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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123 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
170 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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249 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
111 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
2k 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
99 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
289 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
6k 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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1answer
27 views

How to find a MVUE for a certain function of a parameter

The following is one of the exercises from my course in statistics Let $X_1, \ldots, X_n$ be a random sample from a Poisson distribution with parameter $\theta > 0$. Find the MVUE for ...
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1answer
58 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
57 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
59 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
535 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
61 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
166 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
82 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
486 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
127 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
59 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
37 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 ...
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2answers
299 views

Does convergence in probability not imply convergence in distribution for Least Squares estimators?

I have a question relating to convergence in probability and distribution for least squares estimators. Frequently, I see in textbooks that $\hat{\beta} \rightarrow^p b$. Where $b$ is the population ...
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1answer
296 views

Find maximum likelihood estimator, trick?

Let $Y_1, Y_2, \ldots, Y_n$ iid random variables with density $f(y)=\theta\cdot y^{\theta-1}$, $0<y<1$, $\theta >0$. I need to show that the maximum likelihood estimator of $\theta$ is ...
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1answer
98 views

Using Neyman pearson lemma when ratio comes out to be zero.

Consider a Bernoulli random variable: $$X_i= \begin{cases} 1, & \text{with probability }p \\ 0, & \text{with probability }1-p \end{cases}$$ You observe the outcomes of two Bernoulli trials ...
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1answer
546 views

Can we compute confidence intervals for the variance of an unknown distributions from sample variances?

Assume $X_1,\ldots,X_n$ are i.i.d. with unknown distribution $\mathcal D$ - we only know it is not normal and has finite variance. Is there a way to give confidence intervals for the variance of ...
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1answer
523 views

The distribution when combining two samples together?

Suppose $X\sim N(0,{\sigma}^2)$ and $Y\sim N(0,{2\sigma}^2)$ . $X_1, ..., X_m$ are the samples from $X$ and $Y_1, ..., Y_n$ are the samples from $Y$. And then combine two samples as a new sample ...
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1answer
378 views

Using the MSE criterion, which is a better estimator for $\Theta^2$?

Question: Let $T_1$ and $T_2$ be independent unbiased estimators of a parameter $\Theta$. Assume that $\operatorname{Var}(T_2) = \operatorname{Var}(T_1)$. Using the MSE critertion, define which is a ...
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1answer
114 views

Proving that the sum of Good-Turing estimators is $1$

I want to know how to go about proving that the Good-Turing estimator has a total probability of $1$. I have seen this proof (page 2) but I found unclear the first step: $$\sum_j \theta[j] = \sum_r ...
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985 views

Fast variance calculation

Suppose to have a sequence $X$ of $m$ samples and for each $i^{th}$ sample you want to calculate a local mean $\mu_{X}(i)$ and a local variance $\sigma^2_{X}(i)$ estimation over $n \ll m$ samples of ...
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494 views

Gauss-Markov estimator properties

Consider a linear model $$ y = Ab+n, $$ where $b \in \mathbb{R^m}$ is a parameter to be estimated, $n \in \mathbb{R^{n}}$ is a noise with mean $\mathbb{E}n = m_{n}$ and with covariation matrix ...
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1answer
104 views

How to find out the control function of a cosine wave?

I have a system which is sampling at 100Hz. There is only one input for the system. The output is similar to cosine waveforms with varying frequency. I have no clue how to find out the exact formula ...
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1answer
544 views

Maximum Likelihood optimal threshold

I have a decision (detection) problem trying to decide between symbols ${0,2}$. I have the two probability density functions: $$ f(z|s=0) = \begin{cases} 0.25z + 0.5, & -2\le\ z <0 \\ ...
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1answer
136 views

Likelihood function estimating two different means

I completely understand how to find the likelihood functions of simple pdfs. However, how would you attempt to find the likelihood function of a pdf with a negative exponential function with two ...
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31 views

Estimating a sparse vector: Mean squared error when support known

I was reading this paper ("How well can we estimate a sparse vector?" by Candès and Davenport, link: http://arxiv.org/pdf/1104.5246v5.pdf). They consider the problem of estimating a $k$-sparse vector ...
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20 views

Can I bound $P[R > x + \epsilon]$ independently of R?

I have this probability distribution: $P[\Theta < \varphi] = \frac{\varphi}{\pi}$ for $\phi \in [0,\pi]$. Now I have $n$ samples of $D = R\Theta$ i.i.d. ($R>0$) and I want to estimate $R$ as ...
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17 views

How to use MLE method for non-distribution function?

