has a distribution given by $\mathcal{N}(\mu,\Sigma)$. It is known that estimating the graphical model that ...
- 275
1
vote
0
answers
244
views
Distribution of a signal covariance matrix
A common estimation problem in signal processing assumes the following signal model
\begin{equation}
\mathbf{r} = \sum_{i=1}^{Q}\alpha_i\mathbf{s}\left(w_i\right)+\mathbf{n}
\end{equation}
where $\...
- 345
1
vote
0
answers
186
views
Shrinkage (or Stein's phenomenon) in low dimensions, discrete contexts
I am trying to understand shrinkage, or the Stein phenomenon. As someone without a statistics background, the focus in most introductory presentations on normal distributions and squared error loss ...
- 203
0
votes
0
answers
31
views
What is the Fisher information matrix of the von Mises-Fisher distribution?
Assuming the von Mises-Fisher distribution as
$$f_{p}(\mathbf{x}; \boldsymbol{\mu}, \kappa) = C_{p}(\kappa) \exp \left( {\kappa \boldsymbol{\mu}^\mathsf{T} \mathbf{x} } \right),$$
where $\kappa \ge 0$,...
- 287
-1
votes
1
answer
205
views
How to combine estimator with different variances?
Consider independent random variables $X_1,X_2,\ldots,$ that have the same expectation $\mathbb x=\mathbb E[X_1]=\mathbb E[X_2]=\ldots$
Further, assume that we know that $Var[X_i]=\sigma_i^2$.
In the ...
- 127