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 ...

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0
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0answers
39 views

Average of Monte-Carlo estimates: Increasing observations vs. iterations

In the context of Monte Carlo simulations, I would like to understand better the difference between increasing the number of iterations vs. the number of observations. As an example, please consider ...
6
votes
2answers
166 views

Estimating the parameter of a geometric distribution from a single sample

I was surprised not to find anything about this with Google. Consider a geometric distribution with $\text{Pr}[X=k]=(1-p)^{k-1}p$, so the mean is $\sum_{k=1}^\infty k\,\text{Pr}[X=k]=\frac{1}{p}$. ...
4
votes
1answer
78 views

Unbiased estimators of skewness and kurtosis

The skewness and kurtosis are defined as: $$\zeta_3 = \frac{E[(X-\mu)^3]}{E[(X-\mu)^2]^{3/2}} = \frac{\mu_3}{\sigma^3}$$ $$\zeta_4 = \frac{E[(X-\mu)^4]}{E[(X-\mu)^2]^2} = \frac{\mu_4}{\sigma^4}$$ The ...
31
votes
4answers
3k views

What can we say about population mean from a sample size of 1?

I am wondering what we can say, if anything, about the population mean, $\mu$ when all I have is one measurement, $y_1$ (sample size of 1). Obviously, we'd love to have more measurements, but we ...
1
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0answers
28 views

Maximum likelihood estimator for minimum from a vector of random variables

Let $X_i, i =1,\ldots,N$ be vectors of random varaibles. Each of them has $m$ components representing dimension, $X_{ij}, j=1,2,\ldots,m$. Specific values $x_{ij}$ are observations or data. From, ...
4
votes
1answer
93 views

Stein's estimator vs James-Stein estimator

I read a lot of sources concerning stein's estimator and James-Stein estimator. Unfortunately, a lot of sources do not write the correct formulas of each estimator. And so I am now confused!! Kindly, ...
7
votes
1answer
271 views

Should the standard deviation be corrected in a Student's T test?

Using the Student's T test, T-Critical is calculated via: $t = \frac{\bar{X} - \mu_{0}}{s / \sqrt{n}}$ Looking at Wikipedia article on the unbiased Estimation of the standard deviation, there ...
8
votes
1answer
183 views

Other unbiased estimators than the BLUE (OLS solution) for linear models

For a linear model the OLS solution provides the best linear unbiased estimator for the parameters. Of course we can trade in a bias for lower variance, e.g. ridge regression. But my question is ...
1
vote
0answers
22 views

variance and sample confused

When solving (b), is the variance $$ V\bigg(\frac 1 2 (x_1+x_2)\bigg) = \frac 1 4 V(x_1+x_2) = \frac 1 4 \big(v(x_1)+v(x_2)\big)= \frac 1 2 \sigma^2 $$ or should I divide the variance by the sample ...
1
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0answers
37 views

Estimator, unbiased or biased

I am having difficulty with this. My procedure for solving it is that $$ E(\theta)= \frac 1 2 E(X-0.1) + \frac 1 2 E(X+0.1) = \frac 1 2 $$ So, $E(theta)\frac 1 2 - (\theta)\frac 1 2 = 0$, ...
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0answers
48 views

how can I tell whether the estimator is biased or not

$X_i\sim N(\mu,\sigma^2)$. Two independent samples of size $n_1$ and $n_2$, with means $\bar{X}_1$ and $\bar{X}_2$. Two estimators of $\mu$ are proposed: $\hat{\mu}_a = ...
1
vote
1answer
32 views

Biased and Efficient estimators

Is unbiasedness a necessary condition for an estimator to be efficient? For example, if $\hat {\theta}= \frac{\sum_i^n X_i}{3}$, I assume $\hat {\theta}$ can't be efficient in a Cramer-Rao lower ...
4
votes
0answers
43 views

Understanding variance estimation by restricted maximum likelihood (REML)

I'm reading Doug Bates' theory paper on R's lme4 package to better understand the nitty-gritty of mixed models, and came across an intriguing result that I'd like to understand better, about using ...
2
votes
1answer
73 views

Unbiased estimator and sufficient statistics

Let $X_1,..,X_n$ be a random sample of $f(x;\theta)=\theta x^{\theta-1}I_{[0,1]}(x)$ Find a sufficient statistic for $\theta$ and construct a unbiased estimator for $\theta$ as a function ...
3
votes
0answers
54 views

Sufficient statistics and UMVUE for joint poisson, bernoulli

Given a pair $(X,Y)$ of r.v.s such that: $$X \sim \text{Poisson}(\lambda)\quad \text{and}\quad Y \sim B(\frac{\lambda}{1+\lambda})$$ with $X,Y$ independent, determine a one-dimensional ...
0
votes
1answer
45 views

