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Questions tagged [point-estimation]

Point estimation is the application of an estimator to the data in order to learn about a certain population parameter.

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When are Bayes estimators injective as a function of sufficient statistics?

I know that Bayes estimators can be written only as a function of sufficient statistics. When are those functions injectives? That is, when can I say that, given a bayes estimator $\delta (\cdot)$ and ...
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Uncertainties when fitting an image

I know how to fit a straight line to a set of 2d points with uncertainties on both coordinates, in order to obtain estimators, goodness-of-fit, and uncertainties - see for instance Press & ...
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What is the difference between unbiasedness, consistency and efficiency of estimators? How are these interrelated among themselves? [duplicate]

!Efficiency(https://stackoverflow.com/20240427_193105.jpg). Given snapshot of the book states that among the class of consistent estimators, in general, more than one consistent estimator of a ...
Parth's user avatar
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Is this Maximum Likelihood Estimate (MLE) scenario possible? [duplicate]

I'm a stats novice, and I'm currently looking into ML estimates for phylogenetic trees. I've had this "philosophical" problem with the MLE method in general ever since I learned about it. ...
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How does reparametrization of the Fisher information matrix change the variance expression for the sufficient statistics?

If I have an exponential family distribution of the form $$p_{\theta}(x) = e^{\theta^T\cdot t(x) - \psi(\theta)},$$ where $\theta$ is a vector of parameters, $t(x)$ is a vector of sufficient ...
absolutelyzeroEQ's user avatar
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FInding a complete and sufficient statistic

I am attempting to learn how to find a complete and sufficient statistic. So, I am working on this problem for class: Let $X_1, \cdot\cdot\cdot,X_n$ be a random sample from the pdf $f(x_i|u)=e^{-(x-\...
Harry Lofi's user avatar
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Cramér-Rao bound when the samples come from two distributions

Is there a version of the Cramér-Rao bound when samples are independent but not identically distributed? More specifically, I am considering a sample set that is divided in two subsets, each subset ...
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Cramér-Rao / Wolfowitz bound with nuisance parameter

Let $F$ be a distribution with two parameters, $\theta$ and $\phi$, whose values are non-random but unknown. Consider a sampling procedure in which $N$ samples $x_1, \ldots x_N$ are obtained from i.i....
Luis Mendo's user avatar
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Mean Squared Error for point estimation

I am attempting to understand Mean Squared Error when evaluating point estimators for particular parameters of interest. The book we are reading for class states the following: The mean squared error (...
Harry Lofi's user avatar
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Finding the MLE for a piecewise function

I am attempting to find the MLE for the two parameters (i.e. $\alpha, \beta$) in the following piecewise function, but I am slightly confused as to how to proceed through this question: $$ P(X_i \leq ...
Harry Lofi's user avatar
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Efficiency of chi-squared denoising

Suppose my measurement $\theta+\epsilon$ is corrupted by IID additive noise $\epsilon$ distributed as chi-squared with (known) $d$ degrees of freedom, what is the efficiency of pooling multiple ...
Yaroslav Bulatov's user avatar
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The point estimate and the large sample size [closed]

My colleagues and I argued about the point estimate's accuracy. They said that "The point estimate will tend to be accurate if the sample size exceeds 30." While I am saying no, "The ...
Dr. Statistics's user avatar
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What is the variance decomposition method?

For $i = 1, \ldots, m$ and $j = 1, \ldots , n$ we have observations $x_{ij}$. We can assume that $$ x_{ij} = y_{i} + z_{ij}, \qquad y_{i} \sim \mathcal{N}(\mu_{y},\sigma_{y}^{2}), \quad z_{ij} \sim \...
math_space's user avatar
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Extract systematic and random variation

I am working with the data from some manufacturing line. The parts are processed in batches of say 20. In order to control process, we do measurements on manufactured parts. We know that the ...
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Is it legit to use a point estimate along with a conformal predictive interval from a quantile regressor?

I have a quantile regression model that gives me prediction intervals (PI), and I also need to have a point estimate for all sorts of reasons (or at least something as close to a point estimate in a ...
cremebrulee's user avatar
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Can I reasonably estimate the population mean and standard deviation from a large sample all taken at a single percentile?

