Questions tagged [estimators]

A rule for calculating an estimate of a given quantity based on observed data [Wikipedia].

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minimum distance estimation

I need to set up a minimum distance estimator for the uniform distribution $U[0,\theta]$ and take the $\mathcal{X}^{2}$ statistic as distance https://en.wikipedia.org/wiki/Minimum-distance_estimation $...
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estimating transformation of linear function non-parametrically

Suppose I have a regression model as follows $$Y_i = f(X_i + \varepsilon_i), $$ where $\varepsilon_i$ is standard normally distributed. I want to estimate function $f(\cdot)$ non parmetrically. I ...
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Cramér–Rao Lower Bound and UMVUE for $\frac1{\theta}$

Problem: Find the UMVUE of $\frac1\theta$ for a random sample from the population distribution with density $$f(x;\theta)=\theta x^{\theta-1}$$ and show that its variance reaches the Cramér–Rao lower ...
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Difference between rho functions and psi functions for robust estimators? [closed]

I am reading about robust estimators and am confused by the rho functions and psi functions. What's the difference between both of these. https://www.statisticalconsultants.co.nz/blog/m-estimators....
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MLE of median is a consistent estimator - Problem 6.1.4 (Hogg, McKean, Craig)

Suppose $X_1, . . .,X_n$ are iid with pdf $f(x; θ) = 2x/θ^2$, $0 < x ≤ θ$, zero elsewhere. Find: The MLE $\hat\theta$ for $\theta$, The constant $c$ so that $E[c\hat\theta] = \theta$, The MLE of ...
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Covariance and correlation between OLS estimators for slope and intercept [duplicate]

I am a scientist and new to this group so apologies in advance if this has been covered before. In my work, I use experiments to calculate the slope and intercept of experimental data from a simple ...
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Finding a consistent sequence of estimators such that $\lim_{n\to\infty} E_\theta[(W_n-\theta)^2]\ne 0$

There are many ways to check if a sequence of estimators is consistent. By definition, a sequence of estimators $W_n = W_n(X_1,X_2,\ldots,X_n)$ is a consistent sequence of estimators of the parameter ...
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How to calculate the Standard Error of a Wald Estimator?

I am currently working with a dataset about workers and their wages, There are 329,509 observations of workers, and we can observe: lwage : the log weekly wage of a worker educ : the worker’s years ...
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Different regularity conditions for finite population CLT

I am having trouble understanding the different regularity conditions for different versions of the finite population central limit theorem. I would greatly appreciate any help or insight anyone has. ...
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Rewriting integral/summation as weighting estimator

I recently read a biostats paper which featured the following identity: $$ \sum_{y, l, m} y P(y, l, m \mid c, a) \frac{P(l \mid a, c) P\left(m \mid a^{*}, c\right)}{P(l, m \mid c, a)}=E\left(Y \frac{P\...
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understanding the proof that the average of sum of i.i.d cauchy is not a consistent estimator of location parameter

Consider $X_1, X_2, ... ,X_n \sim_{i.i.d} Cauchy(\theta), \bar{X} = \frac{1}{n}\sum_{i=1}^n{X_i}$ To prove that it is inconsistant, consider the characteristic function of $X_i$ and $\bar{X}$, which ...
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Compute a Monte Carlo estimate. Which of the variances (of $\hat{\theta}$ and $\hat{\theta}^{*}$) is smaller, and why?

Compute a Monte Carlo estimate $\hat{\theta}$ of $$ \theta = \int_{0}^{0.5} e^{-x} dx $$ by sampling from Uniform$(0, 0.5)$, and estimate the variance of $\hat{\theta}$. Find another Monte Carlo ...
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Non-adversarial robustness

One measure of an estimator's robustness is the breakdown point, which tells us how many adversarial observations are necessary to make the estimator useless. However, is there a notion of non-...
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GLS estimator - derivation

I'm stuck with the following question: Given the model $$Y_t=\alpha+\beta X_t+u_t\,,$$ where the standard assumptions hold but $Eu_t^2=\sigma^2 X_t^2$, derive the GLS estimator. Basically, all Gauss ...
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bias–variance decomposition related to median?

In evaluating or designing an estimator $\hat\theta$ of a population parameter $\theta$, the most common approach is to look at its bias, $\operatorname{E} \hat\theta - \theta$, its variance, $\...
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Are these statements about the maximum likelihood estimator and efficiency correct?