Minimal sufficiency and UMVUE in a pseudo-Normal distribution

I already asked a (stupid) question about this problem here thinking I wouldn't have problems to continue it but I was pretty wrong. I'm finding several more problems trying to solve it. I'll try to ...
0
votes
0answers
35 views

Estimate the population variance from a set of weighted means

Proviso: I do not have a lot of experience with statistical theory, so please forgive my occasionally poor choice of notation.$$\\$$ My problem is as follows: I have a set of measurements $X_i, ...
0
votes
1answer
24 views

Unbiased estimator and variance

A random sample of n people are asked whether they are against smoking or not. Suppose x are against smoking. What is the distribution of the random variable X (number of those against smoking). State ...
0
votes
0answers
12 views

OLS exam question [duplicate]

have an econometric methods exam coming up and don't have access to any mark schemes, so the more detailed the better! (a) In what sense is the OLS estimator a linear estimator? Distinguish this ...
19
votes
1answer
275 views

Shrunken $r$ vs unbiased $r$: estimators of $\rho$

There has been some confusion in my head about two types of estimators of the population value of Pearson correlation coefficient. A. Fisher (1915) showed that for bivariate normal population ...
4
votes
0answers
164 views

Is the standard explanation for Bessel's correction in unbiased sample variance wrong?

The standard intuition for Bessel's correction $n-1$ instead of $n$ given in many statistical textbooks is something along the lines that you have one degree of freedom fewer because of the presence ...
1
vote
1answer
59 views

Don't understand identity in proof for unbiased sample variance

Wikipedia gives the following proof why to use Bessel's correction for the unbiased sample variance: \begin{align} E[\sigma_y^2] & = E\left[ \frac 1n \sum_{i=1}^n \left(y_i - \frac 1n ...
5
votes
1answer
123 views

What will be the estimator for these parameters

Question: $y_0 = z^d$ is computed from the sum of some recordings by a sensor. Let, there be $k$ sensor nodes. This parameter is calculated by each sensor node and then transmitted to the base ...
9
votes
1answer
223 views

Biased estimator for regression achieving better results than unbiased one in Error In Variables Model

I am working on some syntatic data for Error In Variable model for some research. Currently I have a single independent variable, and I am assuming I know the variance for the true value of the ...
3
votes
1answer
52 views

More than one unbiased estimator for a single unknown parameter?

Is it possible to have more than one unbiased estimator for a single unknown parameter?If "Yes" then how and if "No" the why?
0
votes
0answers
56 views

Unbiased estimator based on minimal sufficient statistic has smaller variance than one based on sufficient statistic

Suppose that $T_1$ is sufficient and $T_2$ is minimal sufficient, U is an unbiased estimator of $\theta$, and define $U_1=\mathbb{E}(U|T_1)$ and $U_2=\mathbb{E}(U|T_2)$ a)Show that ...
19
votes
4answers
1k views

Why shouldn't the denominator of the covariance estimator be n-2 rather than n-1?

The denominator of the (unbiased) variance estimator is $n-1$ as there are $n$ observations and only one parameter is being estimated. $$ ...
0
votes
0answers
25 views

Showing a variance estimator is unbiased

I am trying to show that the variance estimator $ \hat{\sigma}^2 = \sum_{i=1}^{N}(X_{i}^{2}+ X_{i}X_{i-1} + X_{i+1}X_{i})$ is unbiased. $E(\hat{\sigma}^2) = \sigma^2$. I know that ...
2
votes
0answers
51 views

MLE estimate of normal distribution

I am quoting this from Greene's econometrics book: The occasional statement that the properties of the MLE are only optimal in large samples is not true, however. It can be shown that when ...
1
vote
1answer
45 views

Unbiased Estimators

So I've been banging my head against the wall trying to figure out where to go with these problems, and I'm looking for a little direction. Suppose that $Y_1, Y_2, Y_3$ is a random sample where the ...
2
votes
1answer
50 views

Unbiased Estimator for Uniform Distribution

$X_1$ , a sample size 1 is drawn from a uniform distribution over $[0,\theta]$. Find an unbiased estimator for the variance of the population. Find a function for $X_1$, $\tau(X_1)$ such that ...
2
votes
0answers
54 views

Finding MLE with ordered statistics?

Let Y1 < Y2 < ... < Yn be the order statistics of a random sample of size n from the uniform distribution of the continuous type over the closed interval: $$[\theta - \rho, \theta + \rho]$$ ...
-2
votes
1answer
47 views

What is the Mean Squared Error for this estimator?

I understand that to find the MSE, i must find the variance and bias and add them together. I've had trouble calculating either of these so a breakdown would be immensely helpful. The estimator is: ...
1
vote
0answers
45 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
74 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
103 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
76 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: ...
0
votes
0answers
37 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
21 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
33 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
113 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
113 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
340 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
24 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 ...
3
votes
1answer
139 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
87 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
64 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
36 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
55 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
78 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 ...