# 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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### 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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### 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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### 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 ...
### 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 ...