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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120 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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0answers
29 views

How to Derive Relationship to the Gain Constant?

I want to implement a formula but I'm having issues understanding some of the components which make it up. The premise of the equation is to use a modified version of the Kalman filter that ...
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2answers
125 views

Finding the MLE of a function when L'($\theta$) doesn't depend on $\theta$

Here's the problem: Find the MLE of of $\theta$ when $f(x\mid\theta)=(1+x\theta)/2$ for $-1<x<1$, $=0$ otherwise. $0<\theta<1$ Find the maximum likelihood of $\theta$ and find its ...
4
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2answers
74 views

estimation of a parameter

The question is: $x_i = \alpha + \omega_i, $ for $i = 1, \ldots, n.$ where $\alpha$ is a non-zero constant, but unknown, parameter to be estimated, and $\omega_i$ are uncorrelated, zero_mean, ...
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1answer
37 views

Conjugate Bayesian analysis

Suppose that conditional on $\tau$, the random variable $X$ has normal distribution with mean zero and variance $1/ \tau$. The prior distribution for $\tau$ is Gamma with parameter $\alpha$ and $\beta$...
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1answer
27 views

How to show this estimation?

i have this polynom $$p(x) = \sum_{i=0}^{m}a_ix^i$$ I want to show, that if $\tilde{z}$ is the approximation to the simple zero digit $z \neq 0$ in first approximation, the following estimation ...
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1answer
26 views

How to show, that a relative mistake of a special function can be estimated in a given way

how to show, that if you have a function like this $$ y = f(x_1,...,x_m) := c \frac{x_1 *...*x_r}{x_{r+1},...,x_m}, \quad 1 < r \leq m,$$ the relative mistake in first order can be estimated ...
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1answer
101 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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1answer
162 views

Maximum Likelihood

Find maximum likelihood estimator $\hat\theta$ of $f(x;\theta) = (1/2)\exp(-|x-\theta|)$, for $-\infty \leq x < \infty$ and $-\infty \leq x < \infty$. I am confused of how to deal with the ...
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1answer
177 views

Optimal combination of two estimates

I have a set of random variables, $X_1,\dots,X_N$. They are i.i.d. Gaussian with zero mean and $w$ variance. I observe $Y_1,\dots,Y_N$ where $Y_i=\sum_{j=1}^N a_{ij} X_j+N_i$ where all $a_{ij}$s are ...
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1answer
46 views

Statistics: why is this probablility smaller?

a shipment of goods contains two containers, one container has 300 units and the other container has 700 units. A supervisor checks 30 units in the first container and he finds $X_1$ broken units and ...
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1answer
43 views

Overestimate of $|\oint_{|z|=R} f(z) \mathrm{d}z|$ with $f(z)=\frac{z^a}{z^2+1}$, $0<|a|<1\mathrm{with} \;a \in \mathbb R$

How can I overestimate, $|\oint_{|z|=R} f(z) \mathrm{d}z |$ with $f(z)=\frac{z^a}{z^2+1}$, $0<|a|<1 \; \mathrm{with} \;a \in \mathbb R$ ? I tried this: $|\oint_{|z|=R} f(z) \mathrm{d}z | <= \...
0
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1answer
432 views

expectation of Gamma distribution help

If x∼Gamma(1,λ) how would i find the expected value E(e^bx) where b=aλ I'm kinda stuck as to how to approach the question. Some help will be greatly appreciated Thank you in advance
2
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2answers
127 views

Why we always put log() before the joint pdf when we use MLE(Maximum likelihood Estimation)?

Maybe this question is simple, but I really need some help. When we use the Maximum Likelihood Estimation(MLE) to estimate the parameters, why we always put the log() before the joint density? To use ...
0
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1answer
66 views

Suitability of skew normal for rating task and calculation

in an experiment, I ask participants to rate qualities on a continuous scale. I expect the results to be normal distributed and I am confident that assuming a normal works fairly well for most values. ...
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0answers
33 views

A question on Stochastic Approximation

I have just started learning stochastic approximation methods, so the question I'm going to ask may be a trivial one in this field, but I need to know this seriousely. I know, that if $g(x,\xi)$ is a ...
-1
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1answer
426 views

CRLB to find UMVUE

In what situation can one obtain an estimator that reaches the Cramer-Rao lower bound, i.e. an efficient estimator? I know the rules for finding UMVUEs, and I know they are efficient if they reach ...
0
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1answer
579 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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2answers
314 views

Does convergence in probability not imply convergence in distribution for Least Squares estimators?

I have a question relating to convergence in probability and distribution for least squares estimators. Frequently, I see in textbooks that $\hat{\beta} \rightarrow^p b$. Where $b$ is the population ...
2
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0answers
58 views

What are the properties of median-unbiased estimators?

On Wikipedia it says that " A median-unbiased estimator minimizes the risk with respect to the absolute-deviation loss function, as observed by Laplace." How to prove this? Note that I asked on Cross ...
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1answer
73 views

Is it possible to “customize” the multinomial distribution to your specifications?

So according to the multinomial distribution, the probability function $\Pr(X_1 = x_1, X_2 = x_2, \dots, X_k = x_k)$ is equal to $\dfrac{n!}{x_1! x_2! \cdots x_k!} \cdot p_1^{x_1}\cdot p_2^{x_2} \...
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1answer
363 views

How to calculate probability using multinomial distribution?

