Questions tagged [unbiased-estimator]

Refers to an estimator of a population parameter that "hits the true value" on average. That is, a function of the observed data $\hat{\theta}$ is an unbiased estimator of a parameter $\theta$ if $E(\hat{\theta}) = \theta$. The simplest example of an unbiased estimator is the sample mean as an estimator of the population mean.

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Is Bayesian estimation useful for causal analyses?

Is Bayesian estimation useful for causal analyses? For analyses like randomized experiments or even observational studies of natural experiments, we want unbiased estimators of the causal effect (...
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Find the UMVUE of $P\{{X_i < Y_i\}}$ for $X_i,..X_n \sim N(\mu, \sigma^2)$ and $Y_i,..Y_n \sim N(\nu, \sigma^2)$

I have to find the UMVUE of $P\{{X_i < Y_i\}}$ for $X_i,..X_n \sim N(\mu, \sigma^2)$ and $Y_i,..Y_n \sim N(v, \sigma^2$). I know that I have to find the joint distribution of $X$ and $Y$ which I ...
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Can a maximum value of a function be an unbiased estimator?

Given a probability density function f(y) = 2y/(Θ^2) for 0≤y≤Θ, and given Y(n) is the maximum value in the sample, the problem is to determine if Y(n) is an unbiased estimator of Θ. My initial ...
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Is the estimator 0.5X1 + 0.5(n-1)^(-1) * the sum from i=2 to n of Xi an unbiased estimator? Is it consistent?

Let {Xi} from i=1 to n be an i.i.d. sample from a distribution f. I suspect this is unbiased, but is it consistent? I'm not sure how to approach it as I think the variance converges to 0, but won't it ...
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Derivation of unbiased MLE for Gaussian variance

I'm currently studying ML basics with the book Introduction to Machine Learning (Ethem Alpaydin) and had a question regarding checking whether the maximum likelihood estimators (MLE's) for a Gaussian ...
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Unbiased estimator for a parameter from a transformed distribution

I am solving an exercise in which I have to show that a certain estimator is unbiased for a given parameter. However, after a couple lines of computation I got stuck in the following scenario: $$ \...
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(From van der Vaart's Asymptotic Statistics, page 161, U-statistic) Why we can always replace the function $h$ with a symmetric one?

I'm reading the following Chapter from van der Vaart's Asymptotic Statistics, Section 12.1 page 161 (see the screenshot below). For the $h$ function that it mentioned, I have two questions regarding ...
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Using variance of sample to calculate unbiased estimate of population variance

I am trying to follow through the survey sampling chapter of Rice's statistics book. Denote the sample values by $X_1, X_2, \ldots, X_n$ and the population values by $x_1, x_2, \ldots, x_N$ (so the ...
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What is the distinction between bias in prediction and parameter estimation?

I am trying to understand the distinction between bias in prediction and parameter estimation. This example in Gelman, Bayesian Data Analysis, 2nd ed. 2004 pp. 255-256 is very confusing to me. Why do ...
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validation error and test error when limited data is available

In machine learning, to get an unbiased estimate of model performance, we split data 80:20 into train and test set. We use the training set for model training and model selection according to cross-...
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“… because sample mean gets different values from sample to sample and it is a random variable with mean $\mu$ and variance $\frac{\sigma^2}{n}$.”

This answer by user "sevenkul" says the following: The sample mean $\overline{X}$ also deviates from $\mu$ with variance $\frac{\sigma^2}{n}$ because sample mean gets different values from ...
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Citation: Sample mean as consistent and unbiased estimator of the expected value

A reviewer asked for a citation that the sample mean is a consistent and unbiased estimator of the expected value and therefore converges towards the expected value. I know I can easily do the ...
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Intuition behind unbiased OLS estimator derivation

I was going through the derivation of unbiased OLS estimator $$E(\hat{\beta_1}) = \beta_1 + (1/SST_x) \sum_{i=1}^n d_i E(u_i) = \beta_1 + (1/SST_x) \sum_{i=1}^n d_i\cdot 0 = \beta_1$$ My doubt is if $...
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Unbiased estimatior for $\bar{x} $ from a Random Sample with unequal selection probability

I have the following population: Where the left column is the age of our individuals and the right column is their weight (in kg). The exercise tells us that we use Random Sampling with no ...
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Negatively correlated estimators for the AR-1 process

I have the following question. Assume we have a stochastic process \begin{equation} y_t = \gamma + \phi y_{t-1} + \epsilon_t, \ \epsilon_t \sim \mathcal{N}(0, \sigma^2), \end{equation} where $|\phi| &...
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mixed model variance-covariance matrix| parameter estimation

