Questions tagged [estimation]

For questions about estimation and how and when to estimate correctly

Filter by
Sorted by
Tagged with
0 votes
0 answers
6 views

Estimate of the variance of the point estimator

This is probably a very stupid question, but I tried everything and look all over different topics and forums, i just don't get it. A poll of 1500 registered voters is taken. Of these, 600 say they ...
user avatar
0 votes
0 answers
11 views

Sampling distribution of GBM Maximum-Likelihood estimator

Given the geometric Brownian diffusion $$ X_t = \mu X_t dt + \sigma X_t d W_t$$ I learnt that its maximum likelihood estimators are the following as this web article suggests $$\hat \mu = \frac{\delta ...
user avatar
  • 396
2 votes
1 answer
52 views

Method-of-moments estimator for a uniform distribution

I have a sample of data points independently sampled from a uniform distribution with a density function $f(x)=\frac{1}{a}, 0\leq x \leq a$. I need to use the method of moments to estimate $a_{mom}$. ...
user avatar
-1 votes
0 answers
26 views

Method of Maximum Likelihood Estimation

I have this continuous function $$f(x)=(ae^ax)/(e^ax+1)^2, \text{ for} -∞≤x≤∞. $$ And I got this log function I tried the following code in Mathematica n = Length[c]; logL = nLog[a] + Total[Log[...
user avatar
1 vote
1 answer
55 views

Pointwise upper bound on $|f(x+y+z)-[f(y)+f(z)+f'(y)(x+z)]|$ where $f(x) = |x|^{p-1}x$

Let $f(x) = |x|^{p-1}x$ for some $2 \leq p \leq 3$. I've seen the pointwise estimate $$\left| f(x+y+z)-[f(y)+f(z)+f'(y)x + f'(y)z] \right| \lesssim |f(x)| + |f'(x)y| + |f'(z)x| + |f'(z)y|$$ in Lemma 3....
user avatar
  • 1,689
-1 votes
0 answers
22 views

Big O, Big Ω, Big Θ calculations and estimates

I just recently "learned" about Big Theta, Big O, and Big Omega, but I'm not sure how to use them and how to use estimation. Below are a few problems I'm trying to figure out, but I can't ...
user avatar
0 votes
1 answer
24 views

How to calculate the bias when using $\bar{y}^{3}$ as an estimator for $\mu^{3}$?

How do I get the following bias when using $\bar{y}^{3}$ as an estimator for $\mu^{3}$ where $y$ is iid with mean $\mu$ and variance $\sigma^{2}$: $$ \frac{3 \mu \sigma^{2}}{N}+\frac{\mathbb{E}\left[(...
user avatar
1 vote
0 answers
32 views

Combine two position estimates with different accuracies

Hi smart people of StackExchange! I have a question regarding the combination of two position estimates (X and Y coordinates) where each position estimate has a different certainty/accuracy. Example I ...
user avatar
0 votes
1 answer
25 views

Least Square algorithm between two vectors

I have a vector $y \in \mathbb{C}^{N \times 1}$ and it expressed as $y = H\times s$, where $H$ is diagonal $N \times N$ matrix to be estimated and $s \in \mathbb{C}^{N \times 1}$. So I formulated ...
user avatar
  • 7
0 votes
0 answers
11 views

Asymptotic normality of (M) estimators for convex $rho$.

The text from Robust Estimation of Location Parameter (1964), Huber, P.: It will be assumed thad $\rho$ is continuous convex real-valued function of a real variable $t$. tending to $+\infty$, as $t &...
user avatar
1 vote
0 answers
23 views

How does One Come Up with Confidence Intervals For Fermi Estimation Problems?

A common question is estimate some quantity $X$, and give a 95% confidence interval for it. The estimation part, I understand how to think about and attack, but I'm always lost when it asks for a ...
user avatar
  • 49
0 votes
0 answers
13 views

A linear lower bound on sine [duplicate]

I've seen the following inequality stated, but not proven in X.10 of Sarason's Complex Function Theory: If $x \in [0, \pi / 2]$, then $\sin x \geq \frac{2}{\pi} x$. How would I go about proving this ...
user avatar
  • 7,899
-1 votes
0 answers
25 views

unbiased estimator for supremum of expectation

Can we get an unbiased estimator for $$ \sup_{\theta} E_{X\sim P} [\max\{\theta-X, 0\}] \quad ? $$ Let $X_1$ be a sample from $P$. I tried using the $1$-sample estimator: $$ \sup_\theta \max\{\theta-...
user avatar
  • 1,666
0 votes
0 answers
5 views

Infinitesimal Robustness, influence function of $T$ at $F$.

