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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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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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]=...
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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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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 ...
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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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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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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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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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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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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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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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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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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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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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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;\...
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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 ...
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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. ...
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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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1answer
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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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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,...
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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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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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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: ...
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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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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 ...
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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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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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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 ...
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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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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 ...
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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 $$\hat \sigma^2=\frac{\sum_ix_i^2 + n(\bar x)^2}{n-1}=\frac{\sum_i(x_i-\bar x)^2}{n-1}$$ ...
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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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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, ...
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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: "...
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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 ...
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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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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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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 ...
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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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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 ...
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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: ...
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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 (...
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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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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 ...

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