Estimation theory is a branch of statistics and signal processing that deals with estimating the values of parameters based on measured/empirical data that has a random component.

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Conditional distribution [closed]

I am trying to figure out the derivation of Kalman filter based on Bayesian estimator. As we know, the assumption of Gauss-Markov model is used, then, the conditional distribution p(x(t)|Y(t-1))can be ...
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72 views

Near-Application of Cauchy-Schwarz Inequality

I have the following situation: I have two estimators of $\alpha$, both via maximum likelihood of the density: $$ f(x,y\mid \alpha,\beta) = f(y \mid x,\alpha,\beta)f(x \mid \alpha) $$ One uses only ...
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554 views

How to estimate variances for Kalman filter from real sensor measurements without underestimating process noise.

As the title says, I want to estimate the variances needed for a Kalman filter from real sensor measurements only. For example we can take a temperature sensor, but the solution shall be as ...
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53 views

Simulation Velocity of a harmonic oscillator system

I am write a simulation for get true Velocity of a harmonic oscillator system as Where P=[p1 p2;p2 p3] can find using Rung-Kutta Integration method with P(0)=[1 0; 0 1] This is code to find p Now, ...
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40 views

Risk in density estimation: grasping the definition

When generalizing estimators to an entire function what is the space in which we perform the integral to obtain the expected value (with respect to this function)? For example, when estimating ...
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149 views

Polluting an image with Gaussian anisotropic noise and estimate the covariance matrix

Assume that we have a $d\times d$ grey-scale image represented as a vector $$ \mathbf{x}=(x_1,\ldots,x_D)^T\in[0,255]^D, $$ where $D=d\times d$. We would like to import some noise concerning the ...
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48 views

paramter estimation (maximum likelihood) of a mixture density

I have this mixture distribution $f(x) =w \cdot \mathcal{LN}(\mu_1,\sigma) + (1-w)\cdot \mathcal{LN}(\mu_2,\sigma) $ where $\mathcal{LN}(\mu,\sigma)$ is a lognormal distribution. I now have random ...
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1answer
41 views

Bayes - dual estimation of parameter value and parameter growth

I am trying to find an bayesian approach to the following problem: Image a bucket with 100 white balls and an unknown number of red balls During each year, one can take a sample with replacement of ...
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1answer
74 views

How to compute the unique MLE from an Exponential Family of Distributions?

Let $$ f(x;\theta)=\frac{1}{\pi} \frac{e^{\theta x}\cos(\theta \pi/2)}{\cosh(x)}, x\in{\mathbb{R}} $$ be a family of densities and which is clearly exponential family. Then what is the Maximum ...
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64 views

Alternatives to Fisher information

The Fisher information matrix is defined as the following: $$\mathcal{I}(\theta)=E[(\frac{\partial \log f(x;\theta)}{\partial \theta})^2]=-E[\frac{\partial^2 \log f(x;\theta)}{\partial \theta ...
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77 views

Identification of real functions

this my second question, so I'm still new... thanks in advance for any help! Basically, I'm looking for some references and tools to study the following problem. Consider the following function ...
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63 views

Unbiased estimator with conditional expectation.

Suppose that $X$ has a binomial distribution with parameter $N=1$ and $p=1/2$. Y, which is independent of $X$, has a normal distribution with mean $\mu$ and variance 1. Consider the estimator $\mu$ of ...
2
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1answer
26 views

How to show this estimation?

i have this polynom $$p(x) = \sum_{i=0}^{m}a_ix^i$$ I want to show, that if $\tilde{z}$ is the approximation to the simple zero digit $z \neq 0$ in first approximation, the following estimation ...
2
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2answers
86 views

Why we always put log() before the joint pdf when we use MLE(Maximum likelihood Estimation)?

Maybe this question is simple, but I really need some help. When we use the Maximum Likelihood Estimation(MLE) to estimate the parameters, why we always put the log() before the joint density? To use ...
2
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51 views

What are the properties of median-unbiased estimators?

On Wikipedia it says that " A median-unbiased estimator minimizes the risk with respect to the absolute-deviation loss function, as observed by Laplace." How to prove this? Note that I asked on Cross ...
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207 views

Fisher Information and minimum variance estimators

I am trying to understand what can be proved about minimum variance estimators. I have changed the question to make it more specific. Let us assume we have some finite set $S$ of elements and we just ...
2
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1answer
55 views

Parameter optimization in probabilistic models

Task: Suppose we model a variable $y = Wx + \mu$ as a linear transformation of $x$ plus some Gaussian noise $\mu\sim\mathcal N(0,\sigma I)$. Our aim is to minimize the estimation error of $x$ given ...
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97 views

Likelihood Function of Random Process

Given the following data: $$ x(t) = A + \omega(t) $$ where $ \omega(t) $ is an AWGN with zero mean, what would be likelihood function $p(x(t);A)$? I know it could be proven to be: $$ p(x;A) = C ...
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2answers
124 views

