33
votes
What percentage of a population needs a test in order to estimate prevalence of a disease? Say, COVID-19
1) Making some assumptions about the population size (namely that it is large enough that a binomial model is appropriate), the prevalence of a disease in a population at a particular time can be ...
- 39.6k
13
votes
What percentage of a population needs a test in order to estimate prevalence of a disease? Say, COVID-19
It has been answered by Dimitri Pananos, I will only add that in order to estimate the prevalence with pre-set precision you need an absolute sample size which is pretty much invariant with the ...
- 8,718
9
votes
When can't Cramer-Rao lower bound be reached?
There are several instances of (2), namely the case where the variance of a UMVU estimator exceeds the Cramer-Rao lower bound. Here are some common examples:
Estimation of $e^{-\theta}$ when $...
- 11.6k
8
votes
Accepted
Parameter estimation of Gamma Distribution using R
You can compute MLE for the gamma distribution using the dglm package, which is available from the CRAN repository. Here is an example run. Note that the two parameters being estimated in this example ...
- 13.5k
8
votes
Does a Bayes estimator require that the true parameter is a possible variate of the prior?
Yes, it is generally assumed that the true $\theta$ is in the domain of the prior. It is the responsibility of the statistician to see that this is the case.
Usually, yes. For example, when ...
- 41.1k
8
votes
How would a bayesian estimate a mean from a large sample?
There are various flavours of Bayesian statistics. One of them is subjectivist (e.g., according to de Finetti). Subjectivist Bayesians hold that probability applies to an individual's state of belief ...
- 28.3k
8
votes
Accepted
How would a bayesian estimate a mean from a large sample?
With a Bayesian method we could also consider $\bar{X} = \frac{1}{n} \sum_{k=1}^n X_k$ as the observed statistic and it has approximately a normal distribution if we assume that the values have finite ...
- 86.5k
7
votes
Accepted
Trimmed, weighted mean
This is even more complicated than you think.
Let's start with sampling weights: the data are sampled from a larger population and $w_i$ is the reciprocal of the sampling probability for observation $...
- 46.8k
7
votes
Bayesian point estimate of a random sample
I intend to obtain the expectation and variance of my initial data and with them obtain shape1, shape2...
That is wrong, and contradicts the methodology of Bayesian analysis. If you use your data to ...
- 133k
7
votes
Accepted
Confusion about asymptotic distribution of the MLE and of the MAP
Let $g(\theta)$ be the prior distributionfor $\theta$. Let $\mathcal L_n$ be the log-likelhood of the sample,
$$\mathcal L_n = \sum_{i=1}^n \ln f(y_i \mid \theta)$$
The MLE maximizes just $\mathcal ...
- 60.7k
6
votes
Accepted
What is the difference between complete statistics and complete family of distributions?
Suppose $X_1,\ldots,X_n \sim \text{i.i.d. } N(\mu,\sigma^2).$
The family of distributions is $$\left\{ N_n\left(\begin{bmatrix} \mu \\ \vdots \\ \mu \end{bmatrix},\sigma^2 \begin{bmatrix} 1 & 0 &...
- 11k
6
votes
Accepted
Does a Bayes estimator require that the true parameter is a possible variate of the prior?
Very nice question! It would indeed make sense that a "good" prior distribution gives positive probability or positive density value to the "true" parameter $\theta_0$, but from a purely decisional ...
- 108k
6
votes
Accepted
Efficiency, Precision, Accuracy, and Consistency
First, you must realize that precision, accuracy, efficiency, and consistency are technical terms. Ordinary English words have been chosen for each concept because someone thought they were memorable ...
- 57.6k
6
votes
Accepted
Proof that posterior median is the Bayes estimate of absolute loss?
Second derivative yields
$$2 \pi (\delta | x) \geq 0$$
So the original function is convex and hence the median corresponds to a minimum not an inflection point
- 2,554
6
votes
Accepted
Deriving likelihood function of binomial distribution, confusion over exponents
It looks as if you intended $X_1,\ldots,X_n \sim \operatorname{i{.}i{.}d{.}} \operatorname{Binomial}(m,p).$ Then you have
$$
L(p) \propto \prod_{i=1}^n p^{x_i} (1-p)^{m-x_i} = p^{\sum_{i=1}^n x_i} (1-...
