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# 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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### What is the difference between a consistent estimator and an unbiased estimator?

What is the difference between a consistent estimator and an unbiased estimator? The precise technical definitions of these terms are fairly complicated, and it's difficult to get an intuitive feel ...
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### How exactly did statisticians agree to using (n-1) as the unbiased estimator for population variance without simulation?

The formula for computing variance has $(n-1)$ in the denominator: $s^2 = \frac{\sum_{i=1}^N (x_i - \bar{x})^2}{n-1}$ I've always wondered why. However, reading and watching a few good videos about "...
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### How does one explain what an unbiased estimator is to a layperson?

Suppose $\hat{\theta}$ is an unbiased estimator for $\theta$. Then of course, $\mathbb{E}[\hat{\theta} \mid \theta] = \theta$. How does one explain this to a layperson? In the past, what I have said ...
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### Unbiased estimator of exponential of measure of a set?

Suppose we have a (measurable and suitably well-behaved) set $S\subseteq B\subset\mathbb R^n$, where $B$ is compact. Moreover, suppose we can draw samples from the uniform distribution over $B$ wrt ...
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### Different usage of the term "Bias" in stats/machine learning

I think I've seen about 4 different usages of the word "bias" in stats/ML, and all these usages seem to be non-related. I just wanted to get clarification that the usages are indeed non-...
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### Model for population density estimation

A database of (population, area, shape) can be used to map population density by assigning a constant value of population/area to each shape (which is a polygon such as a Census block, tract, county, ...
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### Does a median-unbiased estimator minimize mean absolute deviance?

This is a follow-up but also a different question of my previous one. I read on Wikipedia that "A median-unbiased estimator minimizes the risk with respect to the absolute-deviation loss function, as ...
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### Biased estimator for regression achieving better results than unbiased one in Error In Variables Model

I am working on some syntatic data for Error In Variable model for some research. Currently I have a single independent variable, and I am assuming I know the variance for the true value of the ...
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### Correct equation for weighted unbiased sample covariance

I'm looking for the correct equation to compute the weighted unbiased sample covariance. Internet sources are quite rare on this theme and they all use different equations. The most likely equation I'...
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### Unbiased estimator for the smaller of two random variables

Suppose $X \sim \mathcal{N}(\mu_x, \sigma^2_x)$ and $Y \sim \mathcal{N}(\mu_y, \sigma^2_y)$ I am interested in $z = \min(\mu_x, \mu_y)$. Is there an unbiased estimator for $z$? The simple 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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### Does the biased estimator always have less variance than unbiased one?

Suppose I am estimating one of the parameter. Now if we plot the biased estimator of that and unbiased estimator of that can we say for sure that biased one has less variance than unbiased one always. ...
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### Why is bias equal to zero for OLS estimator with respect to linear regression?

I understand the concept of bias-variance tradeoff. Bias based on my understanding, represents the error because of using a simple classifer(eg: linear) to capture a complex non-linear decision ...
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### Using MSE instead of log-loss in logistic regression

Suppose we replace the loss function of the logistic regression (which is normally log-likelihood) with the MSE. That is, still have log odds ratio be a linear function of the parameters, but minimize ...
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### Should the standard deviation be corrected in a Student's T test?

Using the Student's T test, T-Critical is calculated via: $t = \frac{\bar{X} - \mu_{0}}{s / \sqrt{n}}$ Looking at Wikipedia article on the unbiased Estimation of the standard deviation, there ...
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### Bias / variance tradeoff math

I understand the matter in the underfitting / overfitting terms but I still struggle to grasp the exact math behind it. I've checked several sources (here, here, here, here and here) but I still don't ...
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### Degrees of Freedom In Sample Variance

Recall the formula for sample variance $$s_{n - 1}^2 = \dfrac{1}{n -1} \sum_{i = 1}^n (\bar{x} - x_i)^2,$$ where $\bar{x}$ is the sample mean. There are many proofs for why $s_{n - 1}^2$ is an ...