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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Prove the consistency of Gamma distribution estimators

Given $X$ a random variable in a Gamma distribution, $f(x ; \alpha,\beta)$, and: $E(X) = \alpha \beta$ $Var(X) = \alpha \beta^2$ $\hat \alpha = $$\bar X \over \beta$ $\hat \beta = $$\frac {n \bar ...
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32 views

Help me understand this matrix derivative (for the LS estimation proof)

I'm trying to understand this proof of LS estimation, but I've never studied matrix calculus. I've managed to find a couple of identities on the web and and I see how to get the first part of the ...
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35 views

IMPROVED - Proving that a statistics is not sufficient (Gaussian case).

Let $X=(X_1,...,X_n)$ be i.i.d. $N(0,\sigma^2)$. How to show that $$\frac{2}{n}\sum_{i=1}^{n}X_i$$ is not a sufficient statistic? I have already proven that $\max_{i=1,...,n}X_i$ is a sufficient ...
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26 views

Show that $\hat{\mu}$ has minimal variance

So two independent analyses of a content in a water sample have been made using two different methods, both without systematical errors but with different standard deviations. Method $B$ is assumed to ...
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34 views

Maximum Likelihood Estimator of $\theta$

I have the following question I tried to answer I got answer that same like this answer Is this true answer? (Note that: in the question $0<p<\frac{1}{2}$, but in this answer ...
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95 views

Variance of Variance Estimation Test

I am trying to verify, through numerical simulation, the expression for the variance of the variance estimation, namely: Var(s^2)=2/n sigma^4 where n is the number of samples, and sigma is the ...
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60 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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101 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 ...
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65 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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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, ...
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38 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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40 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 ...
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162 views

questions on bias of estimator

a) Let $X_{1},...,X_{n}$ be i.i.d Uniform$[0,\theta]$. Show that estimator $\beta(X)=max(X_{1},..,X_{n})$ is a biased estimator for $\theta$.Find an unbiased estimator, based on $\theta$. My attempt: ...
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61 views

Literature study for Optimal Estimation Theory

It seems Optimal Estimation/Control Theory requires a lot more than undergraduate maths. Any good book that would help me get started? I have so far referred the following books but found them quite ...
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22 views

Likelihood function for only one trial

I have a trial $\mathbb X = (X_1,X_2,....X_n)$. $X_i$ has specified distribution with unknow parameter $\theta$. I want to find an estimator of this parameter. So I can use methods like Likelihood ...
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1answer
34 views

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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180 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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94 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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1answer
113 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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100 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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56 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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341 views

Finding UMVUE from Lehmann-Sheffe Directly

I am having some trouble with an example from the book I am following. Let $X_1,X_2,...,X_n$ for $n>2$ be an iid set of $N(\mu,\sigma^2)$ random variables with, $\mu\in\mathbb{R}$ and ...
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15 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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54 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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1answer
144 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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8k views

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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1answer
295 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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348 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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106 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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33 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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491 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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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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150 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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1answer
227 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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262 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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1answer
46 views

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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134 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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25 views

Optimal estimation of the fusion of two measurements

Suppose I have a sensor measuring a quantity $\text R$. For example the sensor could be a radar estimating the range of a target. We can write: $$R(t)=r(t)+\nu_0(t)$$ where $r(t)$ is the real range ...
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16 views

An issue related to the expectation maximization algorithm for a coin toss experiment

I just read a very nicely written introduction paper for the expectation maximisation algorithm published in Nature biotechnology by Do and Batzoglou ...
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6 views

Finding the norm of estimation error asymptotically

Let $\theta \in \mathbb{R}^p$ be such that it has uniform distribution on the set of standard unit vectors $\{\tau e_1,\ldots,\tau e_p\}$, for $\tau=\sqrt{(2-\varepsilon)\log p}, \varepsilon>0$. ...
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8 views

Distribution of sample minimum after bivariate selection (double truncation)

Let $X$ and $Y$ be two RVs with joint distribution $$ (X,Y)\sim \text{Normal}(\mu,\Sigma) $$ Suppose that there is selection on $X$ and $Y$, such that we observe a vector of realisations of $X$, ...
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22 views

Number of measurements for least squares and relation to maximum likelihood

I have a simple overdetermined system of equations: $$ y = Xc + e $$ $y, e \in \mathbb{R}^n$, $c \in \mathbb{R}^m$, $X \in \mathbb{R}^{n \times m}$, $e \sim \mathcal{N}(o,\sigma^2)$, $n>>m$ ...
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23 views

Relation of sufficent statistic to random mechanism in constuction of a randomized estimator?

After reading a about randomized estimators and sufficent statistics, the question weather the random mechanism is determined uniquely by the statistic struck me, but I cant seem to get an answer to ...
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22 views

Other method instead of using OLS estimation

enter image description here I get that the resulting beta* is the same as the estimated beta 1. But I think something is wrong. If the researcher is true, then why do we learn the OLS estimation, ...
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48 views

Fitting discrete distribution with normal-inverse gaussian weights

I regard a rather curious discrete distribution $X$ on $(1,2,...)$. Its weights are given by $P(X=i)=P_{NIG}(i)-P_{NIG}(i-1)$ where $P_{NIG}$ denotes the cumulative distribution function of the ...
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13 views

Determining wave algorithm based on sine wave

I have some data that I've noticed conforms to a sine wave and I want to approximate it as closely as I can. In the graph, the blue line is the data I want to model as closely as possible. From ...
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15 views

Intuition behind sampling distributions – specific case

I'm still trying to understand the basics of understanding the intuition of sampling distributions and calculating the sampling distributions of common estimators. For example, I understand the ...
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9 views

Fisher's information better than covariance matrix in estimators

I know Fishers information matrix is the inverse of covariance matrix. But why is it better to use fishers information matrix instead of covariance matrix in the case of distributed sensor networks?
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Maximum Likelihood Estimator of exponential of L2 norm

given the observed data $x = (x_1; x_2; \cdots; x_n)^T$ , the likelihood function p(x; $\theta$) can be charaterized as $$p(x; \theta) = \alpha(x) e^{ ||x - \hat{\theta}||_2} $$ where $\hat{\theta} ...
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43 views

maximum likelihood estimation of exponential and polynomial components model

I tried to find the maximum likelihood estimator and MMSE of the non linear model but I got stuck. Can you help me to explain it? The output of a system can be modeled using a combination of ...