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Questions tagged [mathematical-statistics]

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

1,474 questions with no upvoted or accepted answers
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44 views

Showing that $(l'\Sigma l)\chi_p^2 = (l'S l)\frac{(n-1)p}{n-p} F_{p,n-p}$

Let $X_1,\cdots X_n$ be i.i.d. $N_p(\mu,\Sigma)$. I have that when $\mu$ and $\Sigma$ are unknown that the Scheffe type method gives $(1-\alpha)100$% confidence intervals for all linear combinations $...
2
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1answer
63 views

Calculating portfolio volatility from portfolio returns vs. from covariance matrix

I'm having trouble understanding the difference in calculating portfolio volatility via the portfolio returns vs. via the covariance matrix. To be more specific: I understand that on the individual ...
2
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1answer
42 views

Using a generalised linear model for tennis winning probability

I have a dataset of historical tennis matches, one row per match. Each row has the ELO points of each player, and a calculated ELO difference. And of course a column to indicate if Player 1 won as a 1 ...
2
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48 views

Incorrect computation in Knight and Fu (2000)?

I'm currently reading Knight and Fu's 2000 paper on the asymptotics of "Bridge" estimators with a particular focus on LASSO as a special case. In the proof of theorem 2, they make the claim that under ...
2
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2answers
34 views

Which tests can be used for variables that are non-normal, but have homogenous variances?

I am trying to compare the effect of two treatments (planting distance) on the growth of plants (multiple species), using the variable growth rate in diameter for multiple years. I have 4 sets of ...
2
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50 views

Is this a proper use of the Karlin-Rubin UMP test theorem?

For iid $X_1,...,X_n$ and the unknown parameter $\theta>1$, suppose that the likelihood function of a particular sample is given by: $$L(x;\theta)=log(\theta)^n\theta^{{n-\sum_{{i=1}}^nx_i}} I(x_{(...
2
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1answer
53 views

Calculating CI's using bootstrapping on the holdout test dataset

I’m trying to calculate 95% confidence intervals for the sensitivity and specificity of a decision model that I’m building. I’ve split my dataset into 90/10 train and test sets. I’ve used the 90% ...
2
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1answer
67 views

Finding expression of $n$-th derivative, when $n$ is large

For completeness, assume $C$ is an Archimedean copula with some generator function $\varphi$, which is usually assumed to have nice properties. It is known that $$ C(u_1, u_2, \ldots, u_n)=\varphi^{-1}...
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34 views

Finding the limit of a quotient of hazard functions

Let $\lambda_i(t),S_i(t)$ be the hazard and survival functions of two populations for $i=1,2$ and satisfy that: $\frac{S_2(t)}{1-S_2(t)}=\phi\frac{S_1(t)}{1-S_1(t)}$ (1) I want to proof that $\lim_{t\...
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38 views

Observational Data and Bias - A real problem

I'm hoping you all can provide some guidance. I'm working a problem with the following objectives and data set. I would like to be able to predict, for each unit, at each sampled moment, the expected ...
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30 views

Simulation - problem of maximization inside a circle

I am doing some projects related to statistics simulation using R based on "Introduction to Scientific Programming and Simulation Using R". In the Students projects session (chapter 24), I am doing ...
2
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176 views

Chi-square origin of the name

What are the origins of the names and letters in these distributions: What is the origin of the name in chi-square distribution $\chi_k^2$? And the origin of $t$ in student's $t$-test? And naming $...
2
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34 views

Finding complete statistic from a single sample

How do I find a complete statistic for the sample of size 1 of the following random variable: $f(x; \theta) = \frac{2x}{\theta}I_{[0, \theta]}(x) + \frac{2(1-x)}{1-\theta}I_{[\theta, 1]}(x)$ ? ...
2
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51 views

Bias variance tradeoff when the estimated target function is also random

I'm interested to understand "bias variance tradeoff" notion in a different setting than usually presented. In a setting where target $f$ (see the map $f$ below) is a random map rather than ...
2
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68 views

How to obtain the posterior distribution of a given problem?

