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1 vote

Convolution of two binomial distribution

The cases found by Brent Kirby can also be seen in the likelihood function which is a combination of polynomial functions $p^{z} \, (1-p)^{n_2-z}$ $$\begin{array}{} p_Y(y) &=& \sum_{z=a}^{b} ...
Sextus Empiricus's user avatar
2 votes

Do we ever use the prior predictive distributions of Bayesian Statistics?

Do we ever use the prior predictive distributions of Bayesian Statistics? Yes, frequently. Why would we use it? The prior predictive distribution is routinely used to develop Bayesian models in an ...
Galen's user avatar
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5 votes
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How to arrange the multivariate DKW inequality to an upper bound on n?

You are looking for the largest (real) root of the function $$f(n; p,k,\epsilon) = \log(1 + n) - \log(p/k) - 2\epsilon^2 n$$ where $p$ is the probability on the left hand side and I have taken ...
whuber's user avatar
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1 vote

Understanding the proof of Karlin–Rubin Theorem

Here is a proof of the Karlin-Rubin theorem following Lehmann's classical book. There, the proof omits some details which I try to provide in this posting. Throughout, I assume that The family of ...
Oliver Díaz's user avatar
0 votes

Proof for expressive power of flow-based models: Normalizing Flows for Probabilistic Modeling and Inference by Papamakarios et al

As mentioned in the comments, $p$ is a density so for example $p$ being the normal density or Laplacian densities are positive everywhere and indeed the integral is 1. $F$ is the conditional CDF ...
travelingbones's user avatar
2 votes
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Interpretation of non significant results of the Mann-Kendall Trend Test

Your last paragraph is correct. Maybe someone here can find a set of cutoffs for "small", "moderate" etc. trends, although if you searched then there may not be a set. But, even if ...
Peter Flom's user avatar
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0 votes

Which statistical test should I use for 3 independent variables and one dependent variable?

Are your hypotheses about the IVs' effects main effect hypotheses or do you hypothesize also interactive effects between IVs? In any case it sounds like linear regression with all 3 IVs as predictors ...
Sointu's user avatar
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2 votes
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Why does the SHAP formula not state that |S| = |F|-1?

A key point to note about the Shapley Value is that it is also the average contribution feature $i$ makes to the fit metric/function $f()$ across all possible permutations of the $|F|$ features. Thus, ...
jluchman's user avatar
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0 votes

Diagonal elements of the projection matrix

Since $H$ is symmetric and idempotent, we have $$h_{ii}=\sum_{j=1}^n h_{ij}^2 \quad \forall\, i \in \{1,2,\ldots,n\}.$$ And because there is an intercept in the model, $$HX=X \implies H \begin{pmatrix}...
StubbornAtom's user avatar
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4 votes

When a credible interval coincides with a confidence interval, can one interpret confidence interval as credible interval and vice versa?

No, just because two numbers coincide numerically doesn't mean they allow for the same interpretation. The fundamental principle behind Bayesian statistics is transforming a probabilistic statement ...
Durden's user avatar
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2 votes
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Sample mean of Bernoulli trials is admissible under squared loss

In the other question we encounter an integral like $$r(\pi) = \mathbb{E}_\pi[r(p,d)] = \int_0^1 r(p,d)\pi(p)\text{d}p\\ $$ If the decision rule $d(x) = x/n$ minimizes this integral, then there can ...
Sextus Empiricus's user avatar
0 votes

When does a UMP test fail to exist?

To prove the non-existence of a UMP test for this two-sided hypotheses based on the normal distribution family, use proof by contradiction. The idea is that if such a UMP test for testing the two-...
Zhanxiong's user avatar
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0 votes

The existence of a UMP test in one-parameter exponential families

As noted in the comment, this is Problem 3.54 in Testing Statistical Hypotheses (Third Edition) by E.L. Lehmann and Joseph P. Romano (this reference will be cited later in this answer for a few more ...
Zhanxiong's user avatar
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3 votes

Bayesian Justification of Cross-validation

Several ideas could serve to justify model selection based on the ELPD instead of the model-posterior. Direct relation to frequentist-like MSE comparison for regression models with Gaussian ...
Johan de Aguas's user avatar
3 votes
Accepted

Is it possible that one estimator performs better than others when sample size $n$ is small but performs worse than others when $n$ is large?

I think I have an example that I like well enough to post. The estimators are nice and simple -- mean and median; I look only at odd sample sizes (which is the worse case for the median). The density ...
Glen_b's user avatar
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9 votes

Asymptotic unbiasedness + asymptotic zero variance = consistency?

Your question may be restated as Let $\langle\hat{\theta}_n\rangle_{n\in\mathbb N}$ be a sequence of random variables such that \begin{align*} & \theta_n := E[\hat{\theta}_n] \to \theta & \...
Zhanxiong's user avatar
  • 19.7k
8 votes

Asymptotic unbiasedness + asymptotic zero variance = consistency?

The following general result would be relevant here: Given a measure space $(\Omega,\boldsymbol{\mathfrak A},\mu),$ consider a sequence of measurable functions $\langle f_n\rangle_{n\in \mathbb N}$ ...
User1865345's user avatar
  • 8,812
2 votes

Sample a random subgraph from an undirected, unweighted graph, what's the probability of "every two nodes's distance is at least 3 in the subgraph"?

Suppose there is a bound $b$ on $$\frac{|S|}{\sqrt{|G|/\ln|G|}}$$ and a bound $k$ on the number of vertices within distance 3 of a vertex. Then as the graphs get larger, the probability goes to $1$ ...
Matt F.'s user avatar
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1 vote

Is Pitman-Koopman-Darmois Theorem valid for discrete random variables?

Continuous case I managed to get a proof of the following statement of the Pitman-Koopman-Darmois theorem : Let be $X_1, \ldots, X_n$ be $n$ i.i.d. real random variables from a distribution with ...
Pohoua's user avatar
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1 vote
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Expected value of derivative derivative squared log equivalent to regularity condition

The very relation depends on the fact that interchanging the differentiation under the integral sign is valid. Consider a random variable $\mathbf X$ takes values on the space $(\mathscr X,\boldsymbol{...
User1865345's user avatar
  • 8,812
2 votes
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Proving an Estimator of the sample variance to be MVUE

I'm pretty sure the question was actually asking about the Normal case, but the general case is interesting (if unhelpful). The statement is true, under various much weaker conditions. In order to ...
Thomas Lumley's user avatar
4 votes
Accepted

Understanding the Logistic regression formula

It sounds like you're new to logistic regression as a concept. If your purpose is to properly understand it, and then provide a rigorous justification of its use for an academic manuscript, I suggest ...
cambridgecircus's user avatar

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