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Questions tagged [mathematical-statistics]

Mathematical theory of statistics, concerned with formal definitions and general results.

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2k views

Probability inequalities

I am looking for some probability inequalities for sums of unbounded random variables. I would really appreciate it if anyone can provide me some thoughts. My problem is to find an exponential upper ...
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293 views

Did Deborah Mayo refute Birnbaum's proof of the likelihood principle?

This is somewhat related to my previous question here: An example where the likelihood principle *really* matters? Apparently, Deborah Mayo published a paper in Statistical Science refuting Birnbaum'...
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482 views

Is there a general expression for ancillary statistics in exponential families?

It is known that an i.i.d sample $X_1,\dots,X_n$ from a scale family with c.d.f. $F(\frac{x}{\sigma})$ has $S(X)$ as an ancillary statistic if $S(X)$ depends on the sample only through $\frac{X_1}{X_n}...
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173 views

Intuitive understanding of the Halmos-Savage theorem

The Halmos-Savage theorem says that for a dominated statistical model $(\Omega, \mathscr A, \mathscr P)$ a statistic $T: (\Omega, \mathscr A, \mathscr P)\to(\Omega', \mathscr A')$ is sufficient if (...
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1k views

When does a UMP test fail to exist?

I have a sample $X=(X_1, ...,X_n)\sim N(\mu,\sigma^2)$ with $\sigma^2$ known. The hypotheses are $H_0: \mu=\mu_0, H_1:\mu \neq \mu_0$. I know that in such a case an UMP test does not exist and so ...
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57 views

What is this parameter estimation strategy called?

Let $X_1, X_2, \ldots, X_n$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. Consider the problem of estimating $P(X > 100)$. One way to accomplish this is ...
6
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174 views

Why can't the complete class theorem be easily generalized to all locally-compact spaces?

So I was reading Christian P. Robert's The Bayesian Choice, going through the constellation of results related to complete class theorems, and I don't see why all of them are necessary. In particular, ...
6
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144 views

Time evolution of a Bayesian posterior

I have a question regarding the time evolution of a quantity related to a Bayesian posterior. Suppose we have binary parameter space $\{ s_1, s_2 \}$ with prior $(p, 1-p)$, The data generating ...
6
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55 views

If you know the central moments of the data $X$, find a function $f$ for which $f(X)$ has arbitrary central moments

Say you are given one-dimensional data $X$, with mean $\mu$ and central moments $a_n$ which you know. Can you construct a function $f(x)$ which transforms the data such that $f(X)$ has the central ...
6
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0answers
540 views

quantile of standardized t distribution

How to show that, for any given left tail probability, the corresponding quantile of standardized t distribution is increasing in degree of freedom for left tail probability less than 0.5? This is ...
6
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155 views

Confidence interval of the third moment of normal distribution

How to compute exact confidence interval for the third moment of normal distribution $N(a, \sigma^2)$?
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55 views

Multiple maximum likelihood estimates for discrete parameter

Suppose I have a bivariate likelihood function, $L(\theta ,\lambda |\mathbf{x})$, where $\theta$ can take on continuous values, but $\lambda$ can only take 'count' values $(0,1,2,...)$, and $\mathbf{x}...
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468 views

Are there unbiased, non-linear estimators with lower variance than the OLS estimator?

Consider an ordinary least squares model, $$y = \beta X + \epsilon \qquad \epsilon\sim N(0, \sigma)$$ The Gauss-Markov theorem tells us that the ordinary least-squares (OLS) estimator is the minimum-...
5
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169 views

Intuitive explanation for Marchenko-Pastur law

I am looking for an intuitive reasoning behind the Marchenko Pastur law, which is described as a law of large numbers analog for random matrices. I know the law gives the probability density function ...
5
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214 views

I'm not asking for a conjugate prior. Is there a distribution $p(x|y)$ that satisfies $\int p(x|y)Beta(y|a,b) dy = Beta(x| a', b')$?

I know the result of integrating a Gaussian against another Gaussian is still Gaussian, $$\int N(x|\mu_y,\sigma_y)N(y|\mu,\sigma) dy = N(x|\mu',\sigma')\quad.$$ Can I get the same form for Beta ...
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136 views

Rigorous theory behind overfitting

I am taking an intro to ML class, and in my limited experience, training ML algorithms (validation, overfitting etc.) feels a bit like black magic. For instance, you aren't supposed to touch the test ...
5
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2k views

Help with a proof of Bayes classifier optimality

I have a class assignment to provide a proof that Bayes classifier for the two label version is optimal in that it's error rate is always ${\le}$ any other classifier. I've never worked through a ...
5
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167 views

Distribution of $X'X$ if $X\in\mathbb{R}^{T \times N}$ and $X_i'\sim N(\mu,\sigma^2I_N)$

