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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1answer
200 views

Hypothesis test between two normal distributions

Let $T_1,T_2,\ldots ,T_ n$ be i.i.d. observations, each drawn from a common normal distribution with mean zero. With probability $1/2$ this normal distribution has variance $1$, and with probability $...
0
votes
1answer
185 views

Trajectory estimation

The vertical coordinate (“height") of an object in free fall is described by an equation of the form $x(t) = \theta _0 + \theta _1t + \theta _2 t^2,$ We assume that $\theta_0$ is a known constant. We ...
1
vote
1answer
61 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 ...
1
vote
0answers
74 views

Hammersley–Chapman–Robbins bound for Rice distribution

I am trying to evaluate the Hammersley–Chapman–Robbins bound for the variance of an unbiased estimate $\hat{\alpha}$ of $\alpha$ (for a given $\sigma$) for the Rice distribution: $$p(x|\alpha,\sigma) =...
1
vote
0answers
125 views

Expectation of inverse of a symmetric matrix with gaussian elements

Is there any way to calculate: \begin{equation} \mathbb{E} \; ( H^{T}H )^{-1} \end{equation} assuming that the entries of the matrix $H$ are gaussian random variables with unknown means but same ...
2
votes
1answer
231 views

UMVUE for pdf $f_{\theta}(x) = \theta e^{-\theta x}, x>0$

Let $X_1,\ldots,X_n$ be a random sample from a pdf $f_{\theta}(x) = \begin{cases} \theta e^{-\theta x}, & x>0 \\ 0, & \text{otherwise} \end{cases}$, where $\theta>0$ is an unknown ...
2
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0answers
43 views

Fisher Expected Information for a Gaussian Process model

Suppose I have a two dimensional Gaussian process model (GP), defined by a squared exponential correlation function s.t: $$R(x_{i},x_{j}) = \exp\left(-\frac{|x_{i} - x_{j}|^2}{2}\right).$$ I am ...
0
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1answer
49 views

calculate The maximum likelihood estimator of parameter $\mu$ according to $T$

suppose $X_1,X_2,\ldots,X_n$ be a random sample of $N(\mu,1)$. if $T=\sum_{i=1}^n I_{(X_i<0)}$ how can I calculate The maximum likelihood estimator of parameter $\mu$ according to $T$. ($\Phi$ is ...
-3
votes
1answer
56 views

calculating $\mathbb E\left(\exp\left(\frac{1}{2}\sum_{i=1}^n X_i^2\right)\right)$ [closed]

suppose $X_1,X_2,\ldots,X_n \sim \mathcal N(0,\sigma^2)$. How can I calculate $$\mathbb E\left(\exp\left(\frac{1}{2}\sum_{i=1}^n X_i^2\right)\right)$$
0
votes
1answer
66 views

calculating 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 $(1-\sigma^2)^-\frac{n}{2}$. I know $T=\sum_{i=1}^n X_i^2$ is Sufficient and complete ...
1
vote
1answer
58 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} ...
0
votes
1answer
30 views

Hypothesis Test on Gaussian Mixture

I have data blocks being received at a node which can be presumed to be Gaussian sequence, $X(m_1,sigma_1).$ In some of the blocks a separate Gaussian stream $Y(m_2,sigma_2)$ adds to the original ...
1
vote
1answer
45 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, \...
0
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0answers
42 views

probability distribution estimation from correlated samples

I am looking to solve the following estimation problem. Consider a blackbox where (given below) given an input X, its N observations are recorded as output. These observations are denoted by $Y_1, \...
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0answers
32 views

Rao-Blackwell improvement for a nonrandomized estimator

Context: please consider a parametric statistical model $(\mathcal{Y},\{P_\theta:\theta\in\Theta\})$ and suppose that we are estimating $g(\theta)$. Associated with this is the set of decisions $\...
0
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1answer
434 views

Estimator for second moment for Poisson random variable

Let $X \sim Poiss(\lambda)$. As, $\displaystyle \sum_{i=1}^{N} X_i $ is sufficient statistic for both mean (and variance) of $Y$, so we can define the unbiased estimate for mean as , $ s=\frac{1}{N} \...
0
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0answers
61 views

What is the nonlinear estimator for Gaussian Random variable?

