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 ...

learn more… | top users | synonyms

1
vote
0answers
17 views

Methods of Proving that a UMVUE does not exist?

Are there efficient methods of showing when a UMVUE does not exist? I can think of the trivial case when no unbiased estimators exist at all. But that's not really interesting. I feel like this ...
0
votes
1answer
30 views

bias and sampling

This was an interview question I encountered. can some one answer this When you sample, what bias are you inflicting? How do you control for biases? What are some of the first things that come to ...
2
votes
0answers
46 views

Where is the maximum bias and variance in a histogram as non-parametric density estimator?

I am a little bit confused about bias and variance of non-parametric density estimators and hope you can help me. Assuming a constant bandwidth and sample size, I am wondering at which points of the ...
0
votes
0answers
21 views

Standard Error for Relative Frequency Distribution

I have a weighted distribution with weights $w_i$, such that: $$\sum_i{{w_i}}=1$$ I know that the mean is defined by: $$\sum_i{{w_i}{x_i}}=\mu$$ And that the unbiased variance is defined as: ...
4
votes
0answers
81 views

Unable to reach the CRLB expression as mentioned in the paper

I am unable to calculate the Fisher Information matrix and the CRLB of an autoregressive model from the observations $x$ in the format of the CRLB calculated in the paper "Efficient estimation of ...
0
votes
0answers
23 views

Is there a way to check completeness of certain sufficient statistics?

In general, given a p.d.f. or a p.m.f., is there a method to check if a certain statistic is complete? For example, consider a population $N(\theta,1)$ where $\theta$ is unknown and the statistic ...
0
votes
0answers
18 views

ELO Rating for players that do not learn

Lately there are some AI competitions online, where the bot has no capability to "learn/adapt" (ironically) over games. Thus, each uploaded version of a bot by a player is static. On the website, ...
2
votes
0answers
26 views

A Proof of Tukey's Inequality

Suppose that $W_1,W_2,...,W_n$ are uncorrelated unbiased estimators of a parameter $\theta$. Consider $W=\sum_{i=1}^na_iW_i$ such that $E(W)=\theta$ and $Var(W_i)=\sigma^2_i$, where the $a_i$'s are ...
3
votes
1answer
86 views

Finding an unbiased estimator with the smallest variance

I will state the question then my methodology. Q: We have 3 random variables, $X1,X2,X3$ that are independent and identically distributed (iid). We would like to estimate $\theta = E[X1]$. Suppose ...
2
votes
1answer
79 views

Best OLS estimators

Hi i am stuck on this one, the question is related to Gauss-Markov theorem: Consider a general alternative to the OLS estimator that is also a linear unbiased estimator, say ${\tilde \beta}$. ...
13
votes
2answers
281 views

For which distributions is there a closed-form unbiased estimator for the standard deviation?

For the normal distribution, there is an unbiased estimator of the standard deviation given by: $$\hat{\sigma}_\text{unbiased} = \frac{\Gamma(\frac{n-1}{2})}{\Gamma(\frac{n}{2})} ...
1
vote
0answers
23 views

Unbiased Estimator of Product

Suppose there are stationary times series $\{A_i\}_{i=1}^{T},\{B_i\}_{i=1}^{T},\{C_i\}_{i=1}^{T},\{D_i\}_{i=1}^{T}$, which may not necessarily be independent processes. We know that ...
0
votes
1answer
30 views

Help in finding the pdf for Gaussian distribtuion of time series model

PROBLEM STATEMENT: The original data $y_t$ is a noisy version of a time series obtained from an autoregressive process excited by a deterministic non-linear signal $x_t$. The error terms $u_t$ is : ...
3
votes
1answer
74 views

Unbiased estimator and sufficient statistic from discrete uniform distribution

$z_1,...z_n$ is a sample from a discrete $\{1,...,N\}$ uniform distribution. I have two questions: 1; I want to find an unbiased estimator for N, with the help of $z_1$ 2; I want to find a 1 ...
1
vote
0answers
40 views

Is Mean Squared Error an unbiased estimator to the error variance?

According to the wikipedia page at http://en.wikipedia.org/wiki/Mean_squared_error, it has: Note that, although the MSE is not an unbiased estimator of the error variance, it is consistent, ...
0
votes
0answers
54 views

Best linear unbiased estimator

I have a sample of N stocks. I have the following information: For each stock i, I have an estimate of variance (of returns) $\hat{\sigma}^2_{i}$. I also have a fitted variance, denoted by ...
0
votes
0answers
24 views

How to estimate the bias of a estimator of an unknown length?

