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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Most efficient estimator

$X_1,...X_n$ is a random sample of size $n$ from a population with mean $\mu$ and variance $\sigma^2$.There are three estimators for $\mu$:  $\hat\mu _1=\frac{x_1+x_2}{2}$ $\hat\mu ...
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15 views

Variance estimation of a diffusion process

The framework of this question is a 1 dimensional diffusion process, defined ny the following equation: $dx_t=adt+bdw_t$ Where $w_t$ is a standard berownian motion and and $a$ is a constant drif ...
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4 views

A mix between the Horvitz-Thompson and ordinary estimator

I have asked this question on mathoverflow, but got no answer. Here I have corrected some mistakes and wish to hear any ideas that may bring at least numerical result: The data I have two samples: ...
2
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0answers
41 views
+50

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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2answers
40 views

How to find the bias, variance and MSE of $\hat p$?

If $X_1,\dots,X_n$ are iid $\mathrm{Binomial}(3,p)$, then the maximum likelihood estimator of $p$ is $$\hat p = \frac{1}{n}\sum_i X_i$$ Find the bias, variance and MSE of $\hat p$? We are asked to ...
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1answer
32 views

linear least square estimation with random sum

Let $N$ be a geometric r.v. with mean $1/p$; let $A1,A2,… $be a sequence of i.i.d. random variables, all independent of $N$, with mean $1$ and variance $1$; let $B1,B2,… $be another sequence of i.i.d. ...
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14 views

Second partial derivative of a minimum function

I am reading a book on detection and estimation theory, and the author had this to say in the derivation of the white noise process from the Wiener process: We can formally obtain the covariance ...
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4 views

Function of efficient estimator

Say I have an efficient estimator $\theta$, call it $\hat{\theta}$. If I wanted to estimate another quantity, call it $\delta = g(\theta)$, is there any result which allows me to say that ...
2
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1answer
113 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 ...
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1answer
67 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 ...
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1answer
36 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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0answers
22 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) ...
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0answers
12 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 ...
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0answers
18 views

ML estimator for correlated random vectors

Suppose, we have two random vectors in $\mathbb{R}^2 $ denoted by $(X_1,Y_1)$ and $(X_2,Y_2)$ . The individual random variables can only take value in $\lbrace 0, 1 \rbrace$. We get samples of the ...
1
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1answer
40 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 ...
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0answers
16 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 ...
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22 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 ...
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0answers
10 views

Finding a bayes estimator

Let $X_1,...,X_n|\eta~\exp(1,\eta)$ and $\eta$~$N(\mu,1)$, where $\mu\epsilon\Re$. Find the Bayes estimator $\eta$ under the squared error loss. After finding the joint likelihood of $exp(1,\eta)$ ...
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1answer
53 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 $$\mathbb E\left(\exp\left(\frac{1}{2}\sum_{i=1}^n X_i^2\right)\right)$$
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0answers
17 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 ...
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1answer
16 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
21 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 ...
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1answer
28 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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12 views

What is the test statistic mentioned in the paper by Dai and Singleton?

I have read the paper with title 'Specification Analysis of Affine Term Structure Models' (2000) by Qiang Dai and Kenneth J. Singleton. On page 22 there's a table which include test statistics, these ...
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0answers
32 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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14 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 ...
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1answer
83 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} ...
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24 views

Determining the actual number of observations in a dataset

I have two datasets one is a dataset with doctors in which I have the procedures they have performed at a given hospital where the actual number of procedures is not captured by this data since it is ...
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0answers
21 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 ...
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1answer
17 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 ...
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33 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: ...
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1answer
27 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
42 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 ...
2
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270 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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24 views

Description length in model coding

In class, our professor posted the following: We will discretize $\theta$ (some model) into $1/\sqrt{n}$ distinct values. Intuitive argument: with N data points, our estimation error for $\hat ...
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1answer
39 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 ...
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1answer
14 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, ...
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1answer
16 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 ...
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0answers
14 views

How can I find the nonlinear and linear MS estimates of y in terms of x and the resulting MS errors?

If $y=x^3$, find the nonlinear and linear MS estimates of $y$ in terms of $x$ and the resulting MS errors? This is what I got for the nonlinear MS estimation: Since $e=E\{[y-C(x)]^2\}$, $C(x)=x^3$ and ...
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0answers
11 views

how can I find the resulting MS error with linear estimation?

The problem says: If $\eta_x=\eta_y=0, \sigma_x=\sigma_y=4$ and $\hat{y}=0.2x$ (linear estimate), find $E\{(y-\hat{y})^2\}$. I am doing this, based on Papulis formulas for homogeneous linear estimate: ...
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How Can I show that a=A in this linear MS estimation problem?

How can I show that if the constants A,B and a are such that $E\{[y-(Ax+B)]^2\}$ and $E {\{[(y-\eta_y)-a(x-\eta_x)]^2 \}}$ are minimum, then $a=A$. I am trying to use this: $e=e_m$ is minimum if ...
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48 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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45 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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32 views

Simulation Position of a harmonic oscillator system

I am write a simulation for get true Postion of a harmonic oscillator system as Now, I want to write matlab code to get the true postion z of the system. However, ...
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1answer
31 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 ...
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1answer
24 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 ...
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18 views

Sensitivity analysis of paramaters and input variables

I am trying to perform a sensitivity analysis of an optimization problem $f(x,\alpha)= \min_{ Q} {g(x,\alpha , Q)}$ where $x$ is an input variable for our function, and $\alpha $ is a parameter. ...
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1answer
39 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
35 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 ...
2
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0answers
20 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 ...