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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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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Comparison of samples of optimal objective function values of a deterministic optimisation problem solved by a stochastic optimisation solver

Motivating Problem Assume that I must pack tins of soup into a box, where the box has a fixed volume. I can choose the shape of the soup tin, and the volume of the soup tin, and the packing ...
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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 should one re-formulate hypothesis after hypothesis testing or whether or not to discard data after significance testing or archive it?

@Nalzook summarized what he thinks Ronald Fisher did. Ask a question. Propose a null hypothesis based on the question. Do some experiments, and collect some data. Assuming the null is true, calculate ...
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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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Friendly reference for point estimation that covers the following

I took a mathematical statistics class in college which covered the following: Cramér–Rao bound Lehmann–Scheffé theorem Rao–Blackwell theorem I found these theorems very beautiful and want to ...
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Is it appropriate to use linear regression when the response has a confidence interval?

I am looking to predict a response, $\hat{y}$, and have some information about how the response is generated. I know, for example, that it's a score that is computed from an array of weighted factors. ...
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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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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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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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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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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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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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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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