Let's say I have a ruler of unknown length L. I then measure n objects with the ruler, obtaining a sample x1,..xn. If an object is longer than the ruler, I'll register xi=L. I want an unbiased ...
0
votes
1answer
31 views

Is an WLS estimator unbiased, when wrong weights are used?

It is clear that the WLS estimators are consistent if the "wrong" weights used aren't correlated with the explanatory variables. However, I don't know whether this also holds true for unbiasedness.
2
votes
1answer
61 views

Conceptual question on estimation : How to calculate the variance of estimation error

EDIT/ UPDATE: I have understood CRLB & why we need it. But my problem is something else. In book ...
3
votes
1answer
43 views

Inference about the true intercept of the model and the OLS being BLUE

Consider the following population regression model: $$y_{i} = \beta _{1} + \beta_{2}x_{i} + \epsilon _{i},$$ where $i=1,...,n$. Assume $\epsilon \sim iid$, with the pdf in equation: $f(\epsilon ) = ...
1
vote
0answers
35 views

Unbiased estimator of the variance [duplicate]

I am going through the book The Elements of Statistical Learning and I'm finding it extremely terse. I have a background in probability but not statistics so perhaps that is why. Anyway, in Chapter 3, ...
1
vote
1answer
58 views

Variance of Mean of Samples from Unknown Distribution

I have a bunch of IID samples from a random variable with unknown mean and unknown variance. I now need to know the variance of the average of these samples. I found some references. In decreasing ...
2
votes
0answers
26 views

If we nonlinearly transform the LS estimates, will they still be unbiased estimates of the true value?

So this is an discussion which came up with a friend/colleague who is a physicist postdoc. He has a bunch of data $(x_i,y_i)$ and wants to fit it to the form $y=e^{ax}$. He uses (weighted) nonlinear ...
1
vote
0answers
13 views

Can someone provide an intuitive explanation of how Stein's Unbiased Risk Estimator can be used for signal denosing?

How does SURE just knows what the noise content of a signal is and how can such an estimator eliminate the noise? Note: I'm not very familiar with estimation theory in statistics
2
votes
1answer
49 views

Calculating the expected value and variance of an estimator of a normal quantile

I don't quite understand how to use the estimator function and the variance function and plug in the sample mean. I expected that we would plug in the value $\bar X - 1.645s$ into $E(s)$ and $V(s)$. ...
1
vote
1answer
86 views

Stein unbiased estimate of risk as surrogate for MSE

I am having trouble understanding Stein's unbiased risk estimate (Stein, C. M. (1981). Estimation of the Mean of a Multivariate Normal Distribution. The Annals of Statistics, 9(6), 1135–1151. ...
3
votes
0answers
50 views

Finding $Var(S^2), E(S^4),$ and unbiased estimator for $\sigma^4$ from random, normal samp

Let $X_1,...,X_n$ be a random sample of size $n$ from the normal distribution $N(\mu,\sigma^2)$ and let $S^2$ be the sample variance. (a) Find $V(S^2)$ and derive $E(S^4)$. (b) find an unbiased ...
1
vote
2answers
65 views

$Var(\bar{X})$ for a random sample from Bernoulli Distribution

Let $X_1,...,X_n$ be a random sample of size $n$ from a Bernoulli distribution with parameter $p$ where $0< p< 1$ is unkown. (a) Find $\theta^2=Var(\bar{X}).$ (b) Find the value of $c$ so that ...
1
vote
0answers
38 views

Is this consistent and/or unbiased?

Does OLS produce inconsistent and/or bias estimators? The first example has the beta raised to a power. $$ y = β_0 + (β_1^2 x_1) + (β_2 x_2) + u $$ The second example has (beta + 1)estimator. $$ y ...
1
vote
1answer
53 views

Finding the unbiased variance estimator in high dimensional spaces

The problem comes from linear regression. Assume the regression function is linear, i.e. $$ f(X) = \beta_0+\sum_{j=1}^pX_j\beta_j $$ .Given a set of training data $(x_1, y_1),\ldots,(x_N,y_N)$,we try ...
0
votes
2answers
42 views

efficiency - bias trade-off

Under which conditions would a researcher choose optimally when there is a trade-off between the variance and bias of an estimator? I hope this question is not too broad... Any help would be ...
3
votes
0answers
82 views

Unbiased estimator with minimum variance for $1/\theta$

Let$ X_1, ...,X_n$ be a random sample feom a distribution $Geometric(\theta)$ for $0<\theta<1$. I.e, $$p_{\theta}(x)=\theta(1-\theta)^{x-1} I_{\{1,2,...\}}(x)$$ Find the unbiased estimator ...
2
votes
2answers
85 views

Obtaining an estimator via Rao-Blackwell theorem

Let $X_1,...,X_n$ be iid with pdf $$f(x|\theta) = exp(\theta -x) I(x)_{(\theta, \infty)}$$ It is asked to find an unbiased estimator for$ \theta $ that is function of a sufficient statistical for ...
2
votes
1answer
126 views

Asymptotically unbiased estimator using MLE

I am learning Maximum likelihood estimators for a inference class. And this is a problem I came across. Let $X_1,X_2,X_3,\ldots, X_n$ be a random sample with p.m.f $$p(X)=\theta(1-\theta)^x; ...
5
votes
2answers
102 views

How would one formally prove that the OOB error in random forest is unbiased?

