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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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Can we estimate the mean of an asymmetric distribution in an unbiased and robust manner?

Suppose I have i.i.d. samples $X_1, \cdots, X_n$ from some unknown distribution $F$ and I wish to estimate the mean $\mu=\mu(F)$ of that distribution and I insist that the estimator be unbiased - i.e.,...
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UMVUE for $g(p) = \mathbb{E}_p[X^2]$, where X follows a geometric distribution

I have a random variable X with pmf $$p_\lambda(x) = (1-p)^{x-1}p, \ \ x = 1,2,3,\ldots, \ \ p \in (0,1)$$ and I am trying to find a UMVUE for $$g(p) = \mathbb{E}_p[X^2]$$. Here is my attempt so ...
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Minimum-variance unbiased estimator to estimate quantiles when the errors are normal distributed

What is the minimum-variance unbiased estimator to estimate quantiles when the errors are normal distributed? median When we wish to estimate the median, $\mu$, of a normal distributed variable then ...
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Heteroskedasticity-robust White estimator

The heteroskedasticity-robust White estimator is defined as: \begin{align} V_{\hat{\beta}} = (X'X)^{-1}\left(\sum_{i=1}^n x_i x_i' \hat{e}_i^2 \right)(X'X)^{-1} \end{align} with X the matrix of ...
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Where does linear regression fit into the bias-variance tradeoff?

In ISL, the concept of the bias-variance tradeoff is presented with the rule of thumb that simple models will have high bias and that complex models will have high variance. Given this idea, I would ...
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Does minimizing expected squared loss (MSE) result in an unbiased estimator?

I have heard that the estimator with the lowest expected squared loss (mean squared error) is not always unbiased, but I have also heard that the constant that minimizes the expected squared loss vs. ...
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Unbiased estimator of $\lambda(1 - e^\lambda)$ when $x_1,\ldots,x_n$ are i.i.d Poisson($\lambda$)

Suppose $x_1, x_2, x_3,\ldots, x_n$ are i.i.d. random variables with a common Poisson$(\lambda)$ distribution. I was trying to find an unbiased estimator for $\lambda(1 - e^\lambda)$, but I could not ...
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Is there a mistake in the expression of this variance?

I'm busy reading through an econometrics textbook (page 147), and I don't understand the step $$\mathrm {Var}\left(n^{\frac 12}\left(\hat\beta - \beta\right)\right) = \boldsymbol{A^{-1}}\sigma^2\...
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Estimator for $\frac{1}{\lambda}$ using $\min_i X_i$ when $X_i$ are i.i.d $\mathsf{Exp}(\lambda)$

Let $X_1,\ldots,X_n$ be i.i.d. $\mathsf{Exp}(\lambda)$ random variables, where $\lambda$ is unknown. Consider $f_{\min}(x) = \min_{i}(X_i)=$ $ n \lambda $ Exp$(n\lambda x)$. I am told that $\hat \...
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Visualization of unbiasedness of high dimensional paramter estimates

Assume a statistical model $f_{\theta}(X)$ that allows to estimate a parameter vector $\hat{\theta}\in \mathbb{R}^p$ from data $X$ and assume that $p$ is high dimensional (you may assume something ...
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1answer
31 views

Bootstrap based bias correction

Assume we have a probablistic model $f_{\theta}(x)$ and try to estimate the parameter $\theta$ based on data $x$ with some procedure that yields a biased estimator $$E[\hat{\theta}]=\theta + \eta,$$ ...
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1answer
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Exogeneity Assumption within or across observations

Suppose we have a linear regression model: $$ y_{i}=x_{i}\beta+\epsilon_{i} $$ Where $i$ is an index for individuals $i=1...N.$ Now, the requirement for unbiased estimation of $\beta$ via OLS ...
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What does it mean that the variance is equal to covariance?

I am reading a paper that says that for forecast unbiasdness it is necessary to asusume that Cov(x,x̂)=Var(x̂), where x̂ is the estimate and x is the true value. How should I interpret this assumption,...
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57 views

Variance of unbiased estimator for the shape parameter of Pareto distribution

I'm interested in getting the error bounds of the unbiased estimator of the shape parameter ($\alpha$) using maximum likelihood method of Pareto distribution. The unbiased estimator is known to be ...
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Sample selection in difference-in-differences

I have a dependent variables with a significant amount of zeros (and the share of zeros is different between the control and treatment groups, and changes between the pre- and post-treatment periods). ...
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Why is unbiased estimation from a sample only possible for certain properties?