Problem: Compute the conditional distribution of a random variable $X$ given $Y$. If a random variable $X$ is Bernoulli distributed with probability $q$ for $X = 0$. The conditional ...
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83 views

Expectation of the minimum of dependent random variables

How do we compute the expectation of the minimum of dependent random variables? In other words, what is the value of $\mathbb{E}[Y]$ in the following case: $$ \mathbb{E}[Y]= \mathbb{E}\big[\min(X_1,\ ...
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79 views

How to interpret forestplot with pymc on standard devisions of two groups

I'm using pyMC3 to do Bayesian estimation supersedes the t test (BEST) and I was wondering how to actually interpret this result. I see both groups have significantly different stds because the bar ...
2
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17 views

Conditional correlation, copula, portfolio optimization and diversification

I have a data set which consists of > 500 hedge funds, their historical monthly returns, and their benchmark (index) monthly returns. The number of data points (# of monthly returns) differs from a ...
2
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29 views

Generalization of median, mean for L_p spaces for p different than 1,2

I find it fascinating that the mean and median both minimizing the a measure of error of a point estimate. The median $m_1$ is any (non-unique) $m \in \mathbb R$ which minimizes the $L_1$ norm $\int ...
2
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95 views

Convergence in probability of a multinomial sample correlation coefficient

This problem is from a Ph.D Qualifying Exam on mathematical statistics(also related to probability theory). Let $(X_1,\cdots,X_k)$ be a random vector with multinomial distribution of $n$ trials and ...
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230 views

Why does the variance of a Brownian motion increase linearly with time?

Brownian motion is said to follow a path where each value is normally distributed with mean $\mu t$ and variance $\sigma^2 t$. What is the basis for the relation that variance varies directly ...
2
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50 views

Generalizing on-policy distribution of states

The box on page 199 of Reinforcement Learning: An Introduction, Second Edition talks about distribution of states in on-policy episodic settings. I want to generalize it for continuing tasks as is ...
2
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30 views

how to calculate curvature of multivariate function?

EDIT1: tried to clarify the question Context In the context of an MCMC investigation of non-linear interaction effects in dichotomous models, I am creating data generating processes based on the ...
2
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34 views

How to Estimate a Multi-variable Harmonic Function on a Grid?

What estimation schemes do you suggest for solving the following discrete problem: $$y=f(X)+\epsilon,\\$$ $$\Delta f=0.$$ Here, $X=(x_1,\cdots,x_p)\in\mathbb{R}^{p}$ and $\Delta=\sum_{i=1}^p \frac{\...
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98 views

Sufficiency and completeness of distribution

Let $X=(X_1,...,X_n)$ be drawn from the distribution with pmf $p(x_1,...,x_n)\propto \begin{cases} 1/ {\theta\choose n} & \text{if all } x_i \text{ are different and }1 \le\max(x)\le\theta \\ ...
2
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18 views

How to derive a recursive version of a regularised cost function

I am to derive a recursive version of the following cost function and examine for which choice of D can we have a estimator windup $V(\theta) = \frac{1}{2}\sum_{t=1}^n(y(t)-\phi(t)^T\theta)^2 + \...
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105 views

Percentage of total variation explained in a VAR model

I was studying Campbell, Chen, Viceira (2003) https://dash.harvard.edu/bitstream/handle/1/3163263/campbellnber_assetallocation.pdf?sequence=2 I cannot really understand how they decompose the ...
2
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121 views

Simulation of a random variable given the moment generating function after exponential tilt

The random variable $S$ follows a distribution with moment generating function $$M_S(v)=\frac{\beta\mu v}{1+(1+\beta)\mu v-M_X(v)}$$ I have been looking in some books about this m.g.f and I found ...
2
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244 views

Complete and Sufficient Statistics from Two Samples

Let $X_1,...,X_n$ be a random sample of size $n$ from $N(\epsilon,\sigma^2)$ and let $Y_1,...,Y_m$ be a random sample of size $m$ from $N(\eta,2\sigma^2)$. Find a vector of complete and ...
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45 views

Uniformly minimum variance unbiased estimator

While i was doing some problems, i have encountered this question and i am really stuck. question is the following Suppose that $X_1, \cdots \cdots, X_n$ IID and $X \sim unif\{1, \cdots \cdots , \...
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24 views

Summation of Series involving Exponential terms

I'm currently working on a problem, which involves Poisson-Binomial Distribution. https://en.wikipedia.org/wiki/Poisson_binomial_distribution . The Mean of PBD is given by $M=\sum_{i=1}^{n}p_i$ ....
2
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293 views

Does the definition of regular estimator depend on the rate of convergence? If not, should it?

The definition of regular estimator in my lecture notes is: Let $X_1^{(n)}, \dots, X_n^{(n)} \overset{iid}{\sim} P_n \sim \mathcal{P}(\Theta)$ where $\mathcal{P}(\Theta)$ is a regular parametric ...
2
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157 views

Central limit theorem backwards

Does the fact that some quantity in nature is normally distributed necessarily imply that the quantity can be meaningfully expressed as a sum of smaller iid components (e.g. IQ is a sum of small ...
2
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29 views

In paper “Neighbourhood Components Analysis”, how to determine the optimal number of neighbours (K)?

