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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7 views

Estimating variance or coefficient of variation for population given total and count for four segments of population

My goal is to estimate the coefficient of variation for a population, but the data we have been given is very limited. All we have is the total and count for four segments of the population. The ...
0
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1answer
35 views

If given statistic is unbiased estimator?

I am having trouble finding out and verifying estimators properties ( like unbiased, consistency sufficiency, efficiency ) but in this particular problem given below I have to find a constant such ...
1
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0answers
25 views

Regression method if dependent variable is the absolute value of a continous variable [duplicate]

Suppose we have a dependent variable $Y$ that has normal distribution with a mean of $0$. If I run a regression model using the absolute value of $Y$, $|Y|$, i.e. $|Y| = b_1 + b_2 X + u$, my dependent ...
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0answers
11 views

Efficiency and consistency of unbiased estimators?

For independent random variables (X1, X2, . . . , Xn, where E(Xi) = μ , var(Xi) = σ^2 , i = 1,...,n), how do I find the relative efficiency of the unbiased estimator (X1 + Xn)/2 to the unbiased ...
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1answer
77 views

Unbiased Estimators and Heteroskedasticity

Consider a consumption model with bivariate data points $(Y_i,X_i)$, $i=1,...,n$, with $Y_i$ consumption and $X_i$ income. The univariate model is $$Y_i=\beta X_i+u_i,$$ where $E(u_i|X_i)=0, ...
2
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0answers
17 views

Bias of an estimated Gaussian density

I have an iid sample, $X_1,\dots,X_N \in R^d$, from a multivariate normal density with mean $\mu$ and covariance matrix $\Sigma$. I am estimating the density $p(y) = N(y| \mu, \Sigma)$, using ...
3
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1answer
43 views

Law of Iterated Expectations in Practice

I was wondering if the the following is an example of the law of iterated expectations. Say we observe the entire population of the random variable $X$. Call the mean of the population $\mu$ . ...
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0answers
12 views

Trouble understanding equations with a mix of variances, expected values and means

Let's assume we don't know the real value $\mu$ of the average of a of random variable $x$. We can find an estimator for the variance using the Bessel correction: $$\widehat{V(x)} = ...
2
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1answer
29 views

Bias of method of moments estimator for Pareto distribution with known scale parameter

Let $x$ be a Pareto distribution with a known scale parameter $m>0$, i.e. $x\sim f(x|a)=\frac{am^a}{x^{a+1}}, x>a, a>0$ $\mathrm{E}\left[X\right]=\frac{am}{a-1}$ Using method of moments ...
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0answers
10 views

Best estimate of among groups variance with unequal within groups variances

Goal I have about 100,000 sets of groups. For each set, I would like to measure its among groups variance in order to then make comparisons among sets. Description for each set In each set, I have ...
3
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0answers
22 views

Closed formula for D4 constant calculation? (Moving range chart constant)

I need to build a Moving Range Shewhart control chart given a series of observations. In short, I have to calculate the central line and the upper and lower limits as follows ...
1
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1answer
178 views

Is this an unbiased estimator?

I'm working in the following problem: Let X be a sample of size = 1 from a Poisson distribution with parameter $\lambda$, and let $h(\lambda) = e^{-3\lambda}$. a.) Check if $T = (-2)^X$ is an ...
5
votes
1answer
43 views

why is $1/n \sum_{i} (X_i -10)^2$ unbiased

Let $\{ X_1,X_2,...,X_n \}$ be n observations randomly drawn from normal distribution with mean $10$ and unknown variance. Prove that the estimator $1/n \sum_{i} (X_i -10)^2$ is unbiased. Why is this ...
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0answers
13 views

how can I check the bias between two groups

I have a presentation at the firm... using the stata. I want to check is there any bias between unweighted mean and weighted mean. and I already calculated the means between them..by common method ...
1
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0answers
37 views

Unbiased estimator for AR($p$) model

Consider an AR($p$) model (assuming zero mean for simplicity): $$ x_t = \varphi_1 x_{t-1} + \dotsc + \varphi_p x_{t-p} + \varepsilon_t $$ The OLS estimator (equivalent to the conditional maximum ...
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0answers
6 views

Local Fisher information with 0 probability

I have a question about the definition of the Fisher information. Let's say I have just two conditional probabilities $P(\mu=\pm|\theta)$ that sum to $1$. I want to calculate the Fisher information ...
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2answers
90 views

a question on negative mean square error

for simple random sampling, i have calculated some mean square errors of many types of ratio estimator. Well, i obtained negative mean square error. Is there a mistake? negative MSE is a normal ...
0
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0answers
11 views

How does one show that an extension of an algorithm performs as well or better than the original?

