Questions tagged [mathematical-statistics]

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

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7
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0answers
136 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 $...
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1answer
4k views

pivotal statistic versus distribution free statistic

I was wondering what relations and differences are between pivotal statistic versus distribution free statistic? From Wikipedia a pivotal quantity or pivot is a function of observations and ...
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1answer
6k views

Moment generating function of the inner product of two gaussian random vectors

Can anybody please suggest how I can compute the moment generating function of the inner product of two Gaussian random vectors, each distributed as $\mathcal N(0,\sigma^2)$, independent of each other?...
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2answers
97 views

Why would we want to use average cost as response but not total cost?

Assume I have a data set as follows: ...
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0answers
506 views

monotonicity of power function

Following is Example 4.1.4 from Bickel and Doksum's mathematical statistics book. Given an iid sample $X_1, ..., X_n$. each $X_i$ has a Bernoulli distribution with unknown parameter $\theta$. Given a ...
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0answers
198 views

Expected effect of each single-nucleotide polymorphism (SNP) in a genome-wide association study (GWAS) [closed]

It was mentioned in a genetics class that in a genetic association analyses of a trait with all SNPs, it is possible to compute the expected effect of each SNP with the trait using the correlation ...
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294 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 , ...
4
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1answer
475 views

Orthogonal intersection of linear family and exponential family

I asked the following question in MSE for which I couldn't get any answer yet. I thought this would be a better place for that question. In statistical maniolds $S=\{p_\theta\}$,$\theta=(\theta_1,\...
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2answers
9k views

Why doesn't increasing N guarantee the sample mean will get closer to the population mean?

When sampling from a population, the sample mean will always be closer to the population mean as the sample size increases. Why is this statement wrong? My explanation is that as sample size ...
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3answers
8k views

Connection between Fisher metric and the relative entropy

Can someone prove the following connection between Fisher information metric and the relative entropy (or KL divergence) in a purely mathematical rigorous way? $$D( p(\cdot , a+da) \parallel p(\cdot,...
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69 views

Method of Additional Events for the two-state Markov Process

I'm trying to understand a research paper but am facing a few difficulties, any help would be really appreciated. A three state markov process with transition probabilities for states being: 0 --> ...
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2answers
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Mean of log of cdf

Let $CDF$ be the cumulative distribution function for the standard normal distribution. Let $Z$ be a standard normal random variable. Then $CDF(Z)$ is uniformly distributed on the unit interval, so ...
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834 views

Calculating the n value

I have some practice homework questions. I did the first one I will go over the steps please tell me If I am doing it right. a) As mentioned earlier, it is claimed that 70% of households in Ontario ...
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0answers
869 views

Finding the UMVUE of the variance of a gaussian with mean zero

Given $Z_1, ..., Z_n, \sim\mathcal{N}(0, θ^2), θ>0$. Define $X_i=|Z_i|$ and consider estimation of $\theta$ and $θ^2$ on the basis of the random sample $X_1,...X_n$. Find the uniformly minimum ...
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1answer
109 views

Equations of significance probabilities

Consider a population of independent light bulbs with an exponential lifetime distribution with mean $\mu$. It is claimed that their expected lifetime is 1000 hours. A definition of a 100(1−)% ...
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315 views

Gibbs sampling from full conditionals

I have the following joint density: $p(x_1,x_2,y_1,y_2) \propto \exp\left(−\left(x_1^2+x_2^2+c_1(y_2-y_1)^2+c_2(y_2-y_1)^4\right)\right)$ Can I use Gibbs sampling to sample from that? How can I get ...
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2k views

How to derive the conditional posterior density in hierarchical bayesian models?

I was reading on Gelman's Bayesian Data Analysis - Chapter 5 - Hierarchical model Suppose: data : $y_j$ s parameter: $\theta$ hyperparameter: $\phi$ On page 126, he mentions the analytical ...
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616 views

How to set up a posterior predictive test quantities (Bayesian context) to check for independent Poisson distributions?

