# Tag Info

### Can I use K-Means to group customers based on a single variable?

If you want a data driven clustering, k-means looks promising in the sense that it will produce clusters with similar within-cluster variance, which may make sense in your application. The problem of ...
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### Sufficiency of $|X|$ when $X\sim N(0,\sigma^2)$ without using Factorization theorem

The problem of your derivation is that you misunderstood the concept of conditional distribution. It is not $P_\sigma(X \leq x |T(X) \leq t)$ -- it should be $P_\sigma(X \leq x | T(X) = t)$. For a ...
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### Understanding different approaches to construct confidence intervals with bootstrap

As @dipetkov says in a comment, a full answer requires chapters if not a book. If you want to understand the bootstrap, originally developed by Efron, then you could do worse than to consult the ...
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### Using Jeffreys prior for Bernoulli distribution to find the prior of a transformation on p

You know the Jeffreys' prior density for $p$ is proportional to $\sqrt{\frac{1}{p(1-p)}}$ and that the corresponding odds are $\eta=\frac{p}{1-p}$. This looks as if it is just a change of variables ...
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### Within and Between Variation in fixed and random effects models

RE models rely on within- and between-group variation, while FE models only rely on within-group estimation. However, this is only true if the explanatory variable is independent from group specific ...
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1 vote

### A question about the exchangeability assumption in conformal predictions

There are conformal prediction approaches that explicitly consider time series analysis. Here is an article that discusses this: Conformal prediction for time series. The R caretForecast and the ...
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### Regularity conditions hint

Written in that way, it seems complicated. But what the hint really says is just the following simple inequality: \begin{align} \sup_{\theta \in \Theta}(f(\theta) + g(\theta)) \leq \sup_{\theta \in \...
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1 vote

### How is bandwidth and roughness penalty in nonparametric regression connected?

The relation between spline estimators and spline estimators is far from trivial. The "best starting reference" is likely: Lin et al. (2004) Equivalent kernels of smoothing splines in ...
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1 vote
Accepted

### How would you describe the distribution of this data?

First, you should use a more descriptive title, if possible. You have ordinal data (scores between 1 and 10?), where the data is mostly saturating the scale (getting the top value). Your data seems ...
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### Finding the expected value for the mean squared

Here is a more expanded version of the derivation: \begin{align} E\left[\bar X_n^2\right] &= E\left[\left(\frac{1}{n}\sum^n_{i=1} X_i\right)^2\right] \\ &= E\left[\left(\frac{1}{n}\sum^n_{i=1} ...
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1 vote

### Are the method-of-moments-based normal confidence intervals asymptotically valid and optimal?

For a normal random variable, the moment-matching estimator (MME) for the mean is the maximum likelihood estimate (MLE). For the variance, the MME and the MLE differ just by the bias adjustment ( n/(n-...
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### Which hypothesis test should I use in this scenario?

You probably want to treat your variable Age as an ordinal variable (ordered category). Since Employed is dichotomous, it could be treated as either nominal (nominal category) or ordinal. Since you ...
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### How can I compute the joint distribution function of normal distribution and beta distribution?

I presume your intent is that the marginal distributions are beta and normal respectively, rather than say one conditional and one marginal or both conditional. Specifying the marginal distributions ...
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1 vote

### Samples from a multivariate t distribution

Because you get multivariate Student t as the fraction of the normal vector and a chi-squared random variable. So, of course, the histogram of entries of the vector will be normal, but for every ...
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Accepted

### How to find variance of multivariable expression

This is a partial answer and partially a request for clarification. If I understand correctly, $a$ is a constant. $X_i$ is Bernoulli with $\Pr(X_i=1)=p_i.$ $Y_i$ is Bernoulli with $\Pr(Y_i=1)=q_i.$ ...
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### Statistical test for vector of expected values

The problem can be solved using a slightly modified paired $t$-test in which, the null is $H_0:\mu_1 -2\mu_2 = 0$ and the alternative is $H_1:\mu_1 -2\mu_2 \neq 0$. The procedure is thus to apply a $t$...
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Accepted

### UMVUE for $g(\theta)=\theta^2$ of Poisson random variables

Yes, that's correct. To make it rigorous, you can apply Lehmann–Scheffé Theorem with $Y = T(X)$ and $\varphi(Y) = \frac{1}{n^2}(Y^2 - Y)$. After a closer look at the Wikipedia link above, it seems ...
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### How combine two statistics from different tests?

The simplest approach, but also most conservative, is to use a Bonferroni correction. In your case, this would be simply using $2 \times \text{min}(p_1, p_2)$, where $p_1$ and $p_2$ are the p-values ...
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### Showing that $\frac{\Vert Au\Vert_2^2}{\Vert u\Vert_2^2}\sim \chi_d^2$
If you denote by $a_i$ the rows of $A$, then each element of $a_i$ has a standard normal distribution. Therefore, $$a_i^\top u = a_{i1}u_1 + \cdots +a_{ip}u_p \sim N(0, u^\top u)$$ and thus  \frac{...