If $(X_i)$ are i.i.d. exponential $\lambda$, then $\hat\lambda=n/\sum{X_i}$ is a biased estimator of $\lambda$

the main problem is that i have no clue on calculating $E(\frac{1}{x})$ let $U = \frac{1}{\frac{1}{n}\sum{X_i}}$ then, $E(U) = n*E(\frac{1}{\sum{X_i}})$. I think that i'm supposed to calculate: ...
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2answers
52 views

Variance of sample mean (problems with proof)

Assuming that I have $\{x_1,\ldots, x_N\}$ - an iid (independent identically distributed) sample size $N$ of observations of random variable $\xi$ with unknown mean $m_1$, variance (second central ...
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2answers
119 views

Estimation theory: Finding MSE and Variance of an Estimator

Let $\hat{A}$ be an estimator of $A$ where $a<A<b$ and $\tilde{A}$ be another estimator such that$$\tilde{A} = \cases{a \text{ if } \hat A \leq a\\\hat A \text{ if } a< \hat A <b\\{b ...
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1answer
46 views

Linear Regression with independent but non-identical noise

If I have this linear regression equation: $$y=X\beta+\epsilon $$ ($x$ and $\beta$ are vectors) The likelihood function can be written as $$L= \prod_{n=1}^N N(y_n ;x_n ,\beta ,\sigma^2)=(2\pi ...
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1answer
86 views

Order related to Empirical distribution function and Normal distribution

Let $X_1,\dots,X_n$ are i.i.d with distribution function $F$. Let $\hat F_n$ be its empirical distribution function, i.e., $$ \hat F_n(x)=\frac1n\sum_{i=1}^n1_{\{X_\le x\}}(x) $$ where $1_A(x)$ is the ...
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2answers
833 views

Show that estimates are unbiased

The following is a problem in my book that I don't really understand: We take a random sample: $x_1,x_2,\ldots,x_n$ from a population that is $N(μ,σ)$ where $\mu$ and $\sigma$ are unknown. We build ...
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1answer
219 views

Sufficient Estimators and Generalized Likelihood Ratios

If you can make the assumption that a sufficient statistic exists for some parameter - let's call it $\theta$. How would you show that the critical region of a likelihood ratio test will depend on ...
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2answers
479 views

Question about unbiased estimator

$X_1,X_2,\ldots,X_n$ are iid random variables $B(1,\theta)$ where $0< \theta<1$. Let $w = 1$ if $\sum_i X_i = n$ and $0$ otherwise. What is the best unbiased estimator of $\theta^2$. Attempt: ...
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1answer
34 views

Dice roll, estimator, epsilon

We roll a non-symmetric die. Let $X_n$ be the reulst of $n$-th roll. $$P(X_n = 6)= \frac{1}{6} + \varepsilon, \ P(X_n = 1) = \frac{1}{6} - \varepsilon, \ P(X_n=2) = ... = P(X_n = 5) = \frac{1}{6} $$ ...
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1answer
33 views

Compound distribution with unknown distribution of its hyperparameter

Suppose $X\sim \mathcal{N}(0,\sigma)$, and $\sigma$ is another random variable in a sense that we only know that it is some constant random variable with finite support, i-e $\sigma \in [\sigma_\max, ...
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1answer
96 views

Conditional expectation and Rao Blackwell

Consider a family of densitites $f(x,\theta)=\frac{\exp(-\sqrt{x})}{\theta}$. Let $X_1$ be a single observation from this family. I have shown that $\sqrt{X_1}/2$ is an unbiased estimator. Now ...
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2answers
106 views

Finding the MLE of a function when L'($\theta$) doesn't depend on $\theta$

Here's the problem: Find the MLE of of $\theta$ when $f(x\mid\theta)=(1+x\theta)/2$ for $-1<x<1$, $=0$ otherwise. $0<\theta<1$ Find the maximum likelihood of $\theta$ and find its ...
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1answer
145 views

Optimal combination of two estimates

I have a set of random variables, $X_1,\dots,X_N$. They are i.i.d. Gaussian with zero mean and $w$ variance. I observe $Y_1,\dots,Y_N$ where $Y_i=\sum_{j=1}^N a_{ij} X_j+N_i$ where all $a_{ij}$s are ...
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2answers
163 views

What exactly is simple consistency?