So according to the multinomial distribution, the probability function $\Pr(X_1 = x_1, X_2 = x_2, \dots, X_k = x_k)$ is equal to $\dfrac{n!}{x_1! x_2! \cdots x_k!} \cdot p_1^{x_1}\cdot p_2^{x_2} \...
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1answer
299 views

Find maximum likelihood estimator, trick?

Let $Y_1, Y_2, \ldots, Y_n$ iid random variables with density $f(y)=\theta\cdot y^{\theta-1}$, $0<y<1$, $\theta >0$. I need to show that the maximum likelihood estimator of $\theta$ is $-n\...
3
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2answers
87 views

Estimating time in harmonic signal

I hope someone can help me with the following problem: Assume a periodic signal of the form $$\begin{align} s(t) &= \sum\limits_{p=1}^P \sin(p\Omega_0t)\\ &= \sum\limits_{p=1}^P \sin(\...
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0answers
84 views

Markov Chain : Montonicity of Sample Mean

Let $\{X_n\}_{n\geq1}$ be an irreducible, ergodic Markov chain with discrete state-space $S$, transition probability matrix $P$ and steady state distribution $\pi = \{\pi_j\}_{j\in S}$. Let $f$ be a ...
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0answers
26 views

ML Estimation for number of animals in a park. Hypothesis Testing.

A park of area $S=10 000 km^2$ was surveyed for bears, and out of $n$ disjoint regions of equal area $s=1km^2$, there were $n_k$ regions with $k=0,1,....,N$ bears. On each of these regions, the amount ...
0
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1answer
95 views

Bounds on least squares and weighted least squares estimator

I was wondering if I can get some help in getting bounds on the parameters estimated by least squares (LS) and weighted least squares (WLS) methods. Suppose our observation model is: $\mathbf{y} = \...
1
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1answer
98 views

Using Neyman pearson lemma when ratio comes out to be zero.

Consider a Bernoulli random variable: $$X_i= \begin{cases} 1, & \text{with probability }p \\ 0, & \text{with probability }1-p \end{cases}$$ You observe the outcomes of two Bernoulli trials ...
2
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1answer
51 views

Monotonicity of Sample Mean

$X_1,X_2,\ldots$ are drawn i.i.d. from a distribution with mean $\mu$. Define $\bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i$ Prove that $\forall t \quad E[|\bar{X}_t - \mu|] \geq E[|\bar{X}_{t+1} - \mu|]$
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1answer
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 ...
9
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1answer
10k views

Maximum Likelihood Estimator of parameters of multinomial distribution

Suppose that 50 measuring scales made by a machine are selected at random from the production of the machine and their lengths and widths are measured. It was found that 45 had both measurements ...
0
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1answer
346 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 $\sigma^2\in\...
3
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2answers
157 views

Estimation of the number of prime numbers in a $b^x$ to $b^{x + 1}$ interval

This is a question I have put to myself a long time ago, although only now am I posting it. The thing is, though there is an infinity of prime numbers, they become more and more scarce the further you ...
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2answers
266 views

What exactly is simple consistency?

Now yes, I know the definition: A sequence of estimators $\{T_n\}$of $\tau(\theta)$ are consistent if for every $\epsilon > 0$ $$ \lim_{n\to\infty}P[|T_n-\tau(\theta)|\leq\epsilon]=1 \\\text{ ...
4
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1answer
158 views

Estimate the scale of $e^{-(m+1) t} \sum _{k=0}^{\infty } \frac{t^k}{k!}\left(\sum _{r=0}^k \frac{t^r}{r!}\right)^{m}$

I would like to estimate the scale of the following series, $$S(m,t)=e^{-(m+1) t} \sum _{k=0}^{\infty } \frac{t^k}{k!}\left(\sum _{r=0}^k \frac{t^r}{r!}\right)^{m},$$ where $e$ is the base of ...
3
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1answer
1k views

CRLB/UMVUE estimation of $\theta$

We have a random sample $X_1,X_2,\ldots,X_n$ from a probabilitiy distribution with density $f(x;\theta) = \theta x^{-\theta-1} $ given that $x > 1$, and $0$ else. where $\theta >1 $ is an ...
0
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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})] &=E[\frac{X}{n}]-E[\frac{X^...
2
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1answer
118 views

Finding the MLE of $\theta$ where $\theta \leq x$

consider the following PDF $$ \begin{eqnarray} f(x;\theta) &=& \left\{\begin{array}{ll} 2\frac{\theta^2}{x^3} & \theta \leqslant x\\ 0 & x< \theta; 0 < \theta \end{array}\right.\...
1
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1answer
112 views

find $\theta_{MLE}$ for a function

For $$ f(x;\theta)=(\theta+1)x^{-\theta-2} $$ find the maxmimum likelihood estimators (MLEs) for $\theta$ based on a random sample of size $n$. My work so far: $$ \begin{align} \prod_{i=1}^n \...
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1answer
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 $i=1,\...
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1answer
35 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 $a$, ...
3
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1answer
717 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
2k 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 rao-...
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0answers
31 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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0answers
91 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 $H_1$ : ...
0
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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: $\frac{1}...
0
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1answer
9k 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...
1
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1answer
579 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 $\...
1
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1answer
324 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 ...
2
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1answer
456 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 $$ \mathbb{I}(\theta)=\mathbb{E}\left[\...