I was thinking about finding some monotonic measure of entropy of a sample from a continuous distribution (2D, btw), but couldn't think of any such without making assumptions. Why is it that one can ...
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Variance of an Unbiased Estimator for $\sigma^2$

Let $X_1, X_2,...X_n\sim N(0,\sigma^2)$ independently. Define $$Q=\frac{1}{2(n-1)}\sum_{i=1}^{n-1}(X_{i+1}-X_i)^2$$ I already proved that this Q is an unbiased estimator of $\sigma^2$. Now I'm stuck ...
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1answer
41 views

Distribution function of a biased estimator

$f(y) = ay^{a-1}/θ^a, 0<y<θ$ $ \hat{\Theta} = max(Y_1, Y_2, . . . , Y_n).$ How do I find the $E[\hat{\Theta}]$ ? I'm trying to show that it's a biased estimator, then I'm going to find ...
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Ratio of Unbiased Estimators

If there is a linear regression model as follows: $$y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + u$$ and we want to estimate the ratio of the slope coefficients: $$\theta = \frac{\beta_1}{\...
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1answer
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Error of Bias-Corrected Kurtosis Estimators

Background I've found two different bias-corrected estimators for the kurtosis. The first one is used in various software packages, such as MATLAB, and is called bias-corrected in the respective ...
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Consistency vs. Asymptotic Efficiency of estimator

I'm thinking about the relationship between an asymptotically efficient estimator and a consistent estimator, and I'd like to make sure that my thinking is correct. An estimator is asymptotically ...
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Asymptotically unbiased estimator vs consistent estimator [duplicate]

I'm wondering if there is a difference between an asymptotically unbiased estimator and a consistent estimator. For asymptotically unbiased estimators, the expected value of the estimator converges ...
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Determining variance of UMVUE

Let $X_1,...,X_n$ be iid with pdf given by $f(x;\theta)=\frac{log\theta}{\theta^{x-1}}I(x>1)$. My task is to determine if the $\mu=E[X]=1+\frac{1}{log\theta}$ can be estimated efficiently, i.e. if ...
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Are unbiased efficient estimators stochastically dominant over other (median) unbiased estimators?

General description Does an efficient estimator (which has sample variance equal to the Cramér–Rao bound) maximize the probability for being close to the true parameter $\theta$? Say we compare the ...
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Is there any case in which unbiased but larger MSE estimator preferred to biased and smaller MSE one?

Let saying we are interested in a population mean $\mu$ and we have two estimators $\hat{\mu}_{n}^b$ and $\hat{\mu}_{n}^{u}$ defined on $n$ samples such that $\hat{\mu}_{n}^b$ : biased (i.e, $\mathbb{...
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1answer
199 views

Unbiased estimator of binomial PMF

Is there an unbiased estimator of PMF of a random variable $Y=\sum_{i=1}^{n} X_n $ where $X_i$ are independent Bernoulli trials with probability $p$, that is, the estimator of: \begin{equation}\tag{1} ...
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Unbiased Estimator for $\log\left[\int p(x\mid z)p(z) \, dz\right]$

The naive Monte Carlo estimator is an unbiased estimator for $\int p(x\mid z)p(z) \, dz$, is there a convenient unbiased estimator for $\log \left[\int p(x\mid z)p(z)\,dz \right]$
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Unbiased Estimate of a squared difference between sample of random matrices

In the description below, IER stands for "Inhomogenous Erdos Renyi" random matrix, which is basically saying that the entries in the matrix are Bernoulli distributed and iid. I didn't get the part ...
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The UMVUE of ratio of parameters for two uniform distributions,

Let $X_1,\ldots,X_m$ be i.i.d. having the uniform distribution $U(0, \theta_x)$ and $Y_1,\ldots, Y_n$ be i.i.d. having the uniform distribution $U(0, \theta_y)$. Suppose that $X_i$’s and $Y_j$’s are ...
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Bias-corrected Property for Jackknife's Pseudo Values

I come across the following formula from a note, saying that we could think of jackknife as a bunch of independent pseudo values with the following form: The notes further comment that the sample ...
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Understanding Rao-Blackwell [duplicate]

From Casella and Berger: Let $W$ be an unbiased estimator of $\tau(\theta)$ and let $T$ be a sufficient statistics for $\theta$. Define $\phi(T) = E[W|T]$. Then $E_{\theta}[ \phi(T)] = \tau(\theta)$ ...
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How can I not show the initialization of the estimation in the Extended Kalman Filter?