Now yes, I know the definition: A sequence of estimators $\{T_n\}$of $\tau(\theta)$ are consistent if for every $\epsilon > 0$ $$ \lim_{n\to\infty}P[|T_n-\tau(\theta)|\leq\epsilon]=1 \\\text{ ...
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1answer
91 views

find $\theta_{MLE}$ for a function

For $$ f(x;\theta)=(\theta+1)x^{-\theta-2} $$ find the maxmimum likelihood estimators (MLEs) for $\theta$ based on a random sample of size $n$. My work so far: $$ \begin{align} \prod_{i=1}^n ...
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1answer
1k views

Rao-Blackwell unbiased estimator geometric distribution

I'm looking at review questions and having trouble with this one! Let $X_1,\ldots,X_n$ be i.i.d. geometric R.V.s with the pmf: $(1-p)^{x-1}p$, for $x=1,2,\ldots$ and $0<p<1$. I need to use ...
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2answers
98 views

Filter to obtain MMSE of data from Gaussian vector

Data sampled at two time instances giving bivariate Gaussian vector $X=(X_1,X_2)^T$ with $f(x_1,x_2)=\exp(-(x_1^2+1.8x_1x_2+x_2^2)/0.38)/2\pi \sqrt{0.19}$ Data measured in noisy environment with ...
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1answer
273 views

How to find parameters from an equation

the question could be "stupid" but i don't know if it is feasible or not, please don't kill me :) EDIT WITH NEW FORMULAS! I have an equation like this: (unfortunatly in my first Q&A i cannot ...
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3answers
3k views

How to estimate failure probability from count until first failure?

What would be the formula to estimate the rate of failure of some test as a percentage chance of failure from the number of runs of the test until the first failure was seen? For example, considering ...
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2answers
31 views

Maximum a Posteriori (MAP) Estimator of Exponential Random Variable with Uniform Prior

What would be the Maximum a Posteriori (MAP) estimator for $ \lambda $ for IID $ \left\{ {x}_{i} \right\}_{i = 1}^{N} $ where $ {x}_{i} \sim \exp \left( \lambda \right), \; \lambda \sim U \left[ ...
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1answer
37 views

Unbiased estimator for the sum of numbers

Let $\alpha_1, \dots, \alpha_n \in \mathbb{R}$. We want to approximate the sum as follows $$ S = \sum_{i=1}^{n} \alpha_i \approx \dfrac{n}{c} \sum_{i=1}^{c} \alpha_i, $$ where $\alpha_i$ is picked ...
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1answer
23 views

How to evaluate the goodness of Fit of parameters obtained from EM algorithm

I have a set of observations $\mathcal{Y} = {Y_1, \ldots, Y_T}$. I am running EM algorithm to fit the observations to the following Hidden Markov Model $$A = [a_{ij}]_{N \times N}, a_{ij} = P(X_{k+1} ...
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1answer
41 views

Inference about the true intercept of the model and the OLS being BLUE

Consider the following population regression model: $$y_{i} = \beta _{1} + \beta_{2}x_{i} + \epsilon _{i},$$ where $i=1,...,n$. Assume $\epsilon \sim iid$, with the pdf in equation: $f(\epsilon ) = ...
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1answer
285 views

Maximum likelihood estimate vs likelihood ratio tests?

Can someone explain to me the intuition behind why we need likelihood ratio tests. From my understanding, they make use of maximum likelihood estimators over different parameters space and they are a ...
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2answers
53 views

how to estimate the phase parameter of a complex function

There is a complex serie: $f(t_n)=\alpha_n+\beta_n i$, for $n = 1,...,N$,$t_n,\alpha_n$ and $\beta_n$ are known.When we have know that $f(t)$ has the following form: $$f(t)=Ae^{-iBt}$$ with unknown ...
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1answer
96 views

statistics inequality

Let $\theta$ be a discrete pararmeter and $\gamma_{n}$ be an estimator. Prove that for any $c>0$ we have that $$\text{E}[(\gamma_n-\theta)^2] \ge\Pr[|\gamma_n-\theta|>c]\cdot c^2$$
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1answer
91 views

When is a minimum distance decoder also a maximum likelihood decoder?

It is well known that if we have a binary symmetric channel with crossover probability $\epsilon<0.5$ and we send a word $x$ through it, the most likely word is the one with minimum hamming ...
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1answer
75 views

Finding the MLE of pareto dist., and trouble interpreting $\prod$ notation properly.

I am generally having trouble understanding how to use product notation when calculating Maximum Likelihood Estimators. The example bellow is from a random sample $X_1,...,X_n$. Find the MLE of ...
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1answer
249 views

Find the Method Moment Estimator of parameter $\theta$

Find the MME of parameter $\theta$ in the distribution with the density $f(x,\theta)=(\theta +1)x^{-(\theta+2)}$, for $x>1$ and $\theta >0$. So far I think I have a basic understanding of the ...
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1answer
82 views

Is Sample Covariance Tied to a Specific Distribution

In many sources on data analysis, the author(s) talk about calculating covariance of the data, and the formula is given as such $$ \Sigma = cov(X) = E[(X-E[X])(X-E[X])^T]$$ This formulation is given ...
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1answer
47 views

How to estimate a vocabulary size?

I have a list of the 1 million most common English words ordered by number of times they appear on all books in Google Books. I want for the user to select from a list of 100 words (small sample) ...
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1answer
33 views

Conjugate Bayesian analysis

Suppose that conditional on $\tau$, the random variable $X$ has normal distribution with mean zero and variance $1/ \tau$. The prior distribution for $\tau$ is Gamma with parameter $\alpha$ and ...