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.
774
questions
2
votes
1
answer
26
views
Prove that $T$ is a complete statistic and find a UMVUE for $p$
While preparing for my prelims, I came across this problem:
Let $X_1, X_2,\cdots, X_n$ be a sequence of Bernoulli trials, $n \geq 4.$ It is given that, $X_1,X_2,X_3 \stackrel{\text{i.i.d.}}{\sim} Ber(\...
- 21
0
votes
0
answers
14
views
An example problem of converting a maximum likelihood problem into a restricted maximum likelihood problem
I have a question about this derivation. What is an example value of the actual matrix $A'$ such that $A'X=0$, $A'A=I$, and $\frac{1}{n}\Sigma((A'Y_{i}-mean(A'Y))^{2}=\frac{1}{(n-1)}\Sigma((Y_{i}-...
- 385
0
votes
0
answers
41
views
Unbiased Estimator of Nugget Effect
Question: I am trying the measure the nugget effect, which is parameterized by $(1-\lambda)$ in the following variance-covariance used to describe the multivariate normal distribution of my n-...
- 385
2
votes
0
answers
59
views
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 $X_1,...,X_n$...
- 121
2
votes
0
answers
139
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 ...
- 65
1
vote
0
answers
45
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 ...
- 889
0
votes
1
answer
78
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 ...
- 103
1
vote
0
answers
58
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 ...
- 704
0
votes
0
answers
40
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 ...
0
votes
1
answer
141
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 (...
7
votes
1
answer
154
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 ...
- 20.9k
7
votes
1
answer
69
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 ...
- 68.6k
0
votes
1
answer
60
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 $(\Omega,\Sigma,p_{\theta})$, ...
- 125
1
vote
2
answers
94
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 ...
- 13
4
votes
1
answer
147
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 ...
- 275