Questions tagged [point-estimation]
Point estimation is the application of an estimator to the data in order to learn about a certain population parameter.
204
questions
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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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0answers
19 views
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 ...
1
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1answer
21 views
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]=...
0
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2answers
73 views
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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0answers
12 views
estimating a population-average model with known mean and standard deviation
I have a model with some differential equations describing the effect of a drug. There are 100 rat samples, we only know the mean value and its standard deviation for measured drug response. Now I ...
0
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1answer
86 views
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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1answer
49 views
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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0answers
17 views
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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2answers
30 views
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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3answers
584 views
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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2answers
221 views
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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0answers
14 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 ...
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0answers
28 views
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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0answers
10 views
Hypothesis testing on Simple Regression with sub populations
I am attempting to answer the following question:
In a simple linear regression set up the population regression line is $\mu_y^i = \beta_o + \beta_1x_i$ for $i=1,2,3$. In other words, we have 3 sub-...
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1answer
181 views
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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0answers
21 views
Point estimation of parameters
Let $\theta$ be a parameter with values in $\Theta$ that should be estimated by some given data $X$.
The corresponding estimate is denoted as $\hat\theta = \hat\theta(X)$.
Several times I read that ...
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0answers
22 views
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 ...
0
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1answer
32 views
What Cramer-Rao bound should I use?
I have been researching about the Cramer-Rao bound and I have found two inequalities:
$$\text{Var}\left(\hat{\theta}\right)\geq\frac{1}{\text{E}\left[\left[\frac{\partial}{\partial\theta}\ln f(X;\...
0
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1answer
33 views
Estimation with MCMC [closed]
I would like to ask some high-view questions about MCMC. I do not have a specific example, I just want to get a general intuitive idea.
Suppose I have a data set $X$ and a rather complex model with ...
1
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1answer
36 views
Why Do Distributional Forecasts Need to Produce Normally-Distributed Forecasts to be Ensembled/Combined?
I am forecasting a collection of different types of items, using many different forecasting techniques. Some of the techniques I use take the input data as is to produce a distributional forecast. ...
2
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0answers
54 views
Winsorized mean - trimming furthest points instead of both endpoints
I'm wondering if the Winsorized mean can be improved by trimming the 5% farthest points from the mean instead of trimming 5% on each endpoint. Concretely:
Consider the Winsorized mean, where we ...
2
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1answer
46 views
Estimating Means of a Bivariate Normal Distribution where some parameters are known
I am trying to figure out how to estimate means of a bivariate normal distribution from a sample when some of the parameters are already known.
let
$$
\boldsymbol{x} = \begin{bmatrix}
x\\
y\\
\end{...
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0answers
42 views
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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0answers
31 views
Consider N independent RVs having identical binomial distribution with parameters θ and n=3. Estimate θ by method of maximum likelihood
Consider N independent random variables having identical binomial distribution with the parameters θ and n=3. If n0 of them take on the value of 0, n1 take on the value of 1, n2 take on the value of 2,...
4
votes
1answer
229 views
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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0answers
19 views
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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0answers
29 views
Maximum Likelihood Estimation - parameter estimation
I must find the relation between a group of categorical features and a Target (label) variable T.
A proxy of the dataframe I am using is the following:
...
1
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1answer
32 views
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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0answers
27 views
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 ...
2
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1answer
32 views
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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0answers
10 views
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 ...
0
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1answer
2k views
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 ...
2
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0answers
63 views
Consistent estimator of $p^2$
$(X_1, X_2,...,X_n)$ is a random sample of size $n$ from $Bernoulli(p)$ distribution. $S_n=\sum_{i=1}^nX_i$. I have to check whether $\frac{S_n(S_n-1)}{n(n-1)}$ is a consistent estimator for $p^2$.
$...
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0answers
14 views
Is the population parameter more likely to occur closer to the sample statistic? [duplicate]
When we use the sample statistic to find a confidence interval, is there any reason to using the sample statistic after that, when we can instead refer to the confidence interval?
For example, let's ...
1
vote
1answer
377 views
Most Efficient Estimator and Uniformly minimum variance unbiased estimator
I am studying Estimation theory from "Introduction to theory of statistics" by "Mood and Graybill".