I am fairly new to LMM's and I am trying to undestand how the parameter estimation happens; According to this: Beta is obtained with equation 13.28. Beta is supposed to be the parameters for the ...
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Minimizing mean squared error with one scalar variable

I have an estimator $g$ parameterized by a scalar variable $\gamma\in[0,1]$; thus, given a $\gamma$ value and a dataset $D$ of i.i.d $|D|$ samples, $g(D;\gamma)$ is an estimate of the parameter $\...
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proof of Cramer-Rao lower bound

I am trying to understand the proof for this theorem from the book Casella and Berger (2nd ed.) page 336. $W(X)$ if any estimator for samples $X_1,\ldots,X_n$ on distribution $f(X|\theta)$. I notice ...
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Evaluation of Limit involved in the proof of Asymptotic Unbiasedness

We know that $S^{2}$ is an unbiased estimator of $\sigma^{2}$ and $S$ is a biased estimator of $\sigma$. But if $n\rightarrow\infty$, then $S$ is an asymptotically unbiased estimator of $\sigma$. I ...
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Variance estimator using mixture of scaled and unscaled data

Given two datasets: $X_1, \dots, X_n \sim N(1, \sigma^2)$ and $X_{n+1}, \dots, X_N \sim N(1, 2\sigma^2)$ My proposed estimator for $\sigma^2$ is simply a scaled combination of both classical ...
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For regression: Are clustered standard errors(say specified correctly) only consistent, or both unbiased and consistent estimators?

Basically are clustering standard errors only an asymptotic argument or does it possess finite sample properties as well?
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Minimum Variance Unbiased Estimator of Poisson

Let $X1,X2,.....,Xn$ be iid Poisson random variables with unknown parameter $p>0$. Find the minimum variance unbiased estimator of $e^(-2p)$. I could find two estimators, one by the method of the ...
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Unbiased estimator for normal dis standard deviation

I don't get how to solve this: $\{x_i ,.., x_n\}$ are outcomes of independent measurements of an unknown constant with a normal error. For which $k$ sustain that $\hat\theta=k\Sigma^n_i|x_i-\bar x|$ ...
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What is the difference between bias in a beta coefficient estimate and bias as a property of an estimator?

I was thinking about bias in the context of simulation studies - defined as the average of the difference between the estimated beta parameter estimate and the true parameter estimate across all ...
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Is a convolutional neural network unbiased? Is it a regularized multilayer perceptron?

"Is a convolutional neural network biased?" This came up in an interview I had a few years ago, and I’ve recently thought of it. I think it’s a misguided question. Imagine this related ...
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Point estimator for product of independent RVs

Let $X$ and $Y$ be two independent random variables. Given an (iid) random sample of size $n$ of $X$ and a random sample of size $n$ of $Y$, what is a good way to estimate the mean of their product, $...
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Calculating consistent estimators

Let $X_1, X_2,\dots$ be $iid$ random variables with density $f(x|p), 0<p<1$ being the unknown parameter. Suppose that there exists an unbiased estimator T of $p$ based on sample size 1, i.e. $E(...
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Finding Best Linear Unbiased Estimator

I have the doubt that if Gauss Markov theorem is applicable here since the Variance is not constant in the model. Without Gauss Markov Theorem, how can we obtain BLUE?
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What is the fundamental difference between biased and unbiased estimator in the context of PU learning?

I cant get the intution behind calling an estimator to be "unbiased" in machine learning, as zero bias is mostly impossible in real world datasets, especially in the context of PU (positive ...
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Unbiased Estimator of AR(1) Models?

What are the options for unbiased estimators of AR(1) (or AR(p)) models? Bias reduction techniques may also be included (jack knife would be one). I found one paper called "Bias correction using ...
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How to show that the variance estimator of a gaussian is biased? [duplicate]

The sample variance of $m$ samples from a gaussian is $$ \hat{\sigma}^2_m=\frac{1}{m}\sum_{i=1}^m(x^{(i)}-\hat{\mu}_m)^2$$ How do i show that the sample variance $\hat{\sigma}_m^2$ is biased ? I.e $$...
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Based on the record X1 ,…, Xn what is the unbiased estimation of 1/p [duplicate]

If we investigate $n$ patients for SARS. The indicator of sequence of the trails is $X_i$ ($X_i=1$ is for success and $0$ is not success). And the sequence indicator is available for all n independent ...
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In a weighted least squares regression, can we use the weight as a control variable?

