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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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Unbiased estimator of mean divided by square root of second moment [closed]

Let $X$ be some random variable. Assume that $$\mu = \mathbb{E}X,\,\delta = \sqrt{\mathbb{E}X^2}$$ are well defined and finite (in other words $X$ has first two moments). Now suppose that $X_1,...,X_n$...
Slup's user avatar
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96 views

What estimator and R package can be used for staggered difference-in-difference with (non-panel) cross-sectional data, controls and interactions

I am trying to run a difference-in-difference analysis in R. My data is non-panel, so I am reliant on a TWFE model where I have groups of individuals who are ...
flâneur's user avatar
1 vote
0 answers
44 views

Question on nonlinear least squares

Consider the following equation for $Y>0$: $$ (1) \quad \log(Y)=\log(\gamma)+\log(\alpha+\beta X)+\epsilon. $$ Assume that $E(\epsilon| X)=c\neq 0$. What are the consequences of this assumption on ...
Star's user avatar
  • 849
0 votes
1 answer
75 views

Proving an Estimator of the sample variance to be MVUE

Question: Prove that $\hat{\sigma}_x^2=\displaystyle\frac{1}{N-1}\sum_{i=1}^N(X_i-\overline{X})^2$, with $\overline{X}=\frac{1}{N}\sum_{i=1}^N X_i$ is an unbiased, minimum variance estimator of the ...
Subhasis Biswas's user avatar
1 vote
0 answers
57 views

Degrees of freedom for biased sample autocorrelation function

I want to find the expression for the a biased estimate of the autocorrelation function for a time series $X$, and am doing this from the biased estimated autocovariance function for lag $k$, divided ...
hydrologist's user avatar
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0 answers
30 views

How to compute the bias of the auto-normalized importance sampling estimator

A preceding post has compared auto-normalized importance sampling with ordinal importance sampling. Beginner readers shall be directed there, but I will remind the readers of just enough elements for ...
Fernando Zhu's user avatar
0 votes
1 answer
109 views

Why weighted importance sampling is a biased estimator?

By simple math, we can have $$ E_P[f(X)] = \sum_X f(x)p(x) = \sum_X f(x)\frac{p(x)}{q(x)}q(x) = E_Q[f(X)\frac{P(X)}{Q(X)}], $$ which can be approximated by Monte Carlo sampling in two ways. 1. Normal (...
Fernando Zhu's user avatar
7 votes
1 answer
149 views

When to calculate the bias corrected geometric mean

Most sources give a simple equation to compute the geometric mean (GeoMean) of data samples from a lognormal distribution. GeoMean = exp(m) where m is the mean of ...
Harvey Motulsky's user avatar
7 votes
1 answer
66 views

On unbiasedness of an optimal forecast

Diebold "Forecasting in Economics, Business, Finance and Beyond" (v. 1 August 2017) section 10.1 lists absolute standards for point forecasts, with the first one being unbiasedness: Optimal ...
Richard Hardy's user avatar
0 votes
1 answer
59 views

Unbiased estimator for mean

The question: Given a random sample $X_1,...,X_n$ show that $\frac{1}{n}\sum_{i=1}^n X_i$ is an unbiased estimator for $E(X_1)$. My confusion: Given a statistical model $(\Omega,\Sigma,p_{\theta})$, ...
user124910's user avatar
1 vote
2 answers
89 views

Covariance of Best Linear Unbiased Estimators and arbitrary LUE

I'm working on a problem involving two linear unbiased estimators $T$ and $T'$ of a parameter $\theta$, defined from a sample $\{X_1, \dots, X_n\}$ with mean $\theta$ and finite variance. I aim to ...
Taha Rhaouti's user avatar
4 votes
1 answer
146 views

Why does not this underlying hypergeometric distribution lead to unbiased estimators?

This example is take from Lippman's "Elements of probability and statistics". Let N be the number of fish in a lake the warden wants to estimate. He catches 100 fish, tags them and releases ...
Tryer's user avatar
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0 answers
37 views

Do you change the mean / standard deviation when calculating the unbiased normalised autocorrelation function?

