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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Estimator of a Random Variable

Given a random varable $Y$ where $$ f_Y(y) = \begin{cases}e^{-(y-k)} \quad x>k\\0\quad \text{otherwise}\end{cases} $$ Given $n$ observations of $Y$. Is the sample mean $\bar{Y}$ an unbiased ...
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78 views

calculating mean squared error for the Mean.

Exam Question There are two independent random variables $X_{1}$ $\&$ $X_{2}$ that are having normal distribution with mean $\mu$. Further Var$(X_{1})=1$ and Var$(X_{2})=2$.an unbiased estimator ...
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49 views

Variance of unbiased estimator

Let $Y_1,Y_2,...,Y_N$ be a random sample from a distribution with probability density function $f_Y(y,\theta) = 2y/\theta^2$ if $0<y<\theta$ and $0$ otherwise. (a) Show that $W = 3\bar{Y}/2$ ...
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72 views

is any upper bound for mean square error of an unbiased estimator?

There is always a lower bound for an unbiased estimator called Cramer-Rao Lower Bound. Does any one remember any upper bound for unbiased estimator? The upper bound is used for worst-case analysis of ...
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78 views

A quick chanllenge: height and weight probability problem

The average height and weight of a group of people is 175cm and 70kg; Find the upper bound of the portion of the people who are over 200cm and over 100kg. I thought about Markov inequality, but I ...
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50 views

Estimator in a one dimensional normal setting with only one observation

Let $X$ have the distribution $N(\theta,1)$ where $\theta \ge 0$. Is $T=X$ an admissible estimator with respect to the mean squared error? Construct an estimator that respects the assumption $\theta ...
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13 views

Estimating $\hat{p}$

let $X\sim Bin(n,p)$ and $\hat{p} =\frac{X}{n}$ a) Find a constant c such that $E[c\hat{p}(1-\hat{p})]=p(1-p)$ My work: $$ \begin{align} cE[\hat{p}(1-\hat{p})] ...
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46 views

Random Poisson Sample, Probability in terms of $\vartheta$

If $X_1, X_2, \ldots, X_n$ are a random sample from a Poisson Distribution with mean $\vartheta>0$, how do you find $P(X\le 1)$ in terms of $\vartheta$? I've proven that summing $X_i$ for ...
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120 views

Finding expected value of variance estimator (sum expansion problem)

I am trying to show that variance estimator $\frac{1}{n}\sum_{i=1}^{n}(X_i-\bar{X})^2$ is biased. I have an example in the book, and there is one step of this derivation I cannot understand: ...
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Calculating the variance of an estimator (unclear on one step)

How can you go from $4V(\bar X)$ to $\displaystyle \frac{4}{n}V(X_1)$? I understand the rest of the steps...
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193 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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249 views

Empirical Bayes estimator for a Beta-Binomial parameters

Let $X_t$ be collected from a Binomial distribution with parameters $N_t$ and $P_t$, where $N_t$ is known for $t= 1, 2, \dots , T$. On the other hand, $P_t \sim \operatorname{Beta}(\alpha_t, ...
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235 views

Scale Median for MRE Estimators with Absolute Difference Error Function for Scale Families

Lehmann, in Theory of Point Estimation p.212, defines scale median as the solution to: $${E(X)I(X\le c)} = {E(X)I(X\ge c)}$$ given $X$ is a positive random variable, and ${E(X)}< \infty$. Now ...
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98 views

How to estimate parameters of a normal distribution?

Suppose one knew that 105 workers were evaluated by their boss. Such evaluation is distributed according to a normal distribution with mean $\mu$ and std. deviation $\sigma$. We also know that 20 ...
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26 views

How to obtain estimate of covariance matrix that will be guarantee to be semi-positive define?

How to obtain estimate of covariance matrix that will be guarantee to be semi-positive define ? (Is CrossValidated better place for this question ?)
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325 views

how can I get minimum error probability for this decision problem?

I have the decision problem for 4 hypotheses as follows: $$H_j: Y_k=N_k-s_{jk},\ k=1,2,\ldots,n;\ j=0,1,2,3.$$ where signals are $s_{jk}=E_0\sin(w_cT(k-1)+(j+\frac{1}{2})\frac{\pi}{2}).$ $$$$ In ...
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816 views

Exponential Distribution Maximum Likelihood

I found the following question in a past exam paper and I would like to ask how to solve it as I can't find anything in the notes related to it: ...
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130 views

Proof of convergence of a sum of mean-consistent estimators

After a few weeks off I am back at my self-study of Measure-Theoretic probability. As always, I thank the community for any detail and answers they can provide as I try to work myself through these ...
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173 views

Estimation Theory - Maximum Likelihood Estimation

The below homework question comes from Larsen and Marx, 4th edition. Is the maximum likelihood estimator for $\sigma^{2}$ in a normal pdf, where both $\mu$ and >$\sigma^{2}$ are unknown, ...
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142 views

How to match a discrete distribution to a continuous distribution in information theoretic sense?