I have read this statement many times but have never come across a proof. I would like to try to produce one myself but I'm not even sure on what notation to use. Can anyone help me with this?
1
vote
0answers
59 views

Is it necessary to transform an econometric model in order to have only diagonal elements in the error covariance matrix?

Model and its assumptions I'm working on the methodology of a two-way error component model. Here is the model: $$y_{jis} = x_{jis} \beta + \upsilon_{jis}$$ $j$ refers to school, $i$ refers to ...
24
votes
3answers
2k views

How exactly did statisticians agree to using (n-1) as the unbiased estimator for population variance without simulation?

The formula for computing variance has $(n-1)$ in the denominator: $s^2 = \frac{\sum_{i=1}^N (x_i - \bar{x})^2}{n-1}$ I've always wondered why. However, reading and watching a few good videos about ...
3
votes
0answers
62 views

Combining unbiased estimators with unknown variance

Say we are given a sequence of independently (but not identically) distributed random variables $X_1,...,X_n$ which are known to be bounded, $X_t \in (a,b)$, and to have the same mean, $\mathbb{E}X_t ...
4
votes
1answer
133 views

Should sampling happen with or without replacement

When doing a simple random sample to estimate population mean for some statistic, how do I know whether sampling happens with or without replacement? It feels wrong to use replacement, because 1) my ...
2
votes
2answers
81 views

Why we shouldn't be obsessed with unbiasedness

In my Bayesian statistics class, my professor makes the remark that we should not be obsessed with unbiased estimator. First: I understand this statement in the sense of trading biasedness for ...
4
votes
0answers
37 views

R packages that work with biased samples

I'm working with a biased sample of web users. I'm only able to track responses of users who have navigated my site in a certain way, and I'd like to run an analysis to determine how certain factors ...
5
votes
3answers
213 views

Unbiased estimate of population standard deviation: is sqrt(2) a superior correction?

Background on bias correction constants The standard deviation is calculated like this: $$ SD = \left(\frac{1}{N-constant} \sum_{i=1}^N (x_i - \overline{x})^2\right)^{1/2} $$ Following Wikipedia's ...
0
votes
0answers
13 views

What I have to undergo testing before proceeding to time series analysis? and after?

I have to conduct a time series regression analysis. Before proceeding I tested for stationarity and I verified that the variables are distributed according to the normal probability distribution. ...
0
votes
0answers
7 views

More examples of inconsistent, but unbiased data and vice versa? [duplicate]

I'm starting to understand the distinction, but I'm having trouble envisioning an example where one's estimates have one, but not the other of these properties. Thanks for the help!
0
votes
1answer
36 views

Identifiability and unbiasedness

How do you show that if my model parameter $\theta$ (scalar) is U-estimable (i.e. if there exists an unbiased estimator of $\theta)$, then $\theta$ is identifiable? This makes sense intuitively, but ...
0
votes
0answers
17 views

How to analyze data from two-tiered experiment

I have an experiment where each subject undergoes M sessions of treatment drug & control drug. I have N subjects. How do I 'match/pair' my data together? In particular, lets say that d(n,m) is ...
0
votes
0answers
51 views

Symmetric distribution and unbiased estimator

If we have 10 random variables, each distributed with a $Cauchy(t,1)$ distribution and we have an estimator for $t$. When using the R program to simulate mean ...
1
vote
1answer
105 views

Is the bootstrap estimate of the mean biased when a single extreme value is in the sample?

My sample includes $n$ random observations, while $n-1$ of these observations are in the range (0-1) there is also one observation that gets very high value. For example, a sample of prices where ...
1
vote
0answers
26 views

Admissibility and domination for estimators

Watching a video by the "mathematicalmonk" on the web, I was wondering how to answer this kind of questions: Given $X_1,\ldots,X_n\sim \mathcal{N}\left(\mu,\sigma^2\right)$. Assume that $\mu$ is ...
0
votes
0answers
66 views

Bootstrap and var of median

I've 2 different questions: A. Sample with n variables, while $n-1$ variables are in the range (0-1), and one of the variable is very high: $10^{1000}$. If I'll use bootstrap to estimate the mean, ...