Suppose we are given data $y_j \sim \text{Poi}(\lambda)$ and assume $y_j$ are iid. We can assume the prior distribution for $\theta$ follows $\text{Gamma}(\alpha, \beta)$, The posterior ...
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2k views

Autocorrelation of a Bernoulli sequence

I have some doubts about the following exercise: Given the sequence of Bernoulli (0,1/2) independent random variables $x(n)$, $n\ge 1$, determine the average function and autocorrelation (time-delay) ...
2
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1answer
722 views

Realizations of random variable

Can some of you help me with following exercise? Provide a procedure to generate realizations of a random variable with CDF (cumulative distribution function) $Fx(x)$ given by: $Fx(x)=(x+1)/...
12
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1answer
3k views

Reconciling notations for mixed models

I am familiar with notation such as: \begin{align} y_{ij} &= \beta_0 + \beta_i x_{ij} + u_j + e_{ij}\\ &= \beta_{0j} + \beta_i x_{ij} + e_{ij} \end{align} where $\beta_{0j}=\beta_{0}+u_j$, ...
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1answer
2k views

Bounds for the population variance?

Suppose we have i.i.d. samples $x_1$, $\ldots$, $x_n$ for a (potentially non-normal) random variable $X$ with finite moments. We can use these samples to construct an unbiased estimates of the ...
11
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1answer
13k views

Intuitive understanding covariance, cross-covariance, auto-/cross-correliation and power spectrum density

I'm currently studying for my finals in basic statistics for my ECE bachelor. While I think I have the math mostly down, I lack intuitive understanding what the numbers actually mean.(Preamble: I'll ...
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1answer
7k views

Why does the rank of the design matrix X equal the rank of X'X?

Why does the rank of the design matrix $\boldsymbol X$ equal the rank of $\boldsymbol{X'X}$? Is this true in all circumstances? If X is not linearly independent, what would the rank of X'X be?
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1answer
469 views

Measure of dispersion over unordered set

I'm looking for a measure of dispersion, such as standard deviation, that can be used when distributing to an unordered set. Specifically: A bucket distribution assigns a non-negative value to each ...
4
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1answer
2k views

What is the UMVUE for $\sigma^2$ in $\mathcal N(0, \sigma^2 )$?

By using the exponential class factorization theorem, I came up with $Y = \sum (x_i)^2$ to be the complete and sufficient statistics for $\sigma^2$ . Using this sufficient statistic as a condition, ...
4
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1answer
2k views

The formula for covariance in terms of joint cdf [duplicate]

I want to show that $$\newcommand{\cov}{\operatorname{cov}}\newcommand{\d}{\mathrm{d}}\cov(x,y) = \iint (F_{X,Y}(x,y) - F_X(x)F_Y(y))\,\d x\,\d y$$ However, I have no idea how to start. I know that $...
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1answer
106 views

Calculating chance of event over time from sample data

I am doing the following experiment: I have slots which I monitor, the slot can either be occupied / free. The study is per slot, so we are discussing one slot here. Every now and then I receive the ...
4
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1answer
620 views

Multiplying two entropy values

I have seen formulas that either sum or subtract entropy values (eg Information Gain). However, I have not seen a "multiplication/division of entropy values" anywhere. I would like to know if there is ...
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2answers
2k views

How much calculus is necessary to understand maximum likelihood estimation?

I am trying to plan out a study plan for learning MLE. In order to do this I am trying to figure out what is the minimum level of calculus that is necessary to understand MLE. Is it sufficient to ...
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1answer
2k views

Average waiting time

who can help me to resolution of this statistic exercise? below the track: Caio go in a bank,the number of customers ahead him are described by a Poisson random variable of parameter a>0. Calculate ...
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2answers
44k views

Standard deviation of a ratio (percentage change)

I have 2 data sets. The first data set, let's call it $X$ has an average value of ($\bar X$) and standard deviation of ($STD_X$), the second set of data also has the average value of ($\bar Y$) and ...
0
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1answer
56 views

Analysis of a dataset and tool to get the incidence of an event

I have a dataset and I want to do statistical analysis: ...
2
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1answer
86 views

Relation between different ML models

Is there a paper or book that talks about the mathematical relation between different machine learning models - how they are different and how they can (sometimes) be equivalent? For instance ...
10
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1answer
888 views