I'm trying to understand efficiency and its relation with maximum likelihood estimators so I need someone to confirm or correct these statements I deduced : 1/ If the maximum likelihood estimator ...
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Why isn't this estimator unbiased?

Suppose we have a IID sample $X_1, X_2, \cdots, X_n$ with each $X_i$ distributed as $\mathcal{N}(\mu, \sigma^2)$. Now suppose we construct (a rather peculiar) estimator for the mean $\mu$: we only ...
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Did I correctly apply the factorisation theorem in this example?

Suppose that we have a density $f(x,\theta)=c(\theta)\psi(x)\unicode{x1D7D9}(x \in]\theta,\theta+1[)$ and the random variable $\mathbf{X}=(X_1,\ldots,X_n)$ are independently identically distributed ...
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Efficiency and consistency of an estimator

I recently encountered the following question, which I am struggling with for two days: Let $X_1, \ldots, X_n$ be an i.i.d. sample from a $\text{Geometric}(1/(1 + \theta))$ distribution with pmf $$f(...
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Bias and variance of an estimator of a model mean

I have a binary classification model and I need to use its output to estimate the means of groups of observations. I have two questions: A. Can I compute the the bias and variance of the estimator of ...
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Different formulas for $\hat{\sigma}^2$

My book introduces $\hat{\sigma}^2 = \frac{1}{n} \sum_{i=1}^n \hat{e}_i^2$, which makes sense to me since to my understanding $\hat{\sigma}^2$ is the error variance estimator. However, I have also ...
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Are robustness and generalizability the same thing?

An optimal parameter $\theta^*$ is robust if it does not change much when calculated for different samples of data from a population. $\theta^*$ has good generalizability if its predictive power ...
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What's the maximum likelihood estimation of $\theta$ in this density?

Suppose we have a n-sample $X=(X_1,..,X_n)$ with a distribution $f(x,\theta)=exp(\theta - x)\unicode{x1D7D9}_{x \geq \theta}(x)$. Find the maximum likelihood estimator $T$ of $\theta$ and prove that $...
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Identifiable but has no consistent estimator

Let $P_\theta$ denote the distribution of the random variable $X$. The distribution depends on the parameter $\theta$ that lies in some parameter space $\Theta$. Consider a function $f(\theta)$ of $\...
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Induced Likelihood Function for Max Likelihood Estimators

A small doubt on the notation for the induced likelihood function and invariance properties of maximum likelihood estimators - It says in a textbook I am reading that: Let $X = {X1, X2,..., Xn}$ be a ...
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How to estimate state-transition matrix in Kalman Filter from irregular time lags

Let's assume that we have a simple Kalman model for a multivariate $X_t$ where the observation model is given by $$Y_t = K X_t + \epsilon$$ where $K$ is known and $\epsilon \sim N( 0, S )$, and $S$ is ...
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unbiased estimation of the variance of $p$ (proportion) of a random sample without replacement

Given a random sample without replacement of size $n$ from population of size $N$ and $p$ is the estimator of the proportion $P$. How could one show that: \begin{equation*} \frac{N-n}{N(N-1)}pq \end{...
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Estimator of a censored exponential

I am trying to self-study the MIT OpenCourseware course on Statistics, here: https://ocw.mit.edu/courses/mathematics/18-650-statistics-for-applications-fall-2016/syllabus/ On problem set 4, question 3 ...
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Robustness of ELBO

The ELBO $\mathcal{L}(\phi)$ is used to quantify how good an approximate posterior $q_\phi(z|x)$ is for a dataset $x$ and an (unknown) true posterior $p_\theta(z|x)$. However this is all under the ...
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Robustness of MAP estimate

In Bayesian inference, we have a dataset $x$ and assumed to come from a known parameterized distribution with unknown parameters $\theta$. We then seek to maximize the posterior $P(\theta|x)$ in order ...
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mmse for product of complex gaussian random variables

If $Y= (h_{1}h_{2})X + N$ where $h_{1}, h_{2} \cong \mathcal{C}\mathcal{N}$(0 , $\sigma_{1}^{2}$) $ N \cong \mathcal{C}\mathcal{N}$(0 , $\sigma_{2}^{2})$ Y and X are known beforehand $\mathcal{C}\...
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Implied Loss Function of Estimator

