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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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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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
31 views
Improving related estimates
There are three underlying quantities $x$, $y$, and $a$, where $x$ and $y$ are vectors, and $a$ is a scalar. They are related by $x = ay$.
We get noisy observations, $x_0,y_0$. We want to find $a$, ...
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1answer
57 views
Maximum likelihood estimators, hypergeometric and binomial
I'm trying to solve a two part problem. The set up is as follows: consider a bag with $\theta$ red marbles and $7-\theta$ blue marbles, with $\theta$ being unknown. Let $x$ denote the number of red ...
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1answer
70 views
Rao-Blackwell unbiased estimator geometric distribution
I'm looking at review questions and having trouble with this one!
Let $X_1,\ldots,X_n$ be i.i.d. geometric R.V.s with the pmf: $(1-p)^{x-1}p$, for $x=1,2,\ldots$ and $0<p<1$.
I need to use ...
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17 views
Is there a way to estimate the range of fitting coefficients from only the data?
Considering an approximation $f$ for a set of $N$ data points $(x,y)$ using, for example, $M$ radial basis functions at arbitrary sites in the domain
$f_i = \sum_{j=1} ^M c_j\phi(||x_i-x_j||)$
where ...
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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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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
29 views
Hypothesis testing problem of Normal distributions.
Consider the following Hypothesis Testing problem:
Hypothesis $H_0$ : $X \sim N(\mu_0, \sigma_0)$. Mean $\mu_0$ is known but only upper and lower bounds on $\sigma_0$ are known.
Hypothesis $H_1$ : ...
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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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0answers
33 views
Cramer-Rao bound for $\chi^2$ distribution parameter estimates.
I've stuck in unpleasant problem with noncentral $\chi^2$ distribution.
I work with random variables, distributed as $\chi^2_{\nu}(\lambda)$, where $\nu$ is the degree of freedom and $\lambda$ is ...
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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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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
$$ ...
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0answers
48 views
An estimate of the minimum and the second-smallest element in a set.
Suppose I have $N$ independent random variable $(x_1,x_2,x_3,\ldots,x_N)$ with pdf's $(w_1(x_1,z_1),w_2(x_2,z_2),\ldots,w_N(x_N,z_N))$ parameterizes by $N$ non random unknown parameters $\{z\}_1^N$. ...
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46 views
Cramer-Rao bounds for noncentral $\chi^2$ distribution parameters estimates
In many practical problems I came across the necessity to estimate the parameters of noncentral $\chi^2$ distribution (for example the degrees of freedom or non-centrality parameter etc.). Found some ...
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1answer
75 views
Empirical Bayes estimator for a Beta-Binomial parameters
Let $X_t$ be collected from a Binomial distribution with parameters $N_t$ and $P_t$, where $N_t$ is known for $t= 1, 2, \dots , T$. On the other hand, $P_t \sim \operatorname{Beta}(\alpha_t, ...
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1answer
27 views
Scale Median for MRE Estimators with Absolute Difference Error Function for Scale Families
Lehmann, in Theory of Point Estimation p.212, defines scale median as the solution to: $${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
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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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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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
87 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
279 views
How do I find the MLE of $\theta$ when x is dependent on $\theta$?
Let $X_{1},X_{2},...,X_{n}$ represent a random sample from a distribution with pdf:
$f(x; \theta)=e^{-(x-\theta)}, \theta \le x<\infty, -\infty<\theta<\infty$ | zero elsewhere
I need to ...
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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
119 views
Why should Gaussian noise have fractal dimension of 1.5?
In a paper I'm trying to understand, the following time series is generated as "simulated data":
$$Y(i)=\sum_{j=1}^{1000+i}Z(j) \:\:\: ; \:\:\: (i=1,2,...,N)$$
where $Z(j)$ is a Gaussian noise with ...
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1answer
80 views
The distribution when combining two samples together?
Suppose $X\sim N(0,{\sigma}^2)$ and $Y\sim N(0,{2\sigma}^2)$ . $X_1, ..., X_m$ are the samples from $X$ and $Y_1, ..., Y_n$ are the samples from $Y$. And then combine two samples as a new sample ...
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2answers
180 views
Intuitive explanation of a definition of the Fisher information
I'm studying statistics. When I read the textbook about Fisher Information, I couldn't understand why the Fisher Information is defined like this:
$$I(\theta)=E_\theta\left[-\frac{\partial^2 ...
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2answers
77 views
Expected value of a max
We have a roulette with the circumference $a$. We spin the roulette 10 times and we measure 10 distances, $x_1,\ldots,x_{10}$, from a predefined zero-point. We can assume that those distances are ...
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2answers
89 views
Show that estimates are unbiased
The following is a problem in my book that I don't really understand:
We take a random sample: $x_1,x_2,\ldots,x_n$ from a population that is $N(μ,σ)$ where $\mu$ and $\sigma$ are unknown.
We build ...
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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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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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91 views
Estimate number of distinct items
I have a large array of $n$ integers, some of which may be repeated, and I want to estimate how many distinct integers are in the array. Say the number of distinct integers is $N$. I can sample with ...
2
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0answers
135 views
Fisher Information and minimum variance estimators
I am trying to understand what can be proved about minimum variance estimators. I have changed the question to make it more specific.
Let us assume we have some finite set $S$ of elements and we just ...
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1answer
53 views
biasedness/unbiasedness of an MLE.
To show whether an MLE I just found is biased/unbiased, would I need to find the expectation of the answer? Plus would I do this by integrating $\text{MLE} \cdot \text{pdf}$.
My MLE is $ ...
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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 ...
2
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1answer
43 views
Parameter optimization in probabilistic models
Task: Suppose we model a variable $y = Wx + \mu$ as a linear transformation of $x$ plus some Gaussian noise $\mu\sim\mathcal N(0,\sigma I)$. Our aim is to minimize the estimation error of $x$ given ...
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0answers
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
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
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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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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1answer
156 views
Determine whether a statistic is sufficient, given the probability density
Let $X_1, X_2, \dots, X_n$ be a sample of i.i.d. random variables, with density $$f_\theta=\frac{2}{3\theta}\left(1-\frac{x}{3\theta}\right) $$ for $0 < x < 3\theta$. And
$f_\theta=0$ if $ x ...
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1answer
135 views
Using the MSE criterion, which is a better estimator for $\Theta^2$?
Question: Let $T_1$ and $T_2$ be independent unbiased estimators of a parameter $\Theta$.
Assume that $\operatorname{Var}(T_2) = \operatorname{Var}(T_1)$.
Using the MSE critertion, define which is a ...
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1answer
49 views
Proving that the sum of Good-Turing estimators is $1$
I want to know how to go about proving that the Good-Turing estimator has a total probability of $1$. I have seen this proof (page 2) but I found unclear the first step:
$$\sum_j \theta[j] = \sum_r ...
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2answers
122 views
Fast variance calculation
Suppose to have a sequence $X$ of $m$ samples and for each $i^{th}$ sample you want to calculate a local mean $\mu_{X}(i)$ and a local variance $\sigma^2_{X}(i)$ estimation over $n \ll m$ samples of ...
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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 ...
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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
71 views
Sufficient Estimators and Generalized Likelihood Ratios
If you can make the assumption that a sufficient statistic exists for some parameter - let's call it $\theta$.
How would you show that the critical region of a likelihood ratio test will depend on ...
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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 ...