I know that the best estimator is $g(x)=E\{Y|X=x\}$ and the conditional density for jointly Gaussian random variables is known to be Gaussian with mean and variance given by \begin{equation} \eta_{Y|...
2
votes
0answers
29 views

ML estimation for Weibull

What are the maximum likelihood estimators of $\eta$ and $\beta$ ($\eta>0$ and $\beta>0$) for an i.i.d. sample of size $n$ from the following density: $f(y_i)=\frac{\beta x_i^{\beta-1} }{\eta ^ {...
0
votes
1answer
25 views

Finding an maximum likelihood estimator (Bernouilli problem)

Could someone point me in the right direction? Suppose we compare 2 treatments. For each patient we observe $(Y_i,R_i)$ where $Y_i$ denotes if the treatment was succesfull ($Y_i=1$) or not ($Y_i=0$...
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0answers
42 views

Cramer-Rao-Bound for squared parameter

I have a random sample $X_{i}, \ldots, X_{n} \sim \mathcal{N}(\theta,1)$. I would like to find the Cramer-Rao lower bound for the variance of unbiased estimators of $\theta^{2}$. Here's what I have: ...
0
votes
1answer
108 views

estimate coefficients of $y = \alpha x + \beta y + \gamma z + \epsilon$

I know how to find $m$ and $b$ for $y= mx +b$, which is : $m= \frac{\bar{x}\bar{y}- \bar{xy}}{(\bar{x})^2 - \bar{x^2}}$ and $b= \bar{y} - m\bar{x}$ How can we estimate $\alpha, \beta, \gamma$ and $\...
2
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1answer
44 views

Prove that $\int k(w)o(h^2w^2)dw=o(h^2)$ for $\int k(w)dw=1$

Suppose that $k$ is nonnegative real-valued function satisfying $$ \int k(w)dw=1,\quad\int wk(w)dw=0,\quad\int w^2k(w)dw=\kappa_2<\infty.\tag{$\star$} $$ (The limits of the integrals are all ...
4
votes
2answers
3k 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 ...
0
votes
1answer
68 views

Maximum likelihood estimator for general multinomial

Let $(X_1,\ldots,X_r)\sim\text{multinomial}(n,(p_1,\ldots,p_r))$, where $p_r=1-p_1-\cdots-p_{r-1}$. The random likelihood is $Ap_1^{X_1}\ldots p_r^{X_r}$, for some non-zero $A$. The random log-...
0
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1answer
21 views

Computing likelihood for data corrupted by zero mean noise

The following statement is from a text on Statistical Estimation. I am trying to figure out how the likelihood function was arrived at. By definition of likelihood, $p_{\bf{X}|\theta}p(\bf{X}|\...
0
votes
1answer
17 views

Maximum Likelihood (ML) estimation when 1 estimator is dependent on the other.

Let $\mathcal{L}(\theta_1,\theta_2)$ be the log-likelihood function. If I manage to find an estimator for $\theta_1$, as $\hat{\theta}_1=g(\theta_2,data)$. Then, if I want to find a ML estimator for $\...
3
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0answers
53 views

Trying to show convergence (in probability) of integrals using Taylor expansion

I've been working for a long time now on how to prove a proposition given in a paper about the asymptotic normality of POT-quantile estimators. Hope somebody can help me out. Proposition (i) Let $...
2
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0answers
83 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, ...
0
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1answer
34 views

A Estimation about Hölder condition

Let $p:[0,\inf) \to \mathbb{R}$ be a contionous function such that $p(0)=0$ Fix $a>1/2 , k$ is a positive integer $>\frac{1}{a-\frac{1}{2}}$. Suppose for all $n \in \mathbb{N}$ and $\lambda >...
0
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1answer
42 views

Improving data gaussianity using neural networks

I wanted to know if there is a way to use neural networks (deep neural networks or autoencoders) for a data gaussianization. I wonder how could the output data distribution be monitored and ...
1
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1answer
62 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 ) = \...
1
vote
1answer
42 views

On the finite expectation of a function

Let $y_i$, $i=1,\dots,n$, be independent Gaussian rv's of mean $\theta_i$ and variance $\sigma^2$ and let $\mathbf{y}:=[y_1,\dots,y_n]^\top$. Consider the function $f\colon \mathbb{R}^n\to \mathbb{R}$,...
2
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0answers
55 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 ...
2
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1answer
124 views