- 11k
6
votes
Posterior variance vs variance of the posterior mean
There is no particular reason that $\pi(\theta|\mathbf{y})$ should look anything like $f(\hat{\theta}|\theta)$ as functions of $\theta$. The latter is a sampling distribution for the parameter ...
- 133k
6
votes
What happens if I change the range of a flat prior for Bayesian inference?
Comments:
If the prior distribution has support $[.1,1],$ then the posterior distribution has support contained in or equal to $[.1,1],$ so the posterior distribution could not be any ordinary (two-...
- 57.6k
6
votes
What happens if I change the range of a flat prior for Bayesian inference?
If you start with a uniform prior over the support of the parameter, you get the normalized likelihood back as the posterior (I'm going to restrict my attention to cases where the likelihood can be ...
- 290k
6
votes
Accepted
Finding the MLE for a piecewise function
$\require{cancel}$
$$
L(\theta) = f(x\mid\theta) = \xcancel{ \prod_{i=1}^n \frac{x_i^\alpha} {\beta^{n\alpha}}I\{0<x<\beta\} \cdot \prod_{i=1}^n 1 I\{x>\beta\}}.
$$
First you have the density ...
- 11k
5
votes
Accepted
Minimal sufficient statistic whose dimension is less than dimension of parameter
An example where the minimal sufficient statistic has dimension less than the dimension of parameter: a single observation from $\rm{Beta}(\alpha,\beta)$-distribution.
When the sample size is only 1 ...
- 1,802
5
votes
Accepted
Fitting a gamma distribution to truncated data
The contribution to your log-likelihood function due to the truncation should be $\log P_X(\tau_i;\alpha,\beta)$ not $\log 1 - P_X(\tau_i;\alpha,\beta)$. Thus, I think you just need to change ...
- 2,220
5
votes
Accepted
Simple exercise in point estimation: what did I do wrong?
I believe it's because your priors are very different between the two scenarios. To see this, we first need to put the priors on the same interpretation (The first scenario talks about an individual, ...
- 12.3k
5
votes
4
votes
Accepted
Are there examples of non exponential family distributions with sufficient statistics?
Yes, uniform distribution on $(0,\theta)$ ($\theta>0$) is an example. In this case we can write the density function as
$$
f(x; \theta) = \frac1{\theta}\cdot I(0 < x < \theta)
$$
where $I$ ...
- 82.8k
4
votes
Accepted
Why is the maximum risk of an estimator independent of a prior distribution over the parameter?
It would be so comforting to find an estimator that always does better than any other, as it would end up the debate on which estimator to pick! Sadly, it is almost never the case that there exists a ...
- 108k
4
votes
Accepted
Question about Casella and Berger's proof of MLE invariance
The occurrences of suprema (instead of maxima, which might not exist) are troublesome. Let us therefore isolate the basic underlying idea and rigorously establish it.
Definitions
Suppose $f:\Theta\to\...
- 334k
4
votes
What happens if I change the range of a flat prior for Bayesian inference?
The standard proportionality result for the posterior still holds, but the posterior is now concentrated on the same restricted set as the prior. To see this, consider the general case where you ...
- 133k
4
votes
How would a bayesian estimate a mean from a large sample?
You're essentially asking if you can do Bayesian statistics without a likelihood function. The answer is no. The likelihood function is an essential ingredient in Bayesian statistics. Without a ...
- 2,692
4
votes
How would a bayesian estimate a mean from a large sample?
You seem to be asking about a nonparametric estimator for the mean.
First, let's make it clear: for Bayesian statistics, you always need to make distributional assumptions. You can proceed as ...
- 141k
4
votes
Accepted
Unbiased estimator for $\mu_1/\mu_2$
Let's use basic statistical reasoning to simplify the problem, then solve it.
Because the $X_i$ are independent of the $Y_j$ and the former provide information only about $\mu_1$ and the latter only ...
- 334k
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