Recently, I am reading the paper "Neighbourhood Components Analysis" (Goldberger, Jacob, et al. "Neighbourhood components analysis." Advances in neural information processing systems. 2005.). At the ...
2
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86 views

Identity relating Bayes risk to bias

Let $X\sim f(x\mid \theta)$ and $\theta\sim \pi(\theta)$. The Bayes estimator under squared error loss is $\delta^\pi(x)=E(\theta\mid x)$. How do i show that the Bayes risk of $\delta^\pi$ can be ...
2
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204 views

Does minimum norm solution guarantee generalization in the underconstrained case (in the statistical learning sense)?

Recall that pseudo-inverse can be characterized as follows: Solve $$ \| w \|^2 $$ subject to: $$ Xw = y $$ thus it is plausible since its a constrained optimization problem that the solution ...
2
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25 views

Explicit form of the frobenuis norm of difference between the full matrix and its best rank approximation.

Suppose I have a Gaussian process regression $Y(s)= x(s)\beta +w(s) + \epsilon (s) $ where $w(s)$ is the spatial process and $\epsilon(s)$ is the nugget effect. I have a bit of difficulties ...
2
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1answer
147 views

Covariance of order statistics convergence?

Suppose I have a sample $(X_1 \dots X_n)$ and $(Y_1 \dots Y_n)$, all of which are $N(0,1)$ random variables. I am interested in the asymptotic behaviour of $$\frac{1}{n} \sum_{i=0}^n X_{(i)}Y_{(i)} $$...
2
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1answer
89 views

Why does conditional expectation have this property for independent random variables?

For a reference, please see pp. 53-54 of Boucheron, Lugosi, Massart, Concentration Inequalities: A Nonasymptotic Theory of Independence. Let $f: \mathcal{X}^n \to \mathbb{R}$ be a measurable function (...
2
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1answer
47 views

Optimal wager strategy for high win chance / high odds game

In a rigged game of chance where statistically I am expected to win, is there a difference in terms of expected profits, risk, etc - between betting the same amount of money each time (for example ...
2
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250 views

Derive Expectation/Variance of Histogram

Derive the expectation and variance of the following: $\frac{1}{n(b-a)}$$\sum_{i=1}^n 1$$_{Y_i∈(a,b]}$, b>a. My thoughts with this are to put the summation in an integral and then multiply by n. ...
2
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52 views

Tweedie index parameters are restricted: why?

The Tweedie distribution have variance like var(y) = $\phi \mu^p$ for any real p not between 0 and 1. I read in many places (even wikipedia) that p can take any real value except between 0 and 1. ...
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101 views

How to evaluate the bias-variance tradeoff?

How can you evaluate the bias-variance tradeoff by looking at the train error and at the test error?
2
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68 views

Parametric family problems

I came across such a problem that I cannot solve: Let $\mathcal{P} = \{\mathbb{P}_\theta : \theta \in \mathbb{R}\}$ be a parametric family over $\{0,1\} \times \mathbb{R}$ defined in the following ...
2
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0answers
163 views

How can I interpret the result of the 3D two-sample Kolmogorov-Smirnov test computed in R?

I would like to check if the two three-dimensional samples could come from the same distributions. In my data two dimensions are coordinates and third dimension is distances. In K-S test we have null ...
2
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0answers
121 views

How can I get the cumulant expression from the recursive relation between cumulant and moment?

I am reading some paper about high-order statistics https://link.springer.com/article/10.1007%2Fs11004-009-9258-9?LI=true. The paper gives two recursive expressions relating the multivariate cumulants ...
2
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0answers
37 views

How to see if time series terms are independent

I've already read this question, however I found it too theoretical for me, since I haven't studied time series yet. I have a time series that I have gathered from some measurements. At the current ...
2
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0answers
30 views

Is it bad practice to shift traffic distributions for test experiences during the run of a test?

I have recently come across a testing design that Adobe uses in their "Adobe Target" tool that they call "Auto-Allocation". Here is their article on the test design. https://marketing.adobe.com/...
2
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44 views

Proof help: Coincidences in higher dimensions

Background I recently watched a 2014 Talk by Geoffrey Hinton (a key researcher in Machine Learning literature) where he discusses the concepts behind the recently published Capsule Networks. In the ...
2
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0answers
38 views

How to model data with unequal size of dependent and independent variables?

I have a machine process which takes 6 steps to produce a product. Steps 1 -5 are the machine doing some preparation inside the machine, no real production here. Step 6 is the real production step ...