# 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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### Unbiased estimator of $1 + \mu^{2}$ from a Normal population

Question: If $x_{1}, x_{2}, x_{3},...x_{n}$ is a random sample from a $Normal$ $population$ $N(\mu,1)$ then what is the unbiased estimator of $1 + \mu^{2}$ ? I began finding the mean and variance ...
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### Finding the M.V.U.E of n Bernoulli trials [duplicate]

Let $r$ be the observed number of successes in $n$ Bernoulli trials with probability $\pi$ of success. Then M.V.U.E (Minimum Variance Unbiased Estimator) of $\pi (1-\pi)$ is ? $n$ Bernoulli trials ...
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### Definition of the bias of an estimator

I'm quite confused about the definition of the bias of an estimator. Suppose we have unknown distribution $P(x, \theta)$, and construct the estimator $\hat{\theta}$ that maps the observed data sample ...
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### Why do we prefer unbiased estimators instead of minimizing MSE?

I was thinking about why, usually, $\hat{\sigma}^2=\hat{p}(1-\hat{p})$ is used to estimate the variance in a Bernoulli population instead of $s^2=\hat{p}(1-\hat{p})\frac{n}{n-1}$. $s^2$ is unbiased, ...
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### MSE Proof for an estimator

I am trying to figure out the following proof. The third line is not clear. We all know that (a+b)^2=a^2+2ab+b^2. The term 2ab should be 0, but I can't figure out why. I have found other proofs here ...
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### Asymptotic bias of LASSO vs. none of SCAD

I am reading a paper which says that LASSO is asymptotically biased while SCAD is not. I take asymptotic (un)biasedness to concern the slope estimators from LASSO and SCAD as the sample size goes to ...
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### Are these statements about the maximum likelihood estimator and efficiency correct?

I'm trying to understand efficiency and its relation with maximum likelihood estimators so I need someone to confirm or correct these statements I deduced : 1/ If the maximum likelihood 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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### Plot for unbiasedness

I was reading a paper and the authors describe a plot which they used to determine whether their estimator was unbiased. This plot is described as follows (verbatim): To assess if the probabilities ...
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### unbiased estimation of the variance of $p$ (proportion) of a random sample without replacement

Given a random sample without replacement of size $n$ from population of size $N$ and $p$ is the estimator of the proportion $P$. How could one show that: \begin{equation*} \frac{N-n}{N(N-1)}pq \end{...
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### Standard error of estimate of $\lambda^2$

In a problem, given $n$ observations from $Poisson(\lambda)$ , I have to get an unbiased estimator of $\lambda ^2$ and the corresponding standard error. I used the efficiency test to get the unbiased ...
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### OLS biasedness in AR(1) model [duplicate]

I am trying to show why the OLS estimator in time series models is not conditionally unbiased when using a zero-mean strong AR(1) model. From what I've read so far, this can be done through a Monte ...
### For iid $X_1, \dots, X_n \sim N(0,\sigma^2)$, get sufficient statistic $T = \sum_{i=1}^nX_i^2$, how to find unbiased estimator of $\sigma^a$
For $X_1, \dots, X_n \sim N(0,\sigma^2)$, we define a sufficient statistic $T = \sum_{i=1}^nX_i^2$. There is a positive number $a$. My question is how to find unbiased estimator of $\sigma^a$ using ...