When to use Central Limit Theorem or Cramers Theorem

In for example this paper the authors say The central limit theorem provides an estimate of the probability \begin{align} P\left( \frac{\sum_{i=1}^n X_i - n\mu}{\sigma \sqrt{n}} > x \right) \...
2
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2answers
229 views

Convergence Rate of Sample Average Estimator

Let $X_1, X_2,\cdots$ be i.i.d. random variables with $E(X_1) = \mu, Var(X_1) = σ^2> 0$ and let $\bar{X}_n = {X_1 + X_2 + \cdots + X_n \over n}$ be the sample average estimator. Is there a way to ...
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0answers
41 views

asymptotic unbiasedness of weibull mle

It's known that the MLEs of the two-parameter Weibull distribution scale and shape parameters are not available in a closed form. It is, however, known that they do exist, are unique, and moreover, ...
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votes
1answer
66 views

Cumulative distribution function of exponentials

I have the cumulative distribution function $F(x)=(1-e^{-x})\mathbb{1}_{x≥0}$ and want to write the CDF to $F(\frac{x-\mu}{\sigma})$. I have derived $F(\frac{x-\mu}{\sigma})=1-e^{\frac{-x+\mu}{\sigma}...
2
votes
2answers
128 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 ...
2
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0answers
30 views

Estimation of Linear Projection

Given a linear system: $Y=AX+W$ Where: $X$ is the input signal of size $N \times K$ $Y$ is the output signal of size $M \times K$ $A$ is a projection of size $M\times N$; with $M >> N$ $W$ ...
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1answer
79 views

asymptotic normality and unbiasedness of mle

Suppose $\hat{\theta}_n$ is the MLE for some parameter $\theta$. Suppose also that the MLE is such that the Cramer regularity conditions are fulfilled, and $\hat{\theta}_n$ is asymptotically normal ...
0
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1answer
146 views

Sample size required to estimate population proportion with given precision

It plans to conduct a study on the percentage of homeowners who have at least two TVs. What should be the sample size if we want to ensure that $95\%$ of estimation error is less than $0.01$? ...
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vote
2answers
35 views

How to periodically estimate states of a LTI if the output is measured irregularly?

How can I periodically estimate the states of a discrete linear time-invariant system in the form $$\vec{x}(k+1)=\textbf{A}\vec{x}(k)+\textbf{B}\vec{u}(k)$$ $$\vec{y}(k)=\textbf{C}\vec{x}(k)+\textbf{...
1
vote
1answer
71 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 \...
1
vote
1answer
50 views

Maximum Likelihood Question

The aim is to find the maximum likelihood estimator for theta. $f(x)$ is given and we can assume that $1\le x\le-1$. I have completed the steps seen in the image, however I am having difficulty ...
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0answers
45 views

Unbiased estimator for maximum

Assume $n$ independent random variables with unknown distributions $\{X_1,X_2,...,X_n\}$. Multiple "samples" or observations for each of these variables are given (not necessarily with the same size)...
2
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0answers
153 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 ...
0
votes
1answer
43 views

$|p- \dfrac xn|>|q- \dfrac xn|$ $\implies$ $p^x(1-p)^{n-x}<q^x(1-q)^{n-x}$?

If $p,q \in (0,1)$ , and $ n \in \mathbb N$ be given and $x$ be given integer between $0$ and $n$ such that $|p- \dfrac xn|>|q- \dfrac xn|$ , then is it true that $p^x(1-p)^{n-x}<q^x(1-q)^{n-x}...
4
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0answers
269 views

A late-diverging “approximating solution” for a system of functional equations

Peace be upon you, At the end of this question, I have shown that how computing MLE on an i.i.d Beta distributed data, results in the following system \begin{align*} &\begin{cases} \psi(\alpha)-\...
0
votes
1answer
35 views

Does consistent estimators have in-variance property?

If $(T_n)$ is a sequence of consistent estimators of a parameter $\theta$ ( i.e. for every $ \epsilon >0$ , $\lim_{n \to \infty} P [ \space |T_n -\theta|< \epsilon ]=1$ ) , then is it true that ...
1
vote
1answer
581 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 ...