I understand that maximum likelihood estimation (MLE) method is normally used with distribution function. However is there anyway around I can do to use MLE for a function which is not distribution ...
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46 views

Taylor of $\ln(f(exp(x))))$?

Let $ f(x) = \sum a_n x^n$ Such that The $a_n$ are real and $f(a),f ' (a) , f " (a) > 0 $ for any real $a > 0$. Let $ \ln(f(exp(x))) = \sum b_n x^n $. Let $c_n = a_n - b_n$. For a given $f$ ...
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0answers
20 views

Neyman Pearson rule but not a Bayes rule

Consider a binary hypothesis testing problem of $P_0$ vs. $P_1$ under uniform costs. Let $r(\delta,\pi)$ denote the risk line for any decision rule $\delta$ and prior $\pi$, i.e, $r(\delta,\pi)=\pi ...
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31 views

Higher Order Estimation Errors

I well know estimation measure is the so called minimum mean square error (MMSE) defined as: \begin{align} E[|W-\hat{W}(V)|^2] \end{align} where $W$ is a random variable (that we want to estimate) and ...
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1answer
15 views

Does the correlation between the measurement noise influence the result of the LS?

Consider a linear measurement process with some noise: $$y=Hx+v$$ with $$v \sim \mathcal{N}(0,\Sigma)$$ the covariance matrix $\Sigma$ is not a diagonal matrix. As we know, using LS, the $\hat{x}$ is ...
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1answer
32 views

How do I compute a realization of h(x) given its PDF and covariance?

Desription of problem I've added a picture of what I want to compute. In the nomenclature of the picture, I want to compute a realization of y(x) given the known distributions and constants. Let's ...
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1answer
29 views

How do you show that the estimator for the covariance matrix is unbiased?

So according to Wikipedia (Here) the sample covariance matrix is an unbiased estimator of the covariance matrix, but how do I prove this mathematically?
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23 views

Covariance estimation and Graphical Modelling

I've started reading on Convariance Matrix estimation through Graphical model in high-dimensional situation. But I have several questions. Suppose, $X_i \overset{iid}{\sim} N_p(\mu,\Sigma)$, ...
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59 views

Is $\theta_{mle}$ the UMVUE of $\theta$

I am working through some practice problems before an exam, and am getting stuck. The pdf is $f(x|\theta)=\theta x^{\theta-1}$ where $0 < x < 1$. I have already found my MLE as ...
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45 views

UMVUE of parameter $(1-\sigma^2)^{-\frac{n}{2}}$

suppose $X_1,X_2,\ldots,X_n$ be random sample of $N(0,\sigma^2)$. how can I calculate UMVUE of parameter $(1-\sigma^2)^{-\frac{n}{2}}$
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83 views

Origins of Kalman filter Algorithms in his paper in 1960

Concerning Kalman's original paper published in 1960, "A New Approach to Linear Filtering and Prediction Problems", it seems the majority is to show the orthogonal projection is the optimal estimation ...
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29 views

Estimation of unbiased estimator for population variance in case of poisson distribution

sample variance is normally biased estimator for population variance. but in case of poisson distribution sample variance is unbiased for population variance. how do you prove this?
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32 views

How to bound $E \left[\left(E[Z^2\mid Y] \right)^2\right]- 2E \left[ |E[Z\mid Y]| \sqrt{E[Z^2\mid Y]} \right]$

I am looking for an upper bound on the following quantity \begin{align*} A=E \left[\left(E[Z^2\mid Y] \right)^2\right]- 2E \left[ |E[Z\mid Y]| \sqrt{E[Z^2\mid Y]} \right] \end{align*} where $Z$ is ...
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23 views

Proof of differentialbility in mean square calculus?

let $x_t$ be a mean squared Riemann integrable over $[a, t]$ for every $t\in[a,b]$. Then $y_t=\int\limits_a^t x_\tau d\tau\ $ is mean squared continuous on $[a, b]$. Furthermore, if $x_t $ is mean ...
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263 views

How to calculate Fisher Information (FI) matrix for Multivariate Normal Distribution (MN)

Below is the gradient (score) of the MN log likelihood function L for n=1 observation. I originally attempted to calculate the Hessian matrix but ran into difficulty calculating 2nd order derivatives ...