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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48 views
Estimating the number of observations from a set of samples
I repeatedly measure a value $S_n$ which is the sum of a set of $n$ hidden inputs. The goal is to identify the number of hidden inputs.
All of the hidden inputs are driven by an experimenter ...
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0answers
57 views
Showing that statistic is unbiased
Let $X $ be observed data. Let $\hat{\theta}(X)$ be an unbiased
estimate of $\theta$ and let T be a sucient statistic for $\theta$. Define the new estimator
$\hat\theta^{*}$ of $\theta$,
$$ ...
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0answers
83 views
Worst-case error related to Cramer-Rao bound
I would like to understand the relation (if any) between the Cramer-Rao Lower Bound of estimation theory and the following simple definition of "reconstruction accuracy" which doesn't use any ...
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0answers
99 views
Parameter estimation for a distribution by minimizing its conditional entropy
Let $X$ be a discrete random variable with Laplacian distribution with mean $0$ and scale $\lambda$, as
$$
p(X) = \frac{1}{2\lambda} \exp\left(-\frac{|x|}{2\lambda}\right), \\
X \in ...
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0answers
109 views
The fundamental gaussian identities of bayesian estimation
In bayesian estimation, when the model and plant noise is hold , the optimal estimator is Kalman filter. but I am wondering is there any literature that could prove the following gaussian identities?
...
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27 views
MLE estimation of parameters, converting normalized observations to integers and back
I am fitting a model's parameters to grouped data by maximizing the likelihood equation:
$L(\theta)=N!\prod_{i=1}^{G}\frac{p_i(\theta)^{n_i}}{n_i!}$
$\theta$ is the vector of parameters. $n_i$ is ...
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1answer
105 views
Estimation of discrete random variable
Suppose you have a discrete random variable $X_1$ with known probability mass function. I guess that choosing a variable drawn from the same pmf would be the best way to guess $X_1$ assuming all ...
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1answer
25 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
41 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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1answer
25 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
33 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
27 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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1answer
51 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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1answer
88 views
How to estimate parameters of a uniform distribution?
I have information of the order in which students were classified in regard to their scores in a SAT test. I know the distribution of scores for each student is uniform with support [a,b]. I also know ...
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1answer
23 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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2answers
341 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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1answer
65 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
109 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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1answer
70 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
44 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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3answers
116 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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1answer
220 views
Intuitive explanation of Fisher Information and Cramer-Rao bound [closed]
I am not comfortable with Fisher information, what it measures and how and how is it helpful. Also it's relationship with the Cramer-Rao bound is not apparent to me.
Can someone please give an ...
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0answers
26 views
MVUE for Binomial $p^s$
Let $Y_1, Y_2, ..., Y_n$ be a random sample from a $\text{Bin} (1,p)$ where $0 < p < 1$.
Let $0 < s < n$ be an integer. Find the MVUE for $\displaystyle\phi (p)=p^s$ and ...
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0answers
23 views
Variability in estimations over a non-ergodic/non-regular Markov process
Imagine we have a non-ergodic/non-regular Markov Process with with $n$ states.
Among these $n$ states, there are $k$ absorbing states.
For each of the $n-k$ non-absorbing states, it is not possible ...
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0answers
12 views
Shrinkage estimator's risk function
How do you compute the risk function under squared loss of an estimator of the form
$\begin{align*}
\hat{\mu}(x) &= \bar{x} + \left(1-\frac{k}{||x-\bar{x}||_2^2}\right)(x - \bar{x})
\end{align*}$
...
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0answers
35 views
Numerical calculation of fisher information
I am trying to obtain numerically the fisher information. Given a likelihood function
$$ f(X,\theta),$$
with $X \in [0,1]$.
The fisher information is given by
$$ ...
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0answers
48 views
An estimate of the minimum and the second-smallest element in a set.
Suppose I have $N$ independent random variable $(x_1,x_2,x_3,\ldots,x_N)$ with pdf's $(w_1(x_1,z_1),w_2(x_2,z_2),\ldots,w_N(x_N,z_N))$ parameterizes by $N$ non random unknown parameters $\{z\}_1^N$. ...
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1answer
77 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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1answer
51 views
Efficient method of approximating a distribution with Gaussian
Given a univariate uni-modal density function $f(x)$ (very hard to compute its cumulative distribution function (CDF) $F(x)$, not to mention its inverse CDF $F^{-1}(x)$),
how to find the best ...
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1answer
67 views
Clasification of parameter estimation method
Consider that $P$ is the water pressure coming out from a valve A, therefore, the population is all the valve A pressure values. Let $P_{dif}$ be defined as the difference between the maximum and the ...
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0answers
95 views
Finding the efficiency of an unbiased estimator
I have a random sample drawn from a $N(\theta,\sigma^2)$ distribution with $\sigma^2$ known. I am trying to estimate $\theta$.
I need to calculate the efficiency of the unbiased estimator, ...
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0answers
61 views
Asymptotic normality conditions verification
Let $X_t, X_{t-1}, \dots, X_1$ be a sequence of observations from the process $X_t = \theta X_{t-1} + e_t$, where $e_t$ are independent and identically distributed as $N(\mu, \sigma^2)$. Using the ...
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0answers
117 views
Stationary Zero Mean AR(1) time series?? Yule-Walker equations???
Let {$X$t; $-∞ < t < ∞$} be a stationay zero mean AR(1) time series process given by
$X$t = 0.5$X$t-1 + $V$t
and define $Y$t as
$Y$t = $X$t + $W$t
where $V$t ≈$N$ ...
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1answer
112 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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1answer
32 views
Several Unbiased Estimators
If I have some data set $ D={X_1,...X_N} $ and have an esitmator be "pick the first point" $X_1$, how can I show that this estimator is unbiased? I also have to show why its highly undesirable, and I ...
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0answers
38 views
Identification of parameters problem
I always struggle to get the true essence of identification in econometrics. I know that we state that a parameter (say $\hat{\theta}$) can be identified if by simply looking at its (joint) ...
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0answers
44 views
Estimate the size of a set given random sub sets.
Assuming there is a set $S$ that you are given subsets of, $s_1, s_2, ..., s_n$, estimate $|S|$ (and a confidence interval if possible) making as few assumptions as possible.
I'm not going to quibble ...
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0answers
113 views
Mixture of Normal distributions:Estimating variance& Fisher Information
I want to estimate the variances and the Fisher information of a countable mixture of Gaussians with assumed equal variance and identically spaced means. I thought that the Fourier transforms of an ...
0
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0answers
79 views
Maximum Likelihood Estimator of SNR for a Known Signal Superimposed in AWGN
I would like to evaluate the Maximum Likelihood Estimator for the SNR of a given signal:
$ x(t) = as(t-\tau) + n(t) $
Under the following assumptions (This is the model of Radar Signal):
The input ...
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0answers
407 views
Maximum likelihood estimation,
Given a problem, we are asked to use a statistical method to come up with a conclusion. e.g. XYZ drug decreases heart diseases. (null hypothesis)
After searching on internet, I got a list of these ...
