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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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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48 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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107 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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1answer
131 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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98 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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28 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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2answers
64 views

Asymtptotic limit of $e^x$ [closed]

I am looking for functions $A,B$ such that $$ A < e^x < B.$$ $A,B$ should be as close to $e^x$ as possible. I was trying to find something, but all I found was very distant. Can someone suggest ...
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1answer
36 views

proving unbiasedness of an estimator

Question given independent random variable $X_{1},X_{2},...,X_{n}$ from a geometric distribution with parameter $p$. we have an estimator for $p$, mainly $T=Y/n$ where Y is number of $i$ that ...
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132 views

Does an UMVUE's variance match the Cramer-Rao lower bound?

I know it can match the CRLB, but does it have to, if it is an UMVUE?
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1answer
54 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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1answer
124 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
19 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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17 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
23 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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1answer
43 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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1answer
30 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
45 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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1answer
65 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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45 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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1answer
12 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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42 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
96 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
981 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
106 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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206 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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137 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
90 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
25 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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242 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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557 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
102 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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149 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
108 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
45 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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127 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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15 views

Monte carlo formula to compute the approximation of variance of MLE

In the book of "Monte Carlo Statistical Methods", the book gives an approximation formula for the variance of MLE, Later on, the book mentions that this approximation formula can be written as ...
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19 views

How to compute MAP estimate of y?

Suppose that a scalar random variable y is of the form $y=z+v$, where the pdf of $v$ is $p_{v}(t)=\frac{t}{2}$ on the interval $[0,2]$, and the pdf of $z$ is $p_{z}(t)=2t$on the interval $[0,1]$. Both ...
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15 views

Predicting future outcomes from samples when sample sizes and distributions are not controlled and vary

I'm very stale in my statistics and am trying to calculate my confidence around a certain mean outcome from an investment firm (I'll use lay person terms so that I am not assuming any particular type ...
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1answer
42 views

log likelihood function of a cauchy distribution

What is the log likelihood function of a random varible x with cauchy distribution (0,1)? I've tried to work it out. I think its $\log (1+x)^2$. Is that correct?
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20 views

Sufficient statistic of a geometric rv

Can anyone help me prove the sufficient statistics of geo r.v. I am stuck and cant cancel out the thetaws . thanks.
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80 views

Second derivative wrt complex parameter

I'm facing an estimation problem and I need to calculate the Cramer-Rao Lower Bound of an estimator. So I have 2 unknown parameters: the amplitude of the signal $A$ and its direction of arrival $u$. ...
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15 views

Can we predict the positions of zero entries in a sparse vector given a model

I am wondering if we can predict the positions of the zero entries in a $n$-dimensional sparse vector $x$ given the linear model: $y=Ax$ where $y\in\mathbb{R}^m$ and the matrix ...
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15 views

Looking for hints on how to prove the convergence of this iterative estimator! [duplicate]

Let $X_n$ be a Poisson process with mean $\lambda^*$. The following sequence estimates the parameter of the Poisson process: $ X_{n+1} = \hat{\lambda}_{n+1} + ...
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1answer
23 views

Given 50 IID normals, find the exact SE for the estimate of $\sigma^2$?

Given 50 I.I.D Normal distributions random variables $X_i$, the Maximum Likelihood estimator for $\sigma^2$ is $\hat{\sigma}^2$, as proven in my lecture notes. Find the EXACT SE. My Attempt: ...
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7 views

How to compare two different estimators

What is the correct to compare the 'performance' of an estimator? Suppose I have two estimators $\hat{\pmb \theta_1}$ and $\hat{\pmb \theta_2}$ for the same problem of estimating $\pmb \theta$. For ...
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1answer
25 views

Mean Square Estimate problem

I have to find $\textbf{s}_{MS}$ given $\textbf{r} = h\textbf{s}+\textbf{n}$ where $h$ is a Bernoulli random variable with $Pr(h=1)=Pr(h=0) = 1/2$ and $\textbf{s}$ and $\textbf{n}$ are independent ...
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25 views

Asymptotic result on quadratic variation of a semi-martingale linear functional estimator

In the same context of this previous question. Consider $$ \mathcal E^{(n)}_t := \sqrt{n}(\widehat\Lambda_n(\phi)_t - \Lambda(\phi)_t )$$ I desire to prove that $$ \left \langle \mathcal ...
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9 views

Estimating variance from the sequence

Suppose that we have $\{X_n\}\to X\sim N(0,\Omega)$ where $X_n$ can be obtained from observations. My problem is to estimate $\Omega$ consistently. If $var X_n$ converges to a "finite" matrix, then ...
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27 views

How to simplify conditional probability of union of several events

I have an output binary scalar, $y∈B=[0,1]$, and an input binary vector $x=[x_1, x_2,…x_M]$ where $x_i∈B=[0,1]$. I know that the output $P(y)=1$ depends entirely on the input x. Thus, I want to ...
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21 views

Prove that an estimator is UMVU under the usual “assumptions of regularity”

I'm asked to prove that some estimator is UMVU under the usual assumptions of regularity. I'm not sure what is meant with 'usual assumptions of regularity'. Do they mean with this that I can just ...