Tagged Questions

2
votes
2answers
99 views

Unbiased estimator of variance of binomial variable

$Y_{1...n}\sim \operatorname{Bin}(1,p)$, iid, and I need to find an unbiased estimator for $\theta=\operatorname{var}(y_i)$. I did some calculations and I think that the answer is ...
4
votes
0answers
76 views

Is it possible to have an estimator that is unbiased and bounded?

I have a parameter $\theta$ which lies between $[0,1]$. Let us say that I can run an experiment and obtain $\hat{\theta} = \theta + w$, where $w$ is a standard Gaussian. What I need is an estimate of ...
1
vote
0answers
24 views

Trying to understand an example where unbiased estimators don't exist

I am new to statistics especially in the topic of estimators and sufficient statistic. I am reading a note which says "unbiasedness is a desirable (but not necessary) property of a good estimator". ...
0
votes
0answers
36 views

Bias in EM estimation for a mixture of normal distributions

Are the parameter estimates for a mixture of two normal distribution using EM algorithm biased or unbiased? More specifically, if I use the EM-algorithm to obtain ML estimates of $μ_1$, $μ_2$, ...
3
votes
1answer
50 views

Will two estimators converge to the same answer?

Say I have two estimators for the same quantity and using the same model, $E[f(X)]$. I also know that these two estimators are consistent, meaning, if we have a lot of data, they will be close to the ...
3
votes
1answer
120 views

What is the name of the estimator that takes the mean of likelihood?

Let $X,Y$ be input and output (observed) continuous variables in $\mathbb{R}$. Let $\{y_1,...,y_n\}$ be the set of $n$ observations. Is there a name for the estimator $\hat x = \int_{x \in X} x ...
2
votes
1answer
118 views

Estimate the second moment of a latent variable using a conditionally unbiased proxy

The Setup: Let $X_t$ denote an unobservable stochastic sequence where the first two unconditional moments are finite constants; ie $\mathbb{E} X_t = \mu < \infty$ and $\mathbb{E} X_t^2 = \gamma ...
7
votes
2answers
375 views

Estimating parameters of a normal distribution: median instead of mean?

The common approach for estimating the parameters of a normal distribution is to use the mean and the sample standard deviation / variance. However, if there are some outliers, the median and the ...
9
votes
2answers
330 views

Bias correction in weighted variance

For unweighted variance $$\text{Var}(X):=\frac{1}{n}\sum_i(x_i - \mu)^2$$ there exists the bias corrected sample variance, when the mean was estimated from the same data: ...
8
votes
3answers
359 views

Parameter estimation of exponential distribution with biased sampling

I want to calculate the parameter $\lambda$ of the exponential distribution $e^{-\lambda x}$ from a sample population taken out of this distribution under biased conditions. As far as I know, for a ...
2
votes
2answers
477 views

Why doesn't the Cramer-Rao lower bound apply?

Let $X_1, X_2, \dots, X_n$ be a sample of i.i.d. random variables, with density $$f_\theta=\frac{2}{3\theta}\left(1-\frac{x}{3\theta}\right) $$ for $0 < x < 3\theta$. And $f_\theta=0$ if $ x ...
4
votes
1answer
183 views

Two unbiased estimators for the same quantity

In several situations, I have two unbiased estimators, and I know one of them is better (lower variance) than the other. However, I would like to get as much information as possible, and I would like ...
2
votes
2answers
144 views

Can the ratio importance sampling estimate by made to be unbiased with resampling?

Consider approximating the following integral: $$ \mathcal{Z} = \int h(x) \pi(x) dx $$ Where $\pi$ is known only up to a normalizing constant, that is, $\pi(x) = \hat{\pi}(x)/\mathcal{Z}_\pi$. We can ...
5
votes
1answer
286 views

Finding a minimum variance unbiased (linear) estimator

Here is a basic question that perhaps has a simple answer, but one that I was not able to find by quickly scanning the literature. Suppose that I have a collection of $n$ unopened boxes. Each box ...
5
votes
1answer
424 views

Estimation of probability of a success in binomial distribution

Let's say we have two biased coins. The probability of tossing a head on the first coin is $\alpha$ and the probability of tossing a head on the second coin is $1-\alpha$. We toss both coins $n$ times ...
13
votes
3answers
740 views

Unbiased estimation of covariance matrix for multiply censored data

Chemical analyses of environmental samples are often censored below at reporting limits or various detection/quantitation limits. The latter can vary, usually in proportion to the values of other ...