This text is taken from Introduction to robust estimation and hypothesis testing. Wilcox R. First I will write down the description that leads to definition of relative influence on $T(F)$ and then I ...
user avatar
5 votes
1 answer
188 views

Estimating Lambda in a Poisson population where not all samples can be observed

Let $(x_1, x_2, \dots , x_n)$ be a random sample from a population which follows a Poisson distribution with an unknown mean $\lambda$. If we assume that $C$ is a known constant and we can only ...
user avatar
2 votes
1 answer
52 views

Averaging estimates of probability

Say one weather forecaster says there is an 80 percent chance of precipitation at a given place and time. Another forecaster says there is a 30 percent chance. I have no reason to think one forecaster ...
user avatar
1 vote
0 answers
21 views

How to prove multivariate Bayesian Cramér-Rao inequality?

I came cross multivariate Bayesian Cramér-Rao inequality as follow recently, but I don't know how to prove it. Let $\{f(\cdot ; \theta): \theta \in \Theta\}$ be a family of probability density ...
user avatar
  • 405
1 vote
0 answers
16 views

Let $X\sim$Hyp($N,l,n$) and $h:\{0,1,...,\min(l,n)\}\to\mathbb{R}$. Prove that there is no unbiased estimator $h(X)$ for the unknown parameter $N$.

Let $X$ be hypergeometrically distributed with the parameters Hyp(N,l,n). Of those parameters, $N$ is unknown. I have got to argue (preferably prove) that there doesn't exist an unbiased estimator $\...
user avatar
0 votes
0 answers
5 views

How do I arrive at this classical/typical location of random matrix eigenvalues estimate?

Consider an $ N \times N$ Wigner matrix $H$ and define the classical or typical location of eigenvalue $\lambda_i$ as $ N \int_{\gamma_{i}}^{2} \rho(x) \text{d}x = i - 1/2 $, where $i$ indexes the ...
user avatar
  • 425
2 votes
3 answers
101 views

How can we stay confidence replacing the population standard deviation by it's estimate?

So imagine we take $n$ random samples from a Bernoulli Trial. Thus our random samples are composed by binary random variables $X_1, X_2, ..., X_n$. So by central limit theorem we know that the ...
user avatar
  • 145
5 votes
1 answer
56 views

Lehmann–Scheffé theorem's statement

In my notes I have the following L-S theorem statement: Let $T(X_1,...,X_n)$ be an estimator for $\theta \in \Theta$. If $T$ is: unbiased a function of complete and sufficient statistic $S_c(X_1,......
user avatar
0 votes
0 answers
13 views

How to determine accuracy of area estimate given ground truth point?

I have coordinates of form $(x, y)$ which are the ground truths. My analysis does not directly product coordinates instead produces an area estimate. The area estimate is in the form of a closed ...
user avatar
  • 1
0 votes
0 answers
14 views

Chained Kalman Filters

I have read the term “chaining Kalman filters” and I wanted to know precisely what the chained form of a Kalman filter is. I’ve seen also the term dual Kalman filter framed as a different concept ...
user avatar
  • 145
2 votes
0 answers
30 views

Boltzmann distribution - estimator

I have a discrete random variable $X$, with possible values $\{E_k\}$ , following a Boltzmann distribution, with probability mass function given by: $$ p_k = p(X=E_k) = e^{-\beta E_k}/Z $$ where $Z = \...
user avatar
1 vote
1 answer
64 views

Unbiased estimator of $e^{-\theta}$

$(X_1, X_2,...,X_n)$ follows i.i.d. $N(\theta,1)$. Is $g(\theta)=e^{-\theta}$ estimable? I have to show that there exists an unbiased estimator of $g(\theta)$. Now, $T=\overline{X}$ is unbiased for $\...
user avatar
0 votes
0 answers
19 views