I am trying to calculate the unbiased normalised autocorrelation function. I think this field is a little complicated as different sources appear to use different nomenclature to describe the same ...
Steven Thomas's user avatar
2 votes
0 answers
35 views

How to estimate the age of players correctly?

I have the data of players active on a gaming console and the playtime hours corresponding to the games they have played and their age. I want to analyze the top (say 10) games that the people between ...
Ritik P. Nayak's user avatar
9 votes
1 answer
96 views

Adjusted R2 and bias

Consider the population $R^2$: \begin{equation} \rho^2 = 1- \frac{\sigma^{2}_u}{\sigma^{2}_y} \end{equation} This equation describes the proportion of the variation in $y$ in the population explained ...
Dimitru's user avatar
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1 vote
0 answers
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What are the uniformly minimum variance unbiased estimators (UMVUE) for the minimum and maximum parameters of a PERT distribution?

I believe the answers to this question are the sample minimum and the sample maximum, but I have not been able to find a reference or proof of this.
Nick Stats's user avatar
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21 views

Taking a random sample from a non radom sample and checking it is unbiased

I have a large population and I want select a sample based on some characteristics hence the selected sample is not random since not all members of the population have an equal chance of being ...
Moh's user avatar
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1 vote
0 answers
42 views

Showing that the estimator of the log posterior used in stochastic gradient MCMC is unbiased

In the SGLD paper as well as in this paper it is claimed (paraphrasing) that the following estimator: $$\widetilde{U}(\theta) = -\dfrac{|\mathcal{S}|}{|\widetilde{\mathcal{S}}|} \sum_{{x}\in \...
Tan's user avatar
  • 33
0 votes
1 answer
117 views

Unbiased estimator for parameter of random variables following a uniform distribution [duplicate]

Suppose $X_i$ are i.i.d. and have density $f_\theta(x) = \frac{1}{\theta}$ if $x \in (\theta, 2\theta)$ for positive $\theta$. $(\min_iX_i, \max_iX_i)$ is a sufficient statistic for $\theta$? To ...
johnsmith's user avatar
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4 votes
1 answer
137 views

Unbiased estimator of $\sigma^4$

In the post [here], the user asked the question $\{X_i\}_1^n$ is random sample from $N(\mu, \sigma^2)$ with unknown parameters. Find an unbiased estimator of $\sigma^4$. The solution uses a property ...
sheppa28's user avatar
  • 1,636
0 votes
0 answers
46 views

Misunderstanding on the use of Popoviciu and von Szokefalvi Nagy's inequalities on the variance of a unbiased estimator

Let $X_1,\cdots,X_n$ be (discrete in my case) i.i.d. and bounded between $m$ and $M$. I'm interested in bounding the variance of an unbiased estimator: $$\mathbb{V}\left[\frac1n\sum_{i=1}^nX_i\right]$$...
Tristan Nemoz's user avatar
3 votes
1 answer
130 views

Is the sample mean an unbiased estimator of population mean in the presence of autocorrelation?

I've seen previous questions here that the sample mean can be considered an unbiased estimator of the population mean. e.g.1, 2. While the examples seem to refer to independent sample points, it seems ...
JMenezes's user avatar
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0 answers
26 views

How to prove that the MLE of a uniform distribution is biased using the formula given below? [duplicate]

I've calculated the MLE of the uniform distribution on [0,theta] as maxi{Xi} but don't know how to prove it is biased. The formula I have learned to prove it is unbiased is E(θ^)-θ=0. Was stuck on how ...
meow's user avatar
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2 votes
0 answers
51 views

Unbiased estimate of log-likelihood of Markov bridge

Note: I have cross-posted this question to MathSE. I have the following problem I am trying to solve. I have a parametric family of "transition" distributions $p_\theta(x_{i+1}\mid x_i)$ and ...
Daniel Robert-Nicoud's user avatar
3 votes
1 answer
123 views

Partially Endogenous Regressors

If I have a linear model $$ Y = X_1 \beta_1 + X_2\beta_2 + e$$ where $X_1$ is endogenous to $e$ but $X_2$ is not, then simply performing OLS will yield an unbiased estimate for $\beta_2$ but not $\...
Tommy Tang's user avatar
4 votes
1 answer
196 views

