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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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### 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 ...
1 vote
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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 ...
1 vote
201 views

### 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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1 vote
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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 ...
347 views

### 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 ...
215 views

### Unbiased Estimator of Largest Mean of Two Normal Distributions

Given samples from two normal distributions: $X_i \stackrel{iid}{\sim} \mathcal{N}(\mu_X, \sigma_X^2)$ for $i = 1,...,n$ $Y_i \stackrel{iid}{\sim} \mathcal{N}(\mu_Y, \sigma_Y^2)$ for $i = 1,...,n$ How ...
1 vote
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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 ...
531 views

### 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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### The expected value of $\frac{1}{\sqrt{1-r}}$ where $r$ is Pearson correlation

I am looking to unbias the sample statistic $\frac{1}{\sqrt{1-r}}$ where $r$ is a Pearson correlation. The population is assumued binormal with equal variance $\sigma$ and with true correlation $\rho$....
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### Unbiased estimator of variation of median in spatial bins using bootstrap method

Say I have a satellite that's flying through the atmosphere, over multiple orbits, sampling its density at different altitudes, at say 1 measurement per second (specific numbers are irrelevant). The ...
45 views

### Finding good estimators for a function of bernoulli parameter [duplicate]

Given $m$ i.i.d. Bernoulli( $\theta$ ) r.v.s $X_{1}, X_{2}, \ldots, X_{m},$ l'm interested in finding estimator of $(1-\theta)^{1 / k},$ when $k$ is a positive integer. I am considering the following ...
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### Finding UMVUE for a function of a Bernoulli parameter

Given $m$ i.i.d. Bernoulli( $\theta$ ) r.v.s $X_{1}, X_{2}, \ldots, X_{m},$ I'm interested in finding the UMVUE of $(1-\theta)^{1/k}$, when $k$ is a positive integer. . I know $\sum X_{i}$ is a ...
232 views

### Unbiased estimator of standard deviation

I'm reading "Properties of range-based volatility estimators" where the authors talk about using the range of a distribution ($h$ - $l$) to estimate its volatility. Specifically, they say, Daily ...
328 views

### Are all estimators biased? Is the unbiasedness only a theoretical or approximation case?

The definition of unbiased estimator says that it's expected value has no difference comparing to a true value. So can we say that all estimators are biased (even slightly)? I thought that only in ...
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### Showing that a estimator is biased?

I am solving a exercise who asks me to show that one estimator is biased. Given the function \begin{equation} f(x|\theta) = \left( (1-\sigma) + \dfrac{\sigma}{2\sqrt{x}} \right)I_{[0,1]}(x), \sigma \...
968 views

### Why is temporal difference learning biased in reinforcement learning?

When I learn reinforcement learning from David Silver's online video, I saw "the objective of TD learning, $r_t + \gamma V(s_{t+1})$ is a biased target for learning value function. " I know the ...
1 vote
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### How is the sample mean an unbiased estimator of the population mean via deeplearningbook.org?

So I know that the sample mean is a unbiased estimator of the population mean. Just wondering how the author gets from 5.33 to 5.34 in the below. How do you get from $\mathbb{E}[\mu_m]$ to just $\mu$...
I saw one question in which the sample mean was estimated as follows (I don't know why they divided by $n-1$ instead of $n$ here for estimation), $$\widehat{\mu} = \frac{\sum_{i=1}^n x_i}{n-1}$$ ...