After completing I thought I understood UMVUE (uniformly minimum variance ...
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0answers
66 views
Taking Expectation Over Inverse Sum of Indicator Functions?
I'm working with a zero inflated Poisson distribution that has the following pmf:
$$f(y|w,\lambda)=wI[y=0]+(1-w)\frac{e^{-\lambda}\lambda^{y}}{y!}$$
I would like to find the expectation of the ...
2
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1answer
16 views
Variance Estimator Change if we know Population Mean? (Normal dist. example)
For a normal distribution $N(\mu, \sigma^2)$ a commonly used unbiased and consistent estimator of variance is
$$\hat \sigma^2=\frac{\sum_ix_i^2 + n(\bar x)^2}{n-1}=\frac{\sum_i(x_i-\bar x)^2}{n-1}$$
...
3
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1answer
40 views
Outlier detection in point estimates
I have to perform outlier detection on population estimates for certain variables at the city level. For example, I might be estimating median income for a city and I want to know if there are any ...
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0answers
31 views
How to estimate a proportion as a step function of time
There is a black-box mechanical process that, at any time, may be either succeeding or failing. It is known that after an initial disturbance, the process will fail $X\%$ of the time for $N$ seconds, ...
2
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0answers
21 views
Strong consistency in quantum estimation problem
I'm reading the paper: Strong consistency and asymptotic efficiency for adaptive quantum estimation problems by Akio Fujiwara.
In this paper, describes the next adaptive scheme of estimation:
"...
2
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1answer
870 views
It is possible to find point estimate of population mean and population variance when confidence interval of population mean is given?
Let's say that somehow $100(1-\alpha)\%$ confidence interval of population mean $\mu$ is known as $(a,b)$ and the number of samples is $n$. Is it possible to infer point estimates of population mean ...
4
votes
2answers
249 views
Bayesian Analysis: Point Estimates for a Beta Posterior
I think this is a fairly beginner bayesian analysis question.
I have a Beta Posterior with $\alpha = .32$ and $\beta = 1.35$ (estimated using MCMC), that describes a probability.
My question is: ...
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0answers
47 views
Surprising nonlinear variance-based scale est (bias adj) for Laplace Distribution competes with MLE?
Background:
Using the quantile function (inverse cumulative distribution) for the Laplace distribution supplied with uniform random deviates (per the RAND() spreadsheet function), I examined an ...
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0answers
18 views
Better than expected bias corrected estimator for scale parameter of Logistic random deviates based on sample standard deviation?
Background:
Using the quantile function (inverse cumulative distribution) for the Logistic distribution supplied with uniform random deviates (per the RAND() spreadsheet function), I was testing ...
0
votes
2answers
53 views
Parameter estimation for random variables where a control parameter is another r.v
Let $\{X_i\}$ a sequence of independent random variables.
Each $X_i$ has a p.d.f $p(m, \theta)$. Where $\theta$ is a real unknown parameter and $m$ the outcome of another random variable $M$ with p.d....
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0answers
164 views
What is an intuitive of definition of "point identification" (point identified parameter) in econometrics?
I've recently come across the notion of point identification in several econometric papers.
See, e.g., https://scholar.harvard.edu/files/tamer/files/pie.pdf, who mentions point identification ...
2
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0answers
28 views
Derive summary statistic Grouped Data & Frequency Distribution Table
I have the following data from the 2018 American Community Survey for a number of census block groups:
...
1
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0answers
84 views
Point estimate and 95% credible interval
The text of the problem as follows:
The data follows a normal distribution with
$\mu$ and $\sigma^2$ unknown. We wish to perform inference on the mean selling price $\mu$.
And our sample data are (...
38
votes
3answers
9k views
What percentage of a population needs a test in order to estimate prevalence of a disease? Say, COVID-19
A group of us got to discussing what percentage of a population needs to be tested for COVID-19 in order to estimate the true prevalence of the disease. It got complicated, and we ended the night (...
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1answer
139 views
Calculating Confidence Interval for Estimated Parameters of SEIR model
I used a Log-Likelihood Estimation (Poisson) Objective Function to estimate and fit a curve to a data of reported infected cases of COVID-19 using SEIR model in order to estimate its coefficients. How ...