Unbiased estimators and moment of moments

Following section 7.4 of Rose and Smith "Mathematical Statistics with Mathematica" (book available online here), I'm trying to use the Fundamental Expectation Result (eq 7.15) and other ...
sheppa28's user avatar
  • 1,636
1 vote
0 answers
52 views

Proof of attenuation bias in multiple linear regression model

Consider the case of measurement error with a single explanatory variable measured with error \begin{equation} y=\beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_k x^{\ast}_k + \nu \label{...
Maximilian's user avatar
1 vote
0 answers
50 views

Unbiasness of OLS estimates under Stochastic Regressor

I found although the Gauss-Markov Theorms are so widely used, it has so many different versions. Appreciate it if anyone could help me clarify this specific question I have. Given the OLS estimators: $...
Kay99's user avatar
  • 13
0 votes
0 answers
206 views

UMVUE for a Uniform distribution [duplicate]

How did we derive the PDF and CDF highlighted in green? Thanks
learn_to_code1's user avatar
0 votes
1 answer
81 views

The correct condition for OLS estimates to be unbiased?

For the ordinary least square (OLS) estimates of regression ($\vec{y} =\mathbf{X} \cdot \vec{\beta} + \vec{\epsilon}$) to be unbiased (without considering the efficiency), which one of the three ...
Kay99's user avatar
  • 13
0 votes
4 answers
167 views

Unbiased estimator in no-intercept regression model

On an assignment I've been tasked with finding whether $$\hat{\beta}_1=\frac{\sum_{i=1}^n x_iy_i}{\sum_{i=1}^n x_i^2},$$ the estimator for the slope of a no-intercept regression model $Y_i=\beta_1 X_i+...
Kman3's user avatar
  • 103
1 vote
2 answers
39 views

How to obtain the local unbiased condition for an estimator from global unbiased condition?

A standard problem in classical statistics is to find a good estimator that minimizes a given cost function under certain conditions. Normally we want to require the estimator $\hat\theta$ to be ...
narip's user avatar
  • 187
3 votes
1 answer
133 views

Estimating ratio of regression coefficients

What is the best method of estimating a ratio of regression coefficients $\beta_1/\beta_2$ under the usual assumptions / in practice? I have two relatively well approximated signals $X_1, X_2$ and ...
Magemathician's user avatar
3 votes
1 answer
71 views

Tossing Until First Heads Outcome, and Repeating, as a Method for Estimating Probability of Heads

Consider the problem of estimating the heads probability $p$ of a coin by tossing it until the first heads outcome is observed. Say we get $k_1$ tosses, then $U_1 = \frac{1}{k_1}$ is an estimate for $...
Omid Madani's user avatar
1 vote
0 answers
26 views

How to tell if a function is biased [closed]

I have a function that I want to see if its calculations are biased (I think, but an not certain, in the sense of an unbiased estimator). So, I have (can generate) a set of numbers (drawn from a ...
Christopher Clark's user avatar
4 votes
1 answer
117 views

Combine two estimates of same variable

Suppose I know that $ A = 0.3 \cdot B + 0.7 \cdot C $ And I have these estimates: $$ \hat A = A + \varepsilon $$ $$ \hat B = B + \zeta $$ $$ \hat C = C + \eta $$ For the sake of the argument, $ \...
BlackNinja's user avatar
4 votes
1 answer
287 views

Cramer-Rao lower bound for the variance of unbiased estimators of $\theta = \frac{\mu}{\sigma}$

Let $X_1, \cdots, X_n$ be a sample from the $N(\mu, \sigma^2)$ density, where $\mu, \sigma^2$ are unknown. I want to find a lower bound $L_n$ which is valid for all sample-sizes $n$ for the variance ...
Oscar24680's user avatar
0 votes
0 answers
43 views

How to esimate the mean and variance of data from a Pareto distribution

I have large sample of data that is approximately from a Pareto distribution with unknown parameters. Unfortunately the distribution is sufficiently heavy tailed that just taking the sample mean is ...
Simd's user avatar
  • 2,039
14 votes
3 answers
829 views

