# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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