# 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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### How can I compute the expected value and variance of the 4th power of the sample median?

Given the following parameter estimate, how do I find $E[\hat{a}_{MED}]$ and $Var[\hat{a}_{MED}]$? \begin{equation} \label{eq:Estimator_a_Med} \hat{a}_{MED} = - \left( n_0 \right)^4 \cdot \log(0.5)...
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### Deriving Properties of Estimators (Bias and Variance)

I have the following probability distribution function given by: \begin{equation} \label{eq:function} f(x) = \frac{4a}{x^5} \exp \left[ {- \frac{a}{x^4}} \right] \quad \quad 0 \leq x \leq \infty \...
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### Find a function of $\theta$ so that there exists an unbiased estimator and the variance coincides with Cramér-Rao lower bound

Let $X_1,\dots, X_n$ be a random sample from the geometric distribution $P(X=x)=\theta(1-\theta)^x$ for $x=0,1,2,\dots$ where $0<\theta<1$. Find a function of $\theta$, say $\tau=h(\theta)$ so ...
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### Why is i.i.d. an OLS assumption?

Assume the following linear relationship: $Y_i = \beta_0 + \beta_1 X_i + u_i$, where $Y_i$ is the dependent variable, $X_i$ a single independent variable and $u_i$ the error term. According to Stock &...
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### Why isn't an unbiased Standard Error of the Mean calculated with $n-1$? [duplicate]

The unbaised variance of a population from a single sample ($n\ll N$), $s^2=\sum_i(x_i-\bar{x})/(n_x-1)$. ($n_x$ being the sample size.) However, the standard error of the mean: $SE=s/\sqrt{n_m}$, not ...
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### Do I need statistical test with nested-cv?

I'm working with a small data set and with 4 algorithms. The optimization process showed to be Very important as long as it improves a lot their performances. So, I'm using 10x5 nested-cv to estimate ...
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### Trouble understanding expected value (does it assume infinite sample size?) and bias vs consistent

This might be a dumb question.. but I was wondering if someone can help me out with the concept expectation. This question started from trying to understand bias vs consistent. So when we roll a dice, ...
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### Can variance be calculated with a sample size of n=1?

I am currently analysing inter-rater reliability/agreement data for a single case with multiple raters. For that I am using Gwet's $AC_2$ (as described here) using the irrCAC package in ...
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### Estimating the Absolute Difference in Mean Between Two Normal RVs

I've seen more general questions of this nature, but none that discuss specifically the problem setup that I'm interested in, which I believe is substantially simpler. Suppose that we observe two ...
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### Unbiased estimator from a random sample (Poisson) [duplicate]

"Suppose that $X_1,X_2, \cdots, X_n$ is a random sample from a Poisson distribution with parameter $\lambda$; propose an unbiased estimator for $\theta = exp(-\lambda)$. " Hello everyone, I'...
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### Will taking mean of sample quantiles be an unbiased estimate the population quantile?

Say I have $N$ samples of 100 numbers all drawn IID from the same distribution $\mathcal{D}$. For each sample, I take the 95th quantile to get $N$ sample quantiles $\hat{q}_n$. Will taking the average ...
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### Is $\sum(x^2 - \bar{x}^2)/(n-1)$ an unbiased estimator of variance?

We know that sample variance $\sum(x- \bar{x})^2/(n-1) = \sum(x^2- 2x\bar{x}+\bar{x}^2)/(n-1)$ is an unbiasedd estimator of variance Is $\sum(x^2 - \bar{x}^2)/(n-1)$ also an unbiased estimator of ...
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### How do we know the true value of a parameter, in order to check estimator properties?

For example, we say that an estimator is unbiased if the expected value of the estimator is the true value of the parameter we're trying to estimate. However, if we already know the true value of the ...
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### Offset variable in a Poisson regression measured with error

It is known that measurement error in predictors lead to a bias toward zero, and also bias other predictors’ estimates: The effect of covariate measurement error on coefficients in regression with ...
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### Are these moments estimators asymptotic unbiased?