I have found Weighted Least Squares with Endogenous Weights but the answers primarily tackle the question of when $w_i$ correlates with $\epsilon_i$. I would like to ask if we use $w_i$ as a control ...
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Proving Least-Squares Estimator is Unbiased [duplicate]

Im working my way through the text Mathematical Statistics with Applications 7th edition (Wackerly et al) and I am slightly confused by how they go about proving that the least-squares estimates are ...
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Show that the two estimators are unbiased for $\theta$ [closed]

$X_1$ and $X_2$, one accurate than the other, are subject to the standard deviations, $\sigma$ and 1.25$\sigma$ respectively. $X_1$ occurred 6 independent times, giving a mean of $\bar{x}_1$ while $...
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Prove bias/unbias-edness of mean/median estimators for lognormal

Looking at a problem where X is lognormally distributed from normal distribution Y, which asks me to prove that: 1) $e^{\bar{y}}$ is a biased estimator for the median of X 2) $e^{\bar{y} - \sigma^2 /...
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Uniform distribution, estimates, MVUEs and Cramer Rao Lower Bound

As a revision exercise, I'm going through all of the distributions and deriving estimators. I've gotten to the $Uniform$. I've worked out the MLE and MOM estimators. The next step is to consider ...
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Doubt on derivation of OLS estimators as unbiased estimators of Optimal Linear Predictors

I'm studying from C. Shalizi's lecture notes https://www.stat.cmu.edu/~cshalizi/ADAfaEPoV/ . In the third chapter he introduces the optimal linear estimator of a random variable $Y$ conditioned to ...
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Minimizing Mean Square Error

Suppose we have a random sample $\textbf{X}=(X_1,...,X_n)$ from a shifted exponential distribution with common density $f(x|\theta)=\left\{\begin{matrix} e^{-(x-\theta)} & x\geq \theta\\ 0 & ...
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Why is the linear aspect of the estimator in the Gauss-Markov estimator a big deal?

The Gauss-Markov theorem gives conditions for which the usual OLS estimator $\hat{\beta}=(X^TX)^{-1}X^Ty$ is the best (minimum-variance) linear unbiased estimator. Unbiased estimators are nice. Low ...
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MAE regression gives biased regression parameters for symmetric error?

Consider a linear model, $$ y_i = \beta_0 + \beta_1x_{1i} + \beta_2x_{2i} + \epsilon_i. $$ From the Gauss-Markov theorem, I know that, under nice conditions, the $\hat{\beta}_{OLS}=(X^TX)^{-1}X^Ty$ ...
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Unbiased estimator and biased error

I'm having some trouble relating unbiased estimators and bias error. By bias error, I mean the bias error we talk about when analyzing "bias-variance tradeoffs." Is this bias error and an unbiased ...
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Error Propagation for Unbiased Means

I am reading through https://arxiv.org/pdf/1210.3781.pdf, and do not understand its derivation for propagation of errors with respect to means. According to the text, when trying to estimate a ...
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Estimator of $\log \mathbb{E}[X]$

In many fields of statistic we are faced with quantities of type $\log \mathbb{E}[X]$ where $X$ is a generic random variable. However, I never came across any good estimator for this quantity. The ...
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Does minimizing the mean of Varience of an unbiased estimator by selecting the values of ${\bf{r}}$, imply minimizing the mean of CRLB?

I can minimize the mean of variance of an unbiased estimator of a paramater $\theta$ by selecting the values of a set of parameters, ${\bf{r}}$. So i can minimeze ${\rm{E[Va}}{{\rm{r}}_{\hat \theta }}...
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Combining importance sampling with optimization - does this yield an unbiased estimate?

I'm wondering if it is OK to combine importance sampling with optimization to choose the parameters for the substitute distribution. I have a non-negative random variable $X$ on $\mathbb{R}^d$ with ...
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Unbiased Estimator of Largest Mean of Two Normal Distributions

Given samples from two normal distributions: $X_i \stackrel{iid}{\sim} \mathcal{N}(\mu_X, \sigma_X)$ for $i = 1,...,n$ $Y_i \stackrel{iid}{\sim} \mathcal{N}(\mu_Y, \sigma_Y)$ for $i = 1,...,n$ How ...
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How can I find the BUE of $\theta$ in the simple linear relationship $Y_i=\theta x_i^2+\epsilon_i$?

Let $Y_1,...,Y_n$ be described by the relationship $Y_i=\theta x_i^2+\epsilon_i$, where $x_1,...,x_n$ are fixed constants and $\epsilon_1,...,\epsilon_n$ are iid $N(0,\sigma^2)$. How can I find the ...
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How do I find the UMVUE of $\sqrt{\alpha}$ here?

new user here self-studying some mathematical statistics. I came across this problem and am stuck. Problem: Suppose that for $i = 1, ... , n$, the positive random variables $X_i$ are independent and ...
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Proof of consistency of OLS sample estimator

I am having a hard time understanding equation B1-3 below. Why does the maximum probability limit converge to variance of X? From what I understand, Var(X) = E[X^2] - E[X]^2. E[X]^2 seems to be 0 here ...

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