# Tag Info

Accepted

### What is a Highest Density Region (HDR)?

I recommend Rob Hyndman's 1996 article "Computing and Graphing Highest Density Regions" in The American Statistician. Here is the definition of the HDR, taken from that article: Let $f(x)$ be the ...
• 99.4k
Accepted

### Statistical tests when sample size is 1

Unfortunately, your student has a problem. The idea of any (inferential) statistical analysis is to understand whether a pattern of observations can be simply due to natural variation or chance, or ...
• 99.4k

### Maximum likelihood method vs. least squares method

I'd like to provide a straightforward answer. What is the main difference between maximum likelihood estimation (MLE) vs. least squares estimation (LSE) ? As @TrynnaDoStat commented, minimizing ...
• 5,366
Accepted

### Are there parameters where a biased estimator is considered "better" than the unbiased estimator?

One example is estimates from ordinary least squares regression when there is collinearity. They are unbiased but have huge variance. Ridge regression on the same problem yields estimates that are ...
• 95k
Accepted

### Why is the James-Stein estimator called a "shrinkage" estimator?

A picture is sometimes worth a thousand words, so let me share one with you. Below you can see an illustration that comes from Bradley Efron's (1977) paper Stein's paradox in statistics. As you can ...
• 117k

• 18.7k

### Is p-value a point estimate?

Yes, it could be (and has been) argued that a p-value is a point estimate. In order to identify whatever property of a distribution a p-value might estimate, we would have to assume it is ...
• 293k
Accepted

### Why do we need an estimator to be consistent?

If the estimator is not consistent, it won't converge to the true value in probability. In other words, there is always a probability that your estimator and true value will have a difference, no ...
• 52.5k

### Why do we need an estimator to be consistent?

Consider $n = 10\,000$ observations from the standard Cauchy distribution, which is the same as Student's t distribution with 1 degree of freedom. The tails of this distribution are sufficiently heavy ...
• 51.1k
Accepted

### Bias of moment estimator of lognormal distribution

There is something puzzling in those results since the first method provides an unbiased estimator of $\mathbb{E}[X^2]$, namely$$\frac{1}{N}\sum_{i=1}^N X_i^2$$has $\mathbb{E}[X^2]$ as its mean. ...
• 93.4k

### Statistical tests when sample size is 1

BruceET has described the proper analysis (Two-way ANOVA without interaction), so I'll put a more positive spin on the experiment. I'm assuming that the design was three pairs, where there is ...
• 2,249

### Are there parameters where a biased estimator is considered "better" than the unbiased estimator?

Yes there are plenty of cases; you're beating around the bush that is the topic of Bias-Variance tradeoff (in particular, the graphic to the right is a good visualization). As for a mathematical ...
• 968
Accepted

### Why $\sqrt{n}$ in the definition of asymptotic normality?

We don't get to choose here. The "normalizing" factor, in essence is a "variance-stabilizing to something finite" factor, so as for the expression not to go to zero or to infinity as sample size goes ...
• 53.7k
Accepted

### A description of the mean of the Geometric Distribution - is it unorthodox or just incorrect?

$\exp(\mathbb E[\log(X)])$ is the geometric mean of a positive random variable $X$ not the mean of a geometric random variable. So either the homework directions put the words in the wrong order, or ...
• 31.8k
Accepted

### Are estimates of regression coefficients uncorrelated?

This is an important consideration in designing experiments, where it can be desirable to have no (or very little) correlation among the estimates $\hat a$ and $\hat b$. Such lack of correlation can ...
• 293k
Accepted

### Growing number of Gaussians in a mixture

If your goal is to find the maximum-likelihood mixture of size $n+1$, then you can use the existing solution as an initialization, once you have enlarged it to have one more Gaussian. To enlarge it, ...
• 6,650