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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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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Recall and precision point estimates (statistical inference)
Let's say I have a population of 1M objects. I want to make a binary classifier to use on that data, but I can't manually classify all the 1M to create training data because that would take too long, ...
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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 ...
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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 ...
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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(\...
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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 $\...
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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 ...
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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 ...
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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]
...
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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 ...
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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}{...
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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}...
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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 $...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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]=...
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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 ...
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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 ...
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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&...
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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. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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?$$
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Computing the Bayes estimator under weighted squared error loss - interchanging derivatives and integrals
I am revisiting some self-study assignment questions in elementary theoretical statistics that I previously had difficulty with. I would appreciate some clarity on a few points in the following ...
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Standard error of estimate of $\lambda^2$
In a problem, given $n$ observations from $Poisson(\lambda)$ , I have to get an unbiased estimator of $\lambda ^2 $ and the corresponding standard error.
I used the efficiency test to get the unbiased ...
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What Cramer-Rao bound should I use?
I have been researching about the Cramer-Rao bound and I have found two inequalities:
$$\text{Var}\left(\hat{\theta}\right)\geq\frac{1}{\text{E}\left[\left[\frac{\partial}{\partial\theta}\ln f(X;\...
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Estimation with MCMC [closed]
I would like to ask some high-view questions about MCMC. I do not have a specific example, I just want to get a general intuitive idea.
Suppose I have a data set $X$ and a rather complex model with ...
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Why Do Distributional Forecasts Need to Produce Normally-Distributed Forecasts to be Ensembled/Combined?
I am forecasting a collection of different types of items, using many different forecasting techniques. Some of the techniques I use take the input data as is to produce a distributional forecast. ...
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Winsorized mean - trimming furthest points instead of both endpoints
I'm wondering if the Winsorized mean can be improved by trimming the 5% farthest points from the mean instead of trimming 5% on each endpoint. Concretely:
Consider the Winsorized mean, where we ...
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Estimating Means of a Bivariate Normal Distribution where some parameters are known
I am trying to figure out how to estimate means of a bivariate normal distribution from a sample when some of the parameters are already known.
let
$$
\boldsymbol{x} = \begin{bmatrix}
x\\
y\\
\end{...
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Unbiased estimator and getting estimate from estimator
I got a unbiased estimator but I don't know how to interpret it and adjust it to get estimate.
The original problem is to find out the unbiased estimator for $\lambda$ in Zero-truncated Poisson ...
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Trimmed, weighted mean
The trimmed mean (or truncated mean) is a robust version of the mean, designed to be robust to outliers. I am wondering what is the right trimmed version of a weighted average.
If I have a sample ...
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Mean-square convergence of maximum likelihood estimators: Examples?
From what I've gleaned from the literature, Cràmer, in his 1947 monograph Methods of Mathematical Statistics, proved convergence in probability of an MLE under certain regularity conditions. ...
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Point estimate and confidence interval for the difference in $x_1$ between two groups for which a particular $y$ is achieved
I have two variables (continuous $x_1$, control/treatment $x_2$) that I want to use to predict a probability. Domain knowledge suggests that the relationship is roughly linear in the log-odds, so I am ...
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Expression for the Likelihood Function in Point Estimation
I came across this question in my statistics textbook, but I'm struggling to come up with an expression for the likelihood function. Here is the question:
Assume that there are three possible traits ...
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Determining the minimum number of tosses, for heads to be twice more likely than tails in the next toss
I would like some help with the following statistical problem.
We have a coin with probability $\theta$ for heads, with prior for $\theta$ being a Beta(a,a) distribution (a is a known parameter).
...
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Is there a term for an estimator's probability of estimating an impossible estimand value?
This is similar to Mean Squared Error and Mean Absolute Error but in this case the loss function assigns estimates to $0$ when they are a possible estimand and $1$ when they are impossible.
As a ...
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What's the advantage of a point estimate over an interval estimate?
A point estimate is :
A single numerical value that is used to estimate the corresponding population parameter.
Whereas an interval estimate is :
An estimate that consists of two numerical values ...
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