I am currently looking at a dataset of Fair Market Rents which are determined at different percentiles over the years - for example, nationally in 1983 they were all set at the 40th percentile, and in ...
Dylan's user avatar
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Confusion about asymptotic distribution of the MLE and of the MAP

It's well known that the MLE $\hat{\theta}$ maximizes $f(y\mid\theta)$ and under regularity conditions has asymptotic distribution $$N\left(\theta, \frac{I(\theta)}{J^2(\theta)} \right)$$ where $I(\...
ThighCrush's user avatar
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No unbiased estimator of $\min\{\mu_1,\mu_2\}$ [duplicate]

The problem is stated as: Suppose $X, Y$ are independent and $X \sim \mathcal{N}(\mu_1, 1), Y \sim \mathcal{N}(\mu_2, 1)$ with unknown parameters $\mu_1, \mu_2$. Prove that unbiased estimation of $\...
Soon Princeton's user avatar
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How to solve non-identifiability problem in point estimation

I am working with a normal model $X \sim N(0, \sigma^2(\theta))$, where $\sigma^2(\theta) = \frac{1}{e}\cos^2(\theta)+e\sin^2(\theta)$. My goal is to estimate $\theta$ within the range $[0, 2\pi]$. My ...
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Distribution of the point estimator of a sample

We know that point estimator is defined over a sample and is said to be unbiased if its expectation value is same as that of some parametric function $g(\theta)$ where $\theta$ is a parameter for the ...
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Unbiased estimator for binomial random variable

On internet, I was reading about the point estimators. I am attaching the screenshot of the relevant portion. Suppose we have a sample $X_1,\;X_2,\;...,\;X_n$, then the point estimator is a function ...
Iti's user avatar
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Find an estimator of linear regression when errors variance is correlated with one of the K regressors

I need to answer to the following problem: In an heteroschedasticity setting, let $n$ be the index of the n-th statistical unit with $n=1, \dots, N$. Suppose a multiple linear regression setting with ...
user378274's user avatar
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Nonexistence of UMVUE for non-constant function?

I tried to prove the problem: Suppose X $\sim \ U(\theta-1,\theta+1)$, $\theta \in \mathbb{R}$. Then there is no UMVUE for $g(\theta)$ unless $g$ is a constant function. Here is my attempt: Suppose $...
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Do I need a "likelihood Jacobian" for point estimation?

Scenario 1 Suppose that you work in a lumber yard and are given the following measurements of the weights of various pieces of lumber: x = [10, 26, 28, 13, 16, 7] ...
Max's user avatar
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Getting point estimation and confidence interval for gaussian fit

I have a task where I should fit the data sample with curve_fit and get the peak's position and amplitude. I fitted data and got these values and their standard errors. I also need to find point ...
user373868's user avatar
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Why the variance of Maximum Likelihood Estimator(MLE) will be less than Cramer-Rao Lower Bound(CRLB)?

Consider this example. Suppose we have three events to happen with probability $p_1=p_2=\frac{1}{2}\sin ^2\theta ,p_3=\cos ^2\theta $ respectively. And we suppose the true value $\theta _0=\frac{\pi}{...
narip's user avatar
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Derivation of the formula for the asymptotic relative efficiency of two estimators with different estimands

Background In their book, Huber & Ronchetti (pp. 2-3) compare the efficiency of the mean absolute deviation $d_n$ with the standard deviation $s_n$ with the following formula: $$ \operatorname{ARE}...
COOLSerdash's user avatar
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Unbiased estimator for $\mu_1/\mu_2$

Let $X_1,X_2,\ldots,X_n$ and $Y_1,Y_2,\ldots,Y_n$ be independent random samples from $N(\mu_1,1)$ and $N(\mu_2,1)$ populations respectively with $\mu_2\neq0$. I need to find an unbiased estimator for $...
RiXaTorAgu's user avatar
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4 answers
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How would a bayesian estimate a mean from a large sample?

What would a bayesian do if she wanted to do inference for the mean with a large sample but has no idea of the underlying distributions? A frequentist statitician would use the sample mean as a point ...
Manuel's user avatar
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comparison of proportion to a population CI

I am comparing the % of minorities from my organization to a population % of minorities, to see if it is high or low. I have data for my whole organization (not a sample) so I do not show CIs. The “...
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What is known, in principle, about the possibility of approximating the random discrepancy between a statistical estimate and its parameter?

The difference between the value of a statistical estimate and its parameter's value is almost never exactly $0$. For example, $r - \rho$, for a unique sample $r$, is likely to be some non-zero ...
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An estimation method/algorithm for estimating the value of a specific parameter in a convex function

I am looking for an estimation/iteration process to estimate the value of a specific unobserved parameter of a convex function that fits the observed data of the other variables closely. Specifically, ...
Koula's user avatar
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MLE - CDF vs PDF as the likelihood-function?

Would maximum-likelihood estimation: with the cumulative-distribution function as the likelihood-function and the probability-density function as the likelihood-function, yield the same/equal ...
x.projekt's user avatar
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Large sample properties of classical estimator for single scale parameter

This question was first posted on Math Stackexchange and I was told in the comment it would be a good question on Stats Stackexchange, since it comes from the well-established theory of point ...
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Is this point estimate for mean biased?

