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0 votes
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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 $\kappa \ge 0$,...
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 ...
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 $||\...
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 ...
0 votes
0 answers
106 views

Maximum likelihood estimator for power law with negative exponent

Background I have data that roughly follows a power law with a negative exponent (up to a point; also, the parameters of the "fit" were just guesstimated by eye as a demonstration): Now I ...
1 vote
1 answer
101 views

Estimating the average of two gaussians' mean with minimal squared error

This is a follow-up to my previous question. Assume that $X\sim \mathcal N(\mu_1,\sigma_1^2)$ and $Y\sim \mathcal N(\mu_2,\sigma_2^2)$. I want to estimate $\frac{\mu_1+\mu_2}{2}$ after observing $X,Y$....
2 votes
1 answer
872 views

Estimating the average of two gaussians' mean

Assume that $X\sim \mathcal N(\sigma_1,\mu_1)$ and $Y\sim \mathcal N(\sigma_2,\mu_2)$. I want to estimate $\frac{\mu_1+\mu_2}{2}$ after observing $X,Y$. In my setting, $\sigma_1,\sigma_2$ are known ...
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 ...
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 ...
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 ...
-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 ...
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 ...
5 votes
1 answer
107 views

Estimating the size of the remainder in a random partition

Pick a sequence of real numbers $x_i$ as follows. Put $x_0=1$. If $x_i$ is chosen, then pick $x_{i+1}\in[0, x_i]$ according to the uniform distribution. Obviously we have $x_i\rightarrow 0$ with ...
3 votes
2 answers
566 views

Moments of Matrix Gamma distribution

Matrix gamma distribution (defined for example in http://en.wikipedia.org/wiki/Matrix_gamma_distribution) is one way to generalize Wishart distribution. In our course work that distribution was used ...
3 votes
1 answer
113 views

maximum likelihood estimation of X is better than that of f(X)?

Consider a binary variable $C$ with $\Pr(C=0)=\Pr(C=1)=0.5$. Consider a random vector $X \in \mathbb{R}^d$, characterized by $C$, with PDF $p_m(x)$, $m\in\{0,1\}$. Define a maximum likelihood (ML) ...
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(\...
1 vote
1 answer
124 views

Reconstructing the number of distinct elements from a random projection

Assume we have an unknown sequence $x_1,\ldots, x_n\in \mathcal U$. We get to observe the sequence $h(x_1),h(x_2),\ldots, h(x_n)$, where $h:\mathcal U\to \{1,\ldots, k\}$ is a random function such ...
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, \...
2 votes
1 answer
444 views

Literature question on the convergence rate of the empirical distribution

Assume that given $n$ i.i.d samples $(X_1, X_2, ..., X_n)$ drawn from $p_X$, an unknown probability mass function defined over a finite alphabet $\mathcal{X}$, one wants to estimate $p_X(x)$ for each $...
4 votes
2 answers
12k views

Maximum likelihood estimator for Power-law with Exponential cutoff

Hi, for fitting empirical data to power-law I am aware of the work by Clauset et al. (http://arxiv.org/abs/0706.1062) and how to use maximum likelihood estimation. There exists also a simple maximum ...