I'm making estimates through the Extended Kalman Filter and I have a problem related to the vertical axis of my figure, it's too big, so I can not see population dynamics. However, I wish it did not ...
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1answer
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Estimating standard deviation when observing means of various block sizes

I have a sequence $X_i$ of iid random variables (you may assume gaussian distribution if you like) but I only observe the mean value of disjoint blocks of various sizes of $X_i$. E.g. $M_1 = \frac14 \...
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1answer
37 views

Estimator of ratio of central moments

In the context of Control Variates one has to estimate, for example, the following ratios of central moments: $$\frac{\mu_{1,1}}{\mu_{0,2}} \quad \text{and} \quad \frac{\mu_{1,1}^2}{\mu_{0,2}}$$ ...
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How to find an unbiased estimator of $\mathsf{Uniform}(-\theta/2,\theta/2)$

How to find an unbiased estimator of $\mathsf{Uniform}(-\theta/2,\theta/2)$. Is it a function of the order statistics?
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51 views

Unbiased estimator of p in geometric distribution

The answer to this question given by my professor was statistic T(x)= 1when X=0 and T(x) = 0 otherwise. Can I consider E(x) = (1-p)/p and then cross multiply and take 1/(1+x) as an unbiased estimator ...
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1answer
42 views

Finding UMVUE of a family of continuous random variables

Let $X$ has probability density function $f_X(x;\theta) = a(\theta)b(x)I_{(0, \theta)}(x)$ (where $a(\theta)$ and $b(x)$ are nonnegative). I have to find the UMVUE of $\theta$ or show that one doesn't ...
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Unbiasedness and consistency

Assume the simple regression model satisfying all Gauss-Markov assumptions. Somebody suggests the estimator Why may someone consider such an estimator? Why will this estimator be consistent? Why ...
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1answer
154 views

Why is it important that estimators are unbiased and consistent?

I am clear on the definition of unbiasedness and consistency. But why are these the criteria we use to judge whether an estimator is a good one? There are other criteria, of course, like the variance ...
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1answer
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Why is the birthday problem a biased estimator?

Can anyone tell me why the calculated birthday match probability is a slightly biased estimator when simulated? Taking a group of 30 people, theory tells us that the probability of at least 2 having ...
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328 views

Consistent unbiased estimator for the location parameter of Cauchy (theta, 1)

Given Cauchy distribution with pdf $p(x) = \frac{1}{\pi ((x - \theta)^2 + 1)}$ how can I find a consistent unbiased estimator for $\theta$? My reasoning so far Tried MLE, but there seems to be no ...
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1answer
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Is $\mathbb{E}(\exp(-\hat{\mu})) = \exp(-\mu)$, when $\mathbb{E}\hat{\mu}=\mu$?

Say I have a biased estimator for $\xi$, say $\hat{\xi}$. But what I know is $\mathbb{E}(\hat{\mu}) = \mu$(unbiased), and $\xi = \exp(-\mu)$. So I wish to do the following bypass. Is $\exp(-\hat{...
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Perform a wald's test to check the statistical significance of biasedness of MLE

I have done a simulation to see that MLEs are asymptotically unbiased. I want to know whether I can perform a wald test here to check the statistical significance of the difference between the mean of ...
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1answer
78 views

Expected value of squared least squares estimator

I am trying to prove $E(\hat{\beta} '\hat{\beta}) = \beta'\beta+\sigma^2 *\sum_{k=1}^K\lambda_k^{-1}$ where $\lambda_k$ denotes the eigenvalues of the matrix $(X'X)$ with dimensions $K\times K$. $\...
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Estimator for repeated sampling and fitting

Say I have a Normal distribution $\mathcal{N_1}(\mu_1,\sigma_1)$. Now I will sample $N$ samples $X_1$ from this distribution, and use estimators for $\hat\mu_2$ and $\hat\sigma_2$ to fit a new ...
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366 views

Proof Verification: $\tilde{\beta_1}$ is an unbiased estimator of $\beta_1$ obtained by assuming intercept is zero

Consider the standard simple regression model $y= \beta_o + \beta_1 x +u$ under the Gauss-Markov Assumptions SLR.1 through SLR.5. Let $\tilde{\beta_1}$ be the estimator for $\beta_1$ obtained by ...
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1answer
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Solving an equation to find two unknown weights given an unbiased estimate

Apologies if this is a simple question; I am reviewing out of Seber and Lee's book on regression and I am pretty rusty in my linear algebra Suppose that $X_1, ..., X_n$ have a common mean $\mu$ and ...
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Unbiased estimator for $L^2$ probability distance norm

I am trying to find an unbiased estimator for (what looks like) the $L^2$ Wasserstein distance between two probability measures. I'm pretty sure that by bickel-lehmann, there is an unbiased estimator. ...
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Unbiased estimate of sign of mean

Consider the set $\mathcal{P}$ of probability distributions that have a finite first moment and define the function $\operatorname{sgn} :\mathcal{P} \to \mathbb{R}$ as $$ \operatorname{sgn}(\mu) = \...
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Is an unbiased estimator based off multiple complete sufficient statistics also UMVUE?

If $T(X)$ is a complete sufficient statistic such that $ET(X) = \sigma^2$, then $T(X)$ is the UMVUE estimator of $\sigma^2$. My question is, suppose $\tau (T(X),W(X))$ is an unbiased estimator of $\...