# 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 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 ...
1 vote
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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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### Does an endogenous variable bias the coefficient of the exogenous one?

We have the following model: $$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \epsilon.$$ We know that: \begin{align*} \operatorname{Cov}(x_1, \epsilon) &\neq 0 \\ \operatorname{Cov}(x_2, \epsilon) &...
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### Is the least mean square estimator for jointly gaussian variables necessarily affine?

In his book on adaptive filtering, Sayed mentions a subclass of affine estimators in which not only the predictions y are linearly dependent on the observations x, but x and y are jointly Gaussian. ...
1 vote
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### Least square estimate expected value and variance of linear model

I am practice some exercises. Here it goes. "Assume we fit the simple model \begin{equation} \hskip 5cmy=X_1\beta_1+\epsilon \hskip 5cm (1) \end{equation} however the true model is \begin{...
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### Proof that $g(p)$ unbiasedly estimable only if it is a polynomial (Binomial Distribution)

In Lehmann-Casella (Theory of Point Estimation) they state without proof that if $T \sim Bin(n,p)$, then $g(p)$ is estimable only if it is a polynomial in $p$ of degree $\leq n$. How does one go about ...
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### Calculating confidence interval for binomial distribution [duplicate]

Suppose we have a sample $X_1, X_2, \ldots, X_n \stackrel{\text{iid}}{\sim} Binomial(\theta)$, where $n$ is known to be large. I would like to calculate the 95% confidence interval for $\theta$, and I ...
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### Will removing a regressor from a model reduce the variance of the remaining regressor

Let's say our full model is a mean centered: $$y= B_0 + B_1(x_1-\bar x_1) + B_2(x_2-\bar x_2) + e$$ I know $B_0$ works out to be equal to $\bar{y}$, and so $SS_{Reg}(B_0) = 0$ My question is if we ...
1 vote
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### Existence of unbiased estimator for any $f(X)$? [closed]

Suppose you are handed a function of a random variable $f(X)$, how would you construct/rule out the existence of an unbiased estimator for it? I've read through Halmos (1946) but the characterization ...
1 vote
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### Quantifying the bias of a quantile estimator based on order statistics, and its relation to asymptotic unbiasedness

From what I understand, the quantile estimator based on order statistics is asymptotically unbiased (and follows a Normal distribution). I have been looking for a quantification of the non-asymptotic ...
1 vote
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### How do we select model for causal inference?

I am reading Rubin's Causal Inference Sec 7.5 in context of completely randomized experiment. It says performing linear regression will produce asymptotically unbiased estimate of causal effect, ...
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### Symmetric distribution with defined mean: is the median always unbiased for the mean?

Let $X_1,\cdots,X_n\overset{iid}{\sim} F_X(x)$ be a random sample from a symmetric distribution with a defined mean. If need be, assume that $n$ is odd and that $F_X(x)$ is continuous. Is it always ...
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### Unbiased least squares estimate for GM Theorem

In order to prove the Gauss-Markov Theorem, we first have to show that the OLS estimate $\hat{\theta}$ is an unbiased estimator. From what Im reading on Internet and some textbooks, these are the main ...