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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211 views

Use Held-out estimator for unseen events

Held out estimator is an empiric estimation technique for calculating probabilities of events. (I would put a Wikipedia link, but i couldn't find a wikipedia page on this subject) The main idea is to ...
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305 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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1answer
87 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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80 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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72 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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52 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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125 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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168 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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106 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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32 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
48 views

Biased MLE estimate of mean (expectation)

Please give an example of p.m.f. or p.d.f. , the maximum likely-hood estimate of whose mean (expectation) is a biased estimator . Thanks
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33 views

Does consistent estimators have in-variance property?

If $(T_n)$ is a sequence of consistent estimators of a parameter $\theta$ ( i.e. for every $ \epsilon >0$ , $\lim_{n \to \infty} P [ \space |T_n -\theta|< \epsilon ]=1$ ) , then is it true that ...
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53 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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95 views

Estimator for second moment for Poisson random variable

Let $X \sim Poiss(\lambda)$. As, $\displaystyle \sum_{i=1}^{N} X_i $ is sufficient statistic for both mean (and variance) of $Y$, so we can define the unbiased estimate for mean as , $ s=\frac{1}{N} ...
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107 views

Sample size required to estimate population proportion with given precision

It plans to conduct a study on the percentage of homeowners who have at least two TVs. What should be the sample size if we want to ensure that $95\%$ of estimation error is less than $0.01$? ...
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2answers
59 views

MAP Estimator with Laplacian Noise

I need to calculate the MAP estimator of $ x $ in the following case: $$ \left [ \begin{matrix} {y}_{1}\\ {y}_{2} \end{matrix} \right ] = \left [ \begin{matrix} x\\ x \end{matrix} \right ] + ...
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350 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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70 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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141 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
48 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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1answer
15 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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1answer
30 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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31 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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154 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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43 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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33 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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108 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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59 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
86 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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85 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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55 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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277 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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1answer
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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48 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
130 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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5k 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
271 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
285 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
103 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
30 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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1answer
420 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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926 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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140 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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197 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
182 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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3answers
132 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
23 views

Bayes estimator under squared error loss

Consider one random variable X from the Bernoulli distribution with parameter θ. Let p, the prior density, be equal to 6θ(1 − θ), for θ ∈ (0, 1). Under squared error loss, L(t, θ) = (t − θ)$^2$, the ...
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60 views

Maximum likelihood estimator and confidence interval

Let $\theta$ be an unknown constant. Let $W_1,…,W_n$ be independent exponential random variables each with parameter $1$. Let $X_i=θ+W_i$. First, I need to find $\hat\theta _{ML}(x_1,\ldots ,x_ n)$. ...