Different forms of CRLB

If $X=(X_1,X_2,...,X_n)$ follows i.i.d. $f_\theta()$, and we have to estimate $g(\theta)$, then $CRLB=\frac{{(g'(\theta))}^2}{nE_\theta\ {[\frac{\partial lnf_\theta({X_i})}{\partial \theta}\ ]}^2}$ $= ...
user avatar
0 votes
0 answers
25 views

Estimate polynomial function

Here are my difficulties, is there any method to solve or estimate the following integral? $$\int_0^\infty\left[e^{-2λx}+(1-p) e^{-λx}-1\right]^n dx$$ where $\lambda$ and $p$ are constants. following ...
user avatar
3 votes
1 answer
45 views

"Consistency" vs. "Convergence" of Estimators : Are ALL "MLE's" ALWAYS Consistent?

I have heard the terms "Consistency" vs. "Convergence" being used interchangeably - for example: In Machine Learning applications, I have heard the term "Convergence" ...
user avatar
0 votes
0 answers
15 views

Bayesion estimation with Gamma prior

Suppose $X_1,..,X_n$ are i.i.d from $N(\mu_0, \sigma^2)$, where $\mu_0$ is known. Let $\phi = \frac{1}{\sigma^2}$ denote the precision parameter. Suppose we know $\phi \sim Gamma(\frac{m}{2},\frac{m \...
user avatar
  • 151
0 votes
1 answer
40 views

What does this symbol mean in bayesian estimator example?

Is this symbol equivalent to 'multiplication', or equivalent to 'equal'?
user avatar
  • 151
0 votes
1 answer
20 views

Radius of convergence estimate

Often in math we estimate quantities when we can't find the exact values. For power series, it seems that we (almost) always find the exact value of the radius of convergence. I am interested in power ...
user avatar
0 votes
2 answers
44 views

Is the sum of residuals times a regressor equal to zero?

Suppose I am approximating the following system using ordinary least squares regression $$y = p_0 + p_1 x_1 + p_2 x_2 + ... p_M x_M = \xi P$$ I know that a property of the least squares estimator is ...
user avatar
  • 11
4 votes
1 answer
85 views

Finding a closed formula for the sum $\sum_{i=0}^{n-1}\frac{1}{1-(\frac{i}{n})^2}$

Consider the sum: $F(n)=\sum_{i=0}^{n-1}\frac{1}{1-(\frac{i}{n})^2}$. So far I found these upper and lower bounds using the well known sum-integral inequality: $F(n)\geq1+\int_{1}^{n-1}\frac{1}{1-\...
user avatar
0 votes
0 answers
23 views

Find M-estimator for Cauchy distribution

Having a M-estimator $\phi^*$ for $\psi_\phi(x) = \arctan(x - \phi)$, I need to find variance of $\phi^*$ when sample $X_{[n]}$ taken from Cauchy distribution. So I need to solve the equation $\mathbb{...
user avatar
  • 460
0 votes
1 answer
41 views

Biases of estimators of the exponential random variable

Let $X_1,X_2,...,X_n$ be independent random variables, each distributed as $Exp(\lambda)$. I am given 2 estimators of $\lambda$: $$\hat{\lambda_1}=\frac{1}{n}\sum_{i=1}^{n}\frac{1}{X_i}$$ $$\hat{\...
user avatar
0 votes
1 answer
46 views

If x is y percent, what is the total (100%)?

If x is y percent, what is the total (100%)? I will explain my question with an example scenario for better context of problem Scenarios: Some person in 22 minutes, drove for 67 kilometers, then how ...
user avatar
  • 101
1 vote
1 answer
27 views

Influencing Probabilities with Prior Information

I have always had the following question about Bayesian Probabilities. Suppose you observe the weather for 90 days. You observe that: 65 days it was Sunny 25 days it Rained This means that there is ...
user avatar
  • 1,692
0 votes
0 answers
18 views

Estimating the right singular vector corresponding to an all positive left singular vector.