Let $$ S \sim N(\mu, \sigma^2) $$ be a normally distributed random variable with known $\mu$ and $\sigma^2$. Suppose, we observe $$ X = \begin{cases} T & \text{if $S \ge 0$}, \\ -T & ...
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Estimating a function given a noisy sequence of its output

I am new to this forum. Please forgive me if this question is elementary, but I am somewhat lost and could use a little guidance. Suppose I have an unknown function $f(i)=x_i$. I have a sequence of ...
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129 views

Fair selection of most popular items among separate voting sets

This is a practical problem that arose in real life, which I believe creates interesting mathematical questions. There is a festival of small plays lasting 8 weeks. Each week 10 short plays are ...
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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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Procedure to determine unbiased and consistent estimator of moments

Preliminary definitions I have a random variable $X$ and $N$ independent observation of it ($X_i, i\in\{1, \ldots, N\}$). I know that: $$\mathbb{E}[X_i^r] = \hat{\mu}_r,~ \mathbb{E}[(X_i - ...
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How to find the CRLB for non-Gaussian

I have certain doubts related to the following case where I need to estimate the filter coefficients of an autoregressive model from the observations $x$. The AR process is excited by 0/1 Bernoulli ...
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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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23 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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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 ...
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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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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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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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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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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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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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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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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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8 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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Find the MLE of mu/sigma and its expected value?

The problem: $X_1, \ldots, X_n$ be random sample from a normal distribution $N(\mu, \sigma^2).$ Find the MLE of $\mu/\sigma^2$ and its expected value. My solution: Since $\hat{\mu} = \bar{x}$ and ...
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43 views

Second derivatives of MLE to multivariate normal distribution

I want to calculate the second derivatives to the MLE's $\hat{\mu}$ and $\hat{\Sigma}$ to confirm that the extremums indeed are maximums to the multivariate normal distribution $$ ...
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Find the spectral density of white noise $\omega$~(0,2)

I am find the spectral density of a white noise given by $\omega$ ~ (0,2). Could you help me to find it? Thank all.
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Determine a distribution of a gaussian stochastic at different time

I would like to determine the autocorrelation function of a Gaussian stochastic. Let see my problem So my solution is The distribution of $y=x(t_1)-x(t_2)$ is also a Gaussian stochastic with ...
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Least square estimator: $N( \beta x_i, \sigma^2)$

Let $ Y_1,...,Y_n$ be i.i.d $N(\beta x_i, \sigma^2) $ with known $ x_i's$. It is asked to find the Mean Squared Estimator for $\beta.$ I didn't understandmuch about this method of pbtaining an ...
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Estimate time it takes the minimum mean cycle cancelling algorithm to converge

This particular algorithm solves the circulation problem, equivalent to the minimum-capacitated flow. My question rather than only from this particular algorithm, but for combinatorial solutions in ...
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Estimators of the binomial distribution

This is a follow-up of this previous question and elaborates on the answer I received there. $\def\b{\begin{pmatrix}}\def\e{\end{pmatrix}}\def\a{\alpha}X_1,X_2,\dots, X_n$ are a random sample from ...
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Asymptotic coincidence of the MAP and MMSE estimator

In many works, simulations show that as number of samples increases, the mean-square-error (MSE) of the MAP estimator attains the minimum MSE. Where can I find a theoretical proof to these empirical ...
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What is the problem with this model parameter estimation algorithm?

In a statistical model with parameters $\theta$ and unobserved laten variables $Z$, the model likelihood is $$L(\theta;X)=Pr(X|\theta)=\sum_ZPr(X,Z|\theta)$$ The standard way to estimate $\theta$ ...
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estimation problem for two-parameter weibull distribution

Suppose the two-parameter Weibull distribution is given by the pdf $$ f(x;a,b) = \left(\frac{x}{a}\right)^b\frac{b}{a}\exp\left\{-\left(\frac{x}{a}\right)^b\right\}, $$ where $x,a,b>0$. I am ...
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Optimal combination of multiple estimates of a random variable

For the following estimation problem: y = hx + n, x is the sent data, y is the observation (received data), h is a scaling factor (known), n is an AWGN random variable with zero mean ...
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16 views

OLS estimators in stationary process

Given a stationary process xt=a+b*t+et with et a white noise, how can I find the OLS estimators for a and b? Cheers