I was wondering if this point estimate for mean: $\frac{1}{n+1}\sum_{i = 1}^{n}x_i$ is biased? My first thought was that $\frac{1}{n+1}\sum_{i = 1}^{n}x_i \neq \frac{1}{n}\sum_{i = 1}^{n}x_i$, so then ...
Satan Lucifer's user avatar
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Efficiency of two estimators for a sample from a Bernoulli population

Given a Bernoulli population, I have two estimators for a random sample of size $n$: $T_1=\frac{\sum\limits_{i=1}^n X_i + 2X_n}{n+2}$ $T_2=\frac{\sum\limits_{i=1}^{n-2} X_i + 2X_n}{n+2}$ I want to ...
Sinval's user avatar
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Why do we divide by n when solving for the Cramer-Rao Lower Bound here?

"Let $X_1,...,X_n$ be iid Bernoulli(1,$p$), with $p$ unknown. Find the CRLB for the variances of unbiased estimators of $p$." With pdf $p^x(1-p)^{1-x}$, the derivative of the log function is ...
ttc100's user avatar
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2 votes
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Forming a consistent estimator for the area under the regression line

I am trying to solve the following problem: Take the following simple linear regression model, where $x_i \in \mathbb R$: $y_i=\beta_0 + x_i \beta_1 + \epsilon_i$ Given that: $\mathbb E[\epsilon_i]=...
jmars's user avatar
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Estimating $1/a$ for following pdf using method of moments estimation

A random sample of size $n$ is being drawn from a population with pdf as: $$f(x) = \begin{cases} (a + 1)x^a & \text{for }0<x<1, \\ 0 & \text{otherwise.} \end{cases}$$ Can we express the ...
Kcd's user avatar
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Generalized Bayesian estimator (rule) of θ

Question: Let $X_1, · · · , X_n$ be a random sample from $Poisson(θ)$. The prior for θ is $G(α, β)$ Find the Bayesian estimator (rule) of θ under the SEL(squared error loss). Find the generalized ...
ForestGump's user avatar
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Conjugate Prior for Alpha Power Inverse Weibull Distribution

Let $X$ has Alpha Power Inverse Weibull (APIW) distribution with pdf $f(x) = \frac{\log \alpha}{\alpha - 1} \lambda \beta x^{-(\beta+1)} e^{-\lambda x^{-\beta}} \alpha^{e^{-\lambda x^{-\beta}}}, \; x&...
ccmatyn's user avatar
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A hypothesis test that conditional expectation (i.e. regression line) is above some number in a region of the factor space

In my work we want to know whether some variable of interest satisfies some threshold. Maybe imagine that we ask questions like whether a widget has at least a 60% probability of functioning. ...
cgmil's user avatar
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2 votes
2 answers
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How to combine population proportions and confidence intervals?

Say I've looked at three random samples, one each from three populations of students. I found that a few students in each sample didn't turn in a fieldtrip permission slip, leaving me with the ...
Ireland's user avatar
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3 answers
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What happens if I change the range of a flat prior for Bayesian inference?

I am working through an example on doing Bayesian inference on binomial distribution using a flat prior, and trying to understand the impact of choosing a prior. I know that if we work with a flat ...
Ian's user avatar
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Bayesian point estimate of a random sample

I am new to statistics and some concepts are not clear to me. I have a random sample that is distributed as a Binomial with parameters $k=2$ and $\theta$ unknow. Using a Bayesian approach I must give ...
Luis Alexandher's user avatar
1 vote
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66 views

What are the implications of a low coverage in multiple imputation?

When testing multiple imputation algorithms in simulations, the bias of the examined estimates and the 95% coverage rate are often used as a quality metric. I understand that it is generally ...
joacim022's user avatar
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Is it possible to estimate the Hessian as the covariance of primal and cotangent?

Let's say we have a function $$f: \mathbb R^n \to \mathbb R.$$ Can we numerically approximate the Hessian $f''(x)$ as $$\textrm{Var}(a)^{-1} \textrm{Cov}(a, f'(a))$$ where $$E(a) = x?$$
Neil G's user avatar
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3 votes
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Computing the Bayes estimator under weighted squared error loss - interchanging derivatives and integrals

I am revisiting some self-study assignment questions in elementary theoretical statistics that I previously had difficulty with. I would appreciate some clarity on a few points in the following ...
microhaus's user avatar
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Standard error of estimate of $\lambda^2$

In a problem, given $n$ observations from $Poisson(\lambda)$ , I have to get an unbiased estimator of $\lambda ^2 $ and the corresponding standard error. I used the efficiency test to get the unbiased ...
umm_what's user avatar
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What Cramer-Rao bound should I use?

I have been researching about the Cramer-Rao bound and I have found two inequalities: $$\text{Var}\left(\hat{\theta}\right)\geq\frac{1}{\text{E}\left[\left[\frac{\partial}{\partial\theta}\ln f(X;\...
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