Suggested books on spatial statistics

What are some of the best books for studying i) variability of univariate and multivariate variables (real, count data) across a spatial domain. ii) sampling a univariate or multivariate variable ...
34
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1answer
15k views

Maximum likelihood estimators for a truncated distribution

Consider $N$ independent samples $S$ obtained from a random variable $X$ that is assumed to follow a truncated distribution (e.g. a truncated normal distribution) of known (finite) minimum and maximum ...
4
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2answers
405 views

Suppose that $R^2=0$ . Does this imply that Y and X are unrelated? [duplicate]

Possible Duplicate: Under what conditions does correlation imply causation? Can somebody illustrate how there can be dependence and zero covariance? Or could there still be a relationship? Is ...
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0answers
70 views

How one defines the expected value of two parameter distribution?

I have been given two discrete distributions $X$ and $Y$ and values $\mathbb{P}(x_i,y_j)$ for every $i,j$. What is the formula to compute for example $\mathbb{E}(X+Y)$ and $\mathbb{E}(XY)$? My lecture ...
7
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1answer
221 views

Evaluate $\lim_{n \to \infty} \sum_{j=0}^{n}{{j+n-1} \choose j}\frac{1}{2^{j+n}}$

Using the Central Limit Theorem , Evaluate $$\lim_{n \to \infty} \sum_{j=0}^{n}{{j+n-1} \choose j}\frac{1}{2^{j+n}}$$ My solution: Let $\{X_n\}$ be a sequence of iid R.V's each having $Geo(\frac{1}{2}...
3
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1answer
784 views

A constant as an admissible estimator

This is a homework question so I would appreciate hints. I believe I have the first part correct, but I fail to see how the second part is different. Assume square error loss, $L(\theta ,a)=(\theta -...
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0answers
222 views

Multiple categorical Variables and Multiple Hierarchical Counts- how to infer the effects?

I have the following categorical/count data : ...
4
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0answers
101 views

Is there any way to define a distance metric given a Hidden Markov Model?

Let's say I've gotten a HMM that describes user search strings for my e-commerce website. Let's also say that I've just received a search string from a customer that doesn't have any search results. ...
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0answers
85 views

Is there a common family of weight functions for which the center is zero?

One interesting formulation of the ordinary least squares model coefficient is as a weighted average slope where the weights are given by the squared differences between the $X$ values. $$ \begin{...
4
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3answers
362 views

Mathematical basis for conditional probability

In order to get more fundamental in my understanding of probability I watched mathematicalmonk's lectures involving $\sigma$-algebras etc. - good. One of my main concerns was to better understand the ...
2
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1answer
485 views

Statistical properties of parameters estimated by method of Lagrange multipliers in R

I have time series data that can be fitted by a constrained non-linear function (non-linear in the parameters to be estimated). I'd like to use the method of Lagrange Multipliers (link), specifically ...
0
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0answers
599 views

statistical derivation of Pearson correlation of two variables obtained from regression model

I have asked a stat question on math.SE which now I believe should be here. However, I repeat it here. As a newcomer, I would appreciate it if you could excuse me for any mistakes or repeating the ...
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1answer
207 views

Random forest like procedure for regression or other statistical models

I'm wondering if there exist methods similar to one used in random forest algorithm - I mean taking simultaneously bootstrap sample and random subset of features, then building statistisal model. Have ...
2
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1answer
192 views

How to combine 3 random variables?

I have a homework assignment that is giving me a hard time on the statistics. Lets say you have 3 stocks, all with n expected return (mean) $\mu = 8\%$, a risk (standard deviation) $\sigma = 16\%$ and ...
9
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1answer
3k views

Central Moments of Symmetric Distributions

I am trying to show that the central moment of a symmetric distribution: $${\bf f}_x{\bf (a+x)} = {\bf f}_x{\bf(a-x)}$$ is zero for odd numbers. So for instance the third central moment $${\bf E[(...
1
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0answers
214 views

Comparing two different leagues of similar but not equal distributions around a standard deviation of error of a prediction from a rating system

This query ties a lot of my interests in rating sports teams together, because as I’ve mentioned before I do a version of the Kenneth Massey method (as per his 1997 thesis http://masseyratings.com/...

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