In this paper, authors consider method of moments of fitting Gumbel distribution: We know that maximal likelihood estimators are asymptotic unbiased. But are these moments estimators asymptotic ...
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### Unbiased estimators for uniform distribution

I have a given question that I just cannot figure out. From what I can understand, both estimators would be unbiased (since e_1 is the sample mean and e_2 is an unbiased estimator for uniform ...
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### Intuition behind a 0% central/equal-tailed confidence interval

At a vague intuitional level, I feel like, if we generated a 0%, two-tailed, truly central (or "equal-tailed") confidence interval (CI), the number it would wrap around would be a median-...
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### "Unbiased" (at least ballpark) Estimate of Condition Number of True Covariance Matrix being Estimated & other Symmetric Matrices (e.g.,Hessian)

Are there any known ways of getting an unbiased estimate of the condition number of the true covariance matrix being estimated, or at least correct within a small number of orders of magnitude? For ...
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### Estimating the ratio between the square of the sample mean and the second moment of the sample mean

Suppose $y_1,\dots,y_n$ are i.i.d. and have the $N(\mu, \sigma^2)$ distribution. I'm interested in finding an unbiased estimator for $$\rho := \frac{\mu^2}{\mu^2 + \sigma^2/n}.$$ A naive approach ...
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### If the bias of an estimator is expressed as a difference, what do you call the ratio of the estimator and true value?

If $Bias(\hat{\beta}) = (\beta - \mathbb{E}[\hat{\beta}])$, is there a term to describe the quantity $\frac{\mathbb{E}[\hat{\beta}]}{\beta}$?
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### When is it better to have an unbiased estimator instead of one that has a smaller risk?

I just learned that for $X_1, \ldots X_n \sim N(\mu, \sigma^2)$ i.i.d, the sample variance $\frac{1}{n-1} \sum_{i=1}^n (X_i - \bar X)^2$ is unbiased, and it is in fact UMVUE. However, it is not ...
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### Showing the unbiased estimator of variance for GLS estimator

I have the following regression $$y = X\beta +u$$ where $y$ and $u$ are $(n\times 1)$ and $X$ is a fixed $(n \times k)$ matrix with full column rank and $\beta$ is an unknown $(k\times 1)$ vector of ...
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### Find UMVUE of Ratio of two parametric functions

Let T be UMVUE of $g(\theta)$ and S be UMVUE of $h(\theta)$. Is there any way to find UMVUE of ratio of $g(\theta)$ and $h(\theta)$ i.e , $g(\theta)/h(\theta)$?
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### Using consistency to prove unbiasedness of linear regression

I am not sure if this question has already been asked, but basically in an interview I was told that using consistency, we have that $$E(\frac{Cov(X,Y)}{Var(X)}) = \frac{E(Cov(X,Y))}{E(Var(X))}$$ ...
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### Is asymptotic unbiasedness different from unbiasedness in practice?

Given some estimator T for a parameter θ, by definition T is unbiased if its bias B(T) is 0. It is asymptotically unbiased if B(T) is not 0, but some value that tends to 0 as n goes to infinity. My ...
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### Unbiased estimator of $2\left(\frac{E[x^2]}{E[x]^2}-1\right)^{-1}$

For a random variable $x$, how would I go about creating an unbiased estimator of the following quantity? $$R=2\left(\frac{E[x^2]}{E[x]^2}-1\right)^{-1}$$ For instance, when $x$ comes from from chi-...
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### Confusion regarding proof that the variance estimator is unbiased for finite population

Going through Sharon L. Lohr's Sampling design book (2nd Edition), I have no issues with the content all the way until it goes into the proof in chapter 2 on SRSWOR that $E[s^2] = S^2$, where $S^2$ is ...
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### Linear regression on mean + standard deviation

I have observations made at different times from a normally distributed real random variable whose mean and standard deviation both vary linearly with time. How can I estimate these two linear ...
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### why not using sample variance (instead of MSE) to estimate the error variance in linear regression?

Assuming the true equation for Y is linear as below: $$Y_i =\beta_1X_i +\beta_0 + \epsilon_i$$ Assuming X is fixed, then the variance of each Y is: $$var(Y_i )=var(\epsilon_i)=\sigma^2$$ In order to ...
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