Let $X \in \mathbb{R_+}$ be a random variable and $\alpha \in [0,1)$ a parameter. Suppose $\mathbb{E}f(X) = g(\alpha)$ for known invertible functions $f$ and $g$. Given realization $X = x$, I ...
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MLE estimator for the second parameter of binominal distribution

Let us have a sample $X_1, X_2, ... , X_n$ with $B_{p,m}$ distribution. How to make an estimator for $m$ using MLE ? For simplicity, let us $n=1$ I am a little bit stuck, because I have a derivative ...
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How to derive the bias of an entropy estimate

I am trying to understand the bias-variance trade-off in the context of non-parametric entropy estimation. Specifically using a histogram approach to estimate the entropy of a sample we have: $$\hat{H}...
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How to estimate the sample variance of the estimator of the parameter $P(x≤0)$ where $x \sim N(\mu,\sigma)$?

This question relates to How to estimate $P(x\le0)$ from $n$ samples of $x$? One way to make this estimate is to use estimates $\hat\mu$ and $\hat\sigma$ and compute from those $$ \hat p = \Phi \left(...
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How to estimate $P(x\le0)$ from $n$ samples of $x$?

Suppose, we have $n$ samples $x_i$ of a random variable: $$x \sim \mathcal N(\mu,\sigma^2) $$ Based on the samples, we want to estimate the probability that $x$ is negative: $$P(x\le0)$$ Intuitively, ...
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Is the mundlak estimator equal to the within estimator?

Mundlak has proposed to estimate the following correlated random effects model: $$ y_{i t}=\boldsymbol{x}_{i t}^{\prime} \boldsymbol{\beta}+\overline{\boldsymbol{x}}_{i}^{\prime} \gamma+\omega_{i}+u_{...
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Calculating bias with both overestimated and underestimated values

Can somebody please explain why (using the residual plot) the image below is a biased estimator? Some values are overestimated, some values are underestimated. I understand that if I take any thin ...
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How to weight samples in a subsample of a stratified sample?

Suppose I'm first interested in computing the percentage of people in a certain town with brown eyes. However, due to some constraints I end up with the following stratified sampling set up: $$N = ...
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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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Linear regression with non-normal residuals [duplicate]

As a follow up to this topic OLS assumption normallity of error term really needed? I had a question. When we do OLS we have usually the normal model: $$ y_i=ax_i+\epsilon_i [1]$$ $$ \epsilon_i \sim N(...
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SEM with mixed and ordinal continues data, with missing values, with nonnormality

I have a dataset with missing values (over 10% missing), mixed ordinal and continuous variables, and non-normality. I am trying to fit a Structural equation model (SEM) using this dataset. Could I use ...
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Proving sufficiency of a statistic using the expectation

I am blocked trying to solve the following question. I would appreciate if someone could give me a hint. Let $X_1,\ldots,X_n$ be $n$ independent random variables following a continuous uniform ...
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Checking the consistency of the statistic

Let $X_1, X_2,...,X_n$ be n random samples from $N(\theta, \theta^2)$. We have a statistic $T = \sum Xi^2$. I need to prove the consistency of $\frac{T}{2n}$ for estimating $\theta^2$. I know that ...
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Estimating Sample CDF from 1st Order Statistics

I have a process where I can only measure the 1st order statistics, but would like to know something about the underlying sample CDF. I understand that I can calculate the CDF of my 1st order ...
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Ratio type estimator for estimating population mean when there is non-response and response errors

Please can help how the final equation is obtained? I tried to get it but I still get some extra factors such as (1-e1^2) in the denominator in the right-hand side.
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Estimate a sum using proportional sampling

I have some set of items. Each item has a weight and I can sample the items from the population with probabilities proportional to their weights. I know the size of the population. I want to estimate ...
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Optimization function of the Hodges-Lehmann location estimator

The median minimizes the sum of absolute differences while the mean minimizes the sum of square distances. What is the function which is minimized by the Hodges-Lehmann location estimator?
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General Strategy to show an estimator is admissible?

I am getting into decision theory and I was wondering if there was a general way to check if a an estimator is admissible. (PS This question might have already been asked, sorry if that is the case I ...
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Online Algorithm Implementation for the Median [duplicate]

Context My question is related to the binmedian algorithm which is suggested in this post and its implementation originally in C and its adaptation in python. My issue with these algorithms is that ...

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