Suppose that $A$ is a $k\times n$ real matrix (and for intuition suppose that $k$ is very large) that has SVD $U\Sigma V^\dagger$. Suppose further that we know that there is exactly one column of $U$ ...
user avatar
  • 4,110
1 vote
0 answers
53 views

Finding Monte Carlo estimate $\hat{K}$ using Monte Carlo integration

Let $$f\left(x\right)=K\left[sin^2\left(6x\right)+3cos^2\left(x\right)sin^2\left(4x\right)+1\right]e^{-\frac{x^2}{2}},\:-\infty <x<\infty $$ be the probability density function of a random ...
user avatar
0 votes
0 answers
52 views

Estimating Pi by Throwing Bread

I remember hearing about a story in which an Italian King (hundreds of years ago) drew a circle and randomly threw bread behind his shoulder, and calculated the percent of bread that landed inside the ...
user avatar
  • 1,692
0 votes
1 answer
18 views

Ways to combine multiple probabalistic inputs to improve overall probability of result

Hi Folks I am trying find how to approach this problem: I have four input data points - from four data sources of "lets say price of a stock" - each data point has 90 percent chance of being ...
user avatar
-1 votes
2 answers
79 views

I have a question with "Intoduction to analysis, fourth edition" by Wade, theorem 6.40.

I have a question with the theorem 6.40 ii) from the book "Introduction to Analysis, fourth edition", which is written by Wade. Theorem 6.40 I think if $k>N$, $|a_{N + 1}| \leq |a_ N| \...
user avatar
0 votes
0 answers
5 views

Interpolating noisy measurements taking into account derivative bounds

For context, I have a target moving in space which I am able to locate using vision sensors. I end up obtaining $M$ noisy samples of the trajectory $[x(t), y(t)]$ for some time instants $\{t_k\}$. ...
user avatar
2 votes
0 answers
40 views

Consistently estimate the covariance matrix with weakly correlated observations

Suppose there are T k-dimensional observations following the generating process: $Y_t = \mu + \epsilon_t$, where $\mu$ is the mean and $\epsilon$ is a weak stationary error with zero mean and time-...
user avatar
1 vote
2 answers
103 views

How to solve simultaneous equations of the form $\frac{a}{x}+\frac{b}{y}=1$ really fast?

Context: Taking the major and minor axes of an ellipse as the $x$ and $y$ axes respectively find the equation of the ellipse passing through the points $(1,\sqrt{6})$ and $(3,0)$. Taking the major ...
user avatar
0 votes
0 answers
15 views

K-NN regression

I am new to data mining algorithms, and am currently covering the basics of K-Nearest Neighbors for regressions rather than classification. I get that in the 'simple' version of K-NN, the value of a ...
user avatar
  • 13
1 vote
0 answers
39 views

How to estimate the value of this absolute sum?

(*) $\quad$ $\hspace{1mm} |\sum_{k=0}^{N}(\prod_{i=0}^{k-1}(\frac{n-i}{n}) - 1)\frac{z^{k}}{k!}| \hspace{1mm}$ Consider (*) for some bounded $N < n$. I want to confirm that (*) is bounded by some $\...
user avatar
0 votes
1 answer
67 views

consistency of maximum likelihood estimator

For population with n size and following density function $$f(y, a)= (1/6a^4)y^3e^{-y/a}$$ For that, I have found the maximum likelihood estimator of a which is $\hat{a}= \bar{y}/4$ I have also shon ...
user avatar
  • 6,153
0 votes
0 answers
38 views

Is $\frac{n-1}{n \bar{X}}$ a UMVUE?

Consider $X_1,X_2,...X_n$ iid random sample from exponential distribution with pdf: $$f(x,\theta)=\theta e^{-\theta x},\: x>0$$ I found that, $$E\left[\frac{n-1}{n\bar{X}}\right]=\theta$$ So $$\hat{...
user avatar
4 votes
1 answer
117 views

I'm getting 10-15 tonnes of topsoil delivered, what size tarpaulin do I need?

I want to order 10-15 tonnes of top soil. It'll be delivered by lorry and "tipped" onto the ground. This will make roughly pyramid shape, or perhaps a truncated square pyramid. a cone shape, ...
user avatar
  • 123

1
2 3 4 5
33