Best estimator of the mean of a normal distribution based only on box-plot statistics

Suppose $X_1,\ldots,X_n\sim\operatorname N(\mu,\sigma^2)$ and you can observe only the sample size $n,$ the two extreme values, and the first, second, and third quantiles of the sample. Among unbiased ...
Michael Hardy's user avatar
3 votes
2 answers
532 views

Confusion about the notation in Horvitz-Thompson estimator

I am a bit confused about the terminology used in the context of sampling of populations. The Horvitz-Thompson estimator, as well as the Hansen-Hurwitz estimator, for example, are examples of ...
pompeu's user avatar
  • 33
0 votes
0 answers
15 views

How to decompose the scaled estimation error of the debiased machine learning estimator? [Chernozhukov et al 2018]

In page C4 of Chernozhukov et al (2018) the authors introduce a debiased ML estimator $\check{\theta}_{0}$ for scalar parameter $\theta_{0}$: $$ Y = D\theta_{0} + g_{0}(X) + U, \quad \mathbb{E}[U|X,D] ...
Simón Ramírez Amaya's user avatar
1 vote
2 answers
51 views

estimator for standard error [closed]

I have searched extensively and I fail to find an answer. If you are not confident, please don't answer or modify the question, since it will confuse readers even more. Agree on some definitions Let $...
SDE_Amazon's user avatar
1 vote
1 answer
143 views

Best estimator for a binomial distribution

i have a data set that is being generate by a Bernoulli distribution let say $\mathbf{X} \sim x \in \{ 0,1 \}$, where x=1 with probability p. I dont have any information what is the 'p' parameter of ...
Riga's user avatar
  • 13
0 votes
1 answer
55 views

Statistical Properties of OLS Estimators (Unbiasedness)

In deriving the unbiasedness of OLS Estimators, $$\hat{\beta_1} = \beta_1 + \frac {\sum_{i=1}^{n} (x_i - \bar{x})u_i}{\sum_{i=1}^n (x_i - \bar{x})^2}$$ My professor changes the above to: $$\hat{\...
johnf42's user avatar
  • 55
2 votes
0 answers
25 views

Kriging : when it is said kriging is an unbiaised estimator, is that synonymous with saying it is an exact interpolator?

I feel like they are not synonymous, but I cannot intuitively explain the difference between "unbiased" and "exact." In other words, I am asking about the difference between the ...
user386309's user avatar
0 votes
0 answers
23 views

Performance estimation bias in machine learning

Suppose that we have a dataset $D$ that we split on train and test, denoted as $D_{train}$ and $D_{test}$, respectively. We want to construct a predictive model and also estimate its performance. ...
ado sar's user avatar
  • 471
1 vote
0 answers
25 views

Bias of a linear estimator, different results with different methods

Given the linear model: $y=\beta x+\varepsilon $ an estimator for the slope was suggested: $\widehat{\beta} =\frac{\sum y}{\sum x}$ I wish to determine if it is linear and if it is unbiased. Clearly, ...
BlueSigma's user avatar
  • 486
3 votes
1 answer
41 views

Construction of Estimators for Unbiasedly Estimable Functionals

This answer gives an answer for discrete distributions referencing Halmos (1946). However, I am looking for a more general result. For this, the references provided in the previous answer only dealt ...
Dhamma K's user avatar
1 vote
1 answer
310 views

If an estimator for a parameter is unbiased, is it necessary for its variance to tend towards 0 for it to be consistent?

I have been taught that if an estimator is unbiased, then its convergence in probability can be proven by taking the limit of its variance as the sample size grows to infinity and showing it is equal ...
Francesco Bosco's user avatar
5 votes
0 answers
56 views

No unbiased estimator of $\min\{\mu_1,\mu_2\}$ [duplicate]

The problem is stated as: Suppose $X, Y$ are independent and $X \sim \mathcal{N}(\mu_1, 1), Y \sim \mathcal{N}(\mu_2, 1)$ with unknown parameters $\mu_1, \mu_2$. Prove that unbiased estimation of $\...
Soon Princeton's user avatar

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