If we have an extension of an algorithm (for example, we take OLS and scale the inputs, and solve it as a reformulated OLS problem), how does one show that the extension performs as well or better ...
1
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2answers
60 views

showing the estimator of $\sigma^2$ [closed]

Let's take a one way ANOVA Model as $Y_{ij}=\mu + \tau_i +\epsilon_{ij}$ $i=1,2,3$ and $j=1,...,n_i$ and $\epsilon_{ij} \sim N(0,\sigma^2), \ \ \ \forall (i,j).$ Sample variance for factor ...
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0answers
17 views

Pandey and Dubey estimator.

I am studying sampling theory Pandey and Dubey(1988) proposed the following product estimator. $\bar y_{PD} = \bar y \left( \frac{\bar x + C_x}{\bar X +C_x}\right)$ And its Mean square error is ...
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0answers
18 views

Using the MSE of $c\theta$ to infer $MSE_{c \theta_{hat}}=MSE_{\theta_{hat}}/1+\lambda$

Suppose $\newcommand{\Bias}{\rm Bias}\hat\theta$ is an unbiased estimator of $\theta$ and $\operatorname{Var} \hat\theta =\lambda \theta^2$ for constant $\lambda$. Find the MSE of c$\theta$, where ...
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1answer
29 views

Estimating $X\sim \mathcal{N}(0,\sigma^2)$ given a bunch of observations with correlated noise

How is the minimum variance unbiased estimator derived for some RV $X\sim \mathcal{N}(0,\sigma^2)$, given $\mathbf{Y}=X\cdot \mathrm{ones}_{N\times 1}+\mathbf{w}$, where $\mathbf{w}$ is ...
1
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1answer
31 views

Defintion of the terms “node weight” and “case weight”

In the literature about decision tress and especially the family of tree approaches that avoid selection bias (conditional inference trees e.g. here: ctree: Conditional Inference Trees by Hothorn, ...
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0answers
15 views

Is OLS unbiased for two subpopulations?

I have a sample of size N. I know that the relationship Y=X*b always holds without error, where Y and X are vectors of size (Nx1). I do not observe X. However, for a fraction theta of the sample, I ...
2
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1answer
57 views

For $Y \sim N_n(X\theta,\sigma^2I_n)$ show that an unbiased linear estimator, $b'Y$, of $c'\theta$ is independent of $Y'(I-P_x)Y$

For $Y \sim N_n(X\theta,\sigma^2I_n)$ show that an unbiased linear estimator, $b'Y$, of $c'\theta$ is independent of $Y'(I-P_x)Y$ Ok...first some notation: $'$ means transpose and $P_x$ is the ...
0
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0answers
67 views

Unbiased estimator of variance of normal distribution

I am struggling with the following question about unbiased estimators. I don't know if its the wording, but I would appreciate any guidance. Question: Let $X_1,...,X_n$ denote a random sample from a ...
0
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0answers
11 views

Using Horvitz-Thompson for estimation from a simple random sample with unknown membership probability

I learnt about the Horvitz-Thompson estimator yesterday, and am trying to apply it to the degenerate case where $p$ is uniform for each, but I seem to have run into a bit of a confusing situation. ...
1
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0answers
64 views

Find an unbiased estimator from a random sample from the logistic distribution

So I'm not sure where to start on this one, so any hints will help me because I'm a little lost on finding unbiased estimators. So suppose $X_1, X_2,... X_n$ is a random sample from the logistic ...
0
votes
1answer
42 views

Odds ratio and bias

I have a random sample of size n that are distributed as Bernoulli random variables with parameter $p$. Given $$v=\frac{p}{1-p}$$ we are asked to find the the MLE of $v$, say $\hat{v}$. Further, we ...
1
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1answer
27 views

Homoskedasticity and bias in regression

My teacher said "Violations of homoskedasticity assumption does not lead to bias in the estimated coef." Can someone motivate/explain this, perhaps with a formula?
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0answers
22 views

Asymptotics of the MLE: a different flavor of proof? [Reference request]

I'm currently trying to understand more about the properties of the maximum likelihood estimator. It's known that, in the large data-limit, the MLE becomes an unbiased estimator with almost Gaussian ...
8
votes
1answer
140 views

What is the variance of this estimator

I want to estimate the mean of a function f, i.e. $$E_{X,Y}[f(X,Y)]$$ where $X$ and $Y$ are independent random variables. I have samples of f but not iid: There are iid samples for $Y_1,Y_2,\dots Y_n$ ...
0
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0answers
44 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
205 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 $\sum_{k=1}^\infty k\,\text{Pr}[X=k]=\frac{1}{p}$. ...
5
votes
1answer
396 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 ...
34
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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 ...
2
votes
0answers
47 views

Maximum likelihood estimator for minimum from a vector of random variables

Let $X_i, i =1,\ldots,N$ be vectors of random variables. 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
124 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
389 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 ...
11
votes
1answer
528 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
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0answers
27 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
43 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, $E(theta)\frac 1 2 - (\theta)\frac 1 2 = 0$, ...
0
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0answers
55 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
51 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 ...
6
votes
2answers
130 views

Why does Restricted maximum likelihood yield a better (unbiased) estimate of the variance?

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
83 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
86 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
61 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
42 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
28 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 ...