Let $x_i\in\mathbb{R}^N$ be multivariate normal distributed with mean vector $\mu\in\mathbb{R}^N$ and no correlation: $x_i\sim N(\mu,\sigma^2 I_N)$. Given $T$ iid samples, define the matrix $$X:=\left(...
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744 views

Proof of Kolmogorov-Smirnov test

Could someone provide me a reference, preferably a book, where I can find detailed proofs and explanations of the Kolmogorov-Smirnov test (including the two-sample variant) and the derivation of the $...
5
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0answers
102 views

Relation between asymptotic relative efficiency for tests and estimators

The asymptotic relative efficiency for unbiased estimators is the limit of the ratio of the variances as the $n\rightarrow \infty$. Is there a relation to asymptotic relative efficiency according to ...
5
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1k views

Inverse covariance matrix, off-diagonal entries

Let $\Sigma$ be a covariance matrix. According to the material in this link, If the elements of $\Sigma$ are all positive, most of the off-diagonal elements in $\Sigma^{-1}$ will be negative this ...
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231 views

Restriction matrix for a VAR

In New Introduction to Multiple Time Series Analysis by Luetkepohl (2005), section 5.2.1, it says that one can specify linear restraints for a VAR, $Y = \beta X + U$, in the form $$ \operatorname{vec}{...
5
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0answers
563 views

Sum of independent Wishart with same degrees of freedom but different scale matrices

Is there any result showing that a sum of independent Wishart with same degrees of freedom but different scale matrices is a Wishart? For example, if I have two random variables: $$ Y \sim W_p(n,\...
5
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227 views

Reference Request: Information Geometry for Ridge Regression

I am reading the book "regression estimators" by Gruber 2010 where he uses this technique to compare Ridge Regressors, however he concentrates on deriving the mathematical results without giving any ...
5
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0answers
142 views

Hypothesis test on the Euclidean length of an unknown vector

Question Suppose I observe a vector $\mathbf{x}=[X_1 \ldots X_n]$, where each $X_i=m_i+n_i$, with $n_i$ being an independent zero-mean Gaussian random variable with variance $\sigma^2$ (i.e. $n_i\sim\...
5
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0answers
115 views

Orthogonal intersection in a Riemannian manifold

Let $S$ be the set of all probability distributions on $\mathbb{R}$ and $S_n=\{p_\theta\}$ be an $n$ dimensional submanifold of parameterized family of probability distributions on $\mathbb{R}$ where $...
5
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0answers
273 views

Derivation of prediction intervals for a normally distributed population with unknown population standard deviation

I have via the ISO standard 16269 found the solution to a problem that I've been working on. Based on a couple of independent samples from a normally distributed population, I would like to determine ...
4
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0answers
58 views

Proving an inequality for CDF's

I am working on a proof to show that given $x_1, x_2,\ldots,x_k$ random variables with a joint pdf and joint CDF, show that $$ 1-\sum_{i=1}^k \overline{F_i(x_i)} \leq F(x_1,x_2,\ldots,x_k) \leq \min_i ...
4
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0answers
36 views

If a statistic can be written as a function of a minimal sufficient statistic almost everywhere, is it minimal sufficient?

I know that if $T(X) = f(W(X))$ for one-to-one $f$, where $W(X)$ is minimal sufficient, then $T(X)$ is also minimal sufficient. But my textbook does not include "almost everywhere" or "almost surely" ...
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51 views

Is an inadmissible estimator necessarily dominated by some admissible estimator

Basic example: $X$ has a $p$-variate iid standard Normal distribution; the sample mean is not admissible if $p>2$ and is dominated by the Stein shrinkage estimator. However, the Stein shrinkage ...
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62 views

A possible typo in the textbook?

On page 74 of Lehmann's Testing Statistical Hypothesis, the author writes Let $P_0$ and $P_1$ be probability distributions possessing densities $p_0$ and $p_1$ respectively w.r.t. a measure $\mu$ ...
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0answers
72 views

Maximizing a computationally expensive function

Let $f:[0,1]^{80} \rightarrow [0,1]$ be some function, and say I have a computationally expensive way to calculate $f(x)$ for each $x \in [0,1]^{80}$ (expensive = 40s per query). The goal is to ...
4
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0answers
112 views

Finding the Cauchy Principal Value Mean of the pdf for $Z=XY$ where $\mathbb{E}[Y]$ does not exist

Let's first define the Cauchy Principal Value Mean (PVM). For a continuous random variable $V$ with pdf $f_{V}(v)$, the PVM of $f_{V}$ is \begin{equation} \mathrm{PV}(\mathbb{E}[V])=\lim_{a\to\infty}\...
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48 views

Statistical model with $\Gamma(\alpha_i,1)$ sample

We are given statistical sample of $X=(X_1,X_2,X_3)$, where $X_i\sim\Gamma(\alpha_i,1)$ and independent. Let $Z=X_1+X_2+X_3$ and $T$ three dimensional statistic $T:=(\frac{X_1}{Z},\frac{X_2}{Z},\frac{...
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0answers
82 views

Prove that the joint density of independent multivariate normal variables is a matrix-normal

Let $X_1,...,X_n \sim N_p(\mu_i,\Sigma_i)$ be Multivariate Normal a.v. independent. Show that $W = (X_1,...,X_n) \sim MN(M,\mathbb{I},\Sigma)$ where $M = [\mu_1 \mu_2...\mu_n]$ and $\mathbb{I}$ ...
4
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48 views

Why is the $\chi^{2}$ approximation for deviance GLM $\sim \operatorname{Binomial}(n_{i},\pi_{i})$ not valid when $n_{i} = 1$?

I know from McCullagh & Nelder's text (p.118) that the $\chi^{2}$ approximation for deviance for the binomial family is based on a limiting operation in which $n$, the number of observations, is ...
4
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0answers
160 views

Calculating Formula with logs and Weighted Average

I am trying to figure out an equation from the following paper by Cadena and Kovak (2016): http://pubs.aeaweb.org/doi/pdfplus/10.1257/app.20140095 on pages 264 and 265. I don't think the context is ...
4
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0answers
64 views

Hausdorff (fractal) Dimension of a Stochastic Process

It is well known that Brownian motion (BM) has a Hausdorff dimension (ie fractal dimension) of 2, for topological dimension >= 2. In other words, BM always "behaves like" a plane surface, no matter ...
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120 views

Does clustering lead to overdispersion?

TL;DR Clustering is often cited as a source of overdispersion in count data. However, I seem to arrive at the conclusion that clustering actually reduces the dispersion. Could someone confirm this ...
4
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0answers
46 views

Data Transformation Question - Multiplying data proportional to demographics

I have a bunch of data that is tied to demographic variables (Age, Sex, Income, Education, etc.). However, the data is sent by one person in a household for the entire house. It's numerical data and I ...
4
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0answers
99 views

Cox's Theorem: the necessity of (un)countably additivity

I've been trying to understand Cox's Theorem and the problems surrounding it. There's so much information on this topic that I've become confused as to the exact state of the theorem. I've gathered ...
4
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0answers
187 views

Find pdf of a multivariate function with log-normal distributions

Consider the following function: $f(x,t) = \dfrac{1}{2} \dfrac{1}{\sqrt{\pi(d_0+\alpha v) t}}\exp\left(-\dfrac{x^2}{4(d_0+\alpha v)t}-\lambda t\right)$ where the parameters $d_0,\alpha,\lambda$ are ...
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0answers
605 views

Kurtosis, bias, unbiased and statistics

I apologize ahead of time if this is too vague or meta to be a valid question. I've been looking at Algorithms (Sedgewick & Wayne). They define a class stdstats. In that they define min, max, ...
4
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119 views

Truncated trivariate normal - conditional expectation

I am working on a paper in which I'd need to use the two following conditional expectations: $E(X_{1}|a \leq X_{2} \leq b)$ $E(X_{1}|a \leq X_{2} \leq b, a \leq X_{3} \leq b)$ where $X_{1}, X_{2}, ...
4
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0answers
370 views

Conditional expectation everywhere non-zero while unconditional one zero?

I have a real-valued random variable $X$ that takes on positive and negative values, and $$E(X)=0 \tag{A}$$ There is also another real-valued random variable $Y$, not independent from $X$, neither $...
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0answers
2k views

Deriving the maximum likelihood for a generative classification model for K classes

In Christopher Bishop's book "Pattern Recognition and Machine learning", there is the following question: Consider a generative classification model for $K$ classes defined by the prior class ...
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228 views

What properties of a likelihood function are required for quasi-likelihood estimation?

Quasi-likelihood seems like a great way to use Iteratively Weighted Least Squares to fit linear models with a very general class of likelihoods. But what is that class? Obviously the distribution ...
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218 views

Duda, Hart, Stork No Free Lunch Discussion

Please see this question regarding Duda, Hart, and Stork's No Free Lunch Theoremm Discussion Hi all, I was having trouble understanding the description of the NFL theorem in Duda, Hart, and Stork. My ...
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0answers
280 views

The distribution of the scalar product of multivariate complex normal vectors

Assume two independent random complex vectors (with real and imaginary parts): $$ \vec{\dot{Z}}=(\dot{z}_1,\dot{z}_2,\ldots,\dot{z}_n)~~~~~~~~~~ \begin{cases} \Re(\dot{z}_i)&=x_i \\ \Im(\dot{z}_i)...
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0answers
269 views

Likelihood ratio test: $f(x)=2x$ vs $f(x)=3x^2$: $2n$ degrees of freedom?

Suppose $X_1, . . . , X_n$ are i.i.d. with pdf $f(·)$. We want to test the hypotheses \begin{align} H_0 &: f(x) = 2x , \;\, \text{ for } 0 \le x \le 1, \text{ against}, \\ H_1 &: f(x) = 3x^2 , ...