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

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Asking help with Taylor approximation of expectation of ratio

I am trying to understand how I should approach the problem of a Taylor approximation to the expectation of the ratio of two random variables. In my particular problem I am concerned with the ...
2
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17 views

Bayesian importance sampling as an answer to a “paradox” by Wasserman

Both in his book and on his blog, Larry Wasserman has discussed an example in which naive application of the Bayesian methods gives nonsensical answers. Intro The problem is to estimate the ...
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1answer
30 views

Is statistics is the same as mathematical modeling or is it about sophisticated guessing of a pattern between the data? [duplicate]

"Is statistics is the same as mathematical modeling" or "is statistics about sophisticated guessing of the relationships between the data" or "is it meant to create a pattern between the data"? Which ...
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2answers
77 views

What is a statistical book that focus on showing that statistical test work?

What is a book that tells how can we evaluate how well hypothesis tests work, and and compare them? For example: 1)To investigate the significance of the difference between a frequency distribution ...
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0answers
29 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}, ...
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1answer
20 views

How do I split a normal distributed sample into groups of percentiles but with an additional random noise component for uncertainty?

I have a sample of students that I want to divide into smaller groups based on a their IQ but with a certain random noise component - how can I do that? I need to cluster the best, the average and ...
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0answers
14 views

Who can explain this formula for Stein's Unbiased Risk Estimate to me?

Please can any one explain this formula to me, because I don't have enough information about statistic. where d is the length of the signal. This equation represents Stein's Unbiased Risk ...
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38 views

Expectation Maximization clarification questions

I know there are many questions related to EM, but I still could not find the answer I seek. I have learned Expectation Maximization from this turtorial: Andrew Ng. The lecture is easy to follow. ...
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20 views

Conjugate for a special normal distribution

As we know, when the Poisson rate $\lambda$ is large, we can approximate the Poisson distribution as a normal distribution with equal mean and variance. Is there a conjugate family for this normal ...
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1answer
32 views

Is Representer theorem valid with constraints on coefficients?

By the representer theorem, we have that in a Reproducing Kernel Hilbert Space the function being learnt in a regularizer + loss function problem under some conditions, can be represented as $\sum_i ...
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6 views

Hypothesis Function of Linear Regression with Multiple Features

Consider the House Price Prediction (Single Feature) Input Variable = x (Size) ...
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23 views

Comparison of 2 data sets with different sample sizes

I have two data sets that have different sample sizes and I am unsure on how to compare them. Dataset 1 has 51 participants but dataset 2 has only 20 participants. The answers to the questions are ...
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1answer
26 views

What is the standard way to distinguish the errors associated with sampling and measurement in statistics?

This is probably a very basic, yet not easy, question in applied science. I was just wondering, what are usually the standard ways to deal with it? Any pointers to further references are greatly ...
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1answer
30 views

How to prove the identifiability of a likelihood

Consider the likelihood function for parameter vector ...
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0answers
6 views

Predicting score in the presence of latent variables

Given a dataset with the attributes (hour_of_day, day_of_week, performance) where performance is a function of hour_of_day and day_of_week and several other latent variables. How can a prediction ...
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0answers
9 views

Test proportionality between curves [closed]

Is there a test in statistics which examines if the proportion (or ratio) of 2 similar curves is a constant. For example if $y_{1} = sin(x+1.7) - 1$ and $y_{2} = x^{2}$ for domain $x$ in $[-1, 1]$
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1answer
25 views

Standard Deviation [closed]

Assume that we have 10 BIG boxes and each box contains 1000 SMALL boxes. If I know the standard deviation of 1 SMALL box let's say is 3, and I want to average out over the 1000 SMALL boxes and the 10 ...
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0answers
12 views

Math notation for bayesian hierarchical models with covariance matrix with LKJ priors

I am fitting a simple hierarchical bayesian gaussian model of the form: $$ Y = b_0 + b_1 X_1 + b_2 X_2 + e $$ $$ b_{0:2} \sim N(\theta_{0:2}, \sigma_{0:2}) $$ $$ \sigma_{0:2} \sim Gamma(.001 , .001) ...
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1answer
28 views

Converting a 5-point Likert scale to 0 to 30 range

Hi thanks for viewing my question. In my research I want to adopt a measure from prior work and apply it to a new context. The question related to a 5- point likert-scale, however, further reading ...
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1answer
35 views

Understanding a characterization of minimal sufficient statistics

I have some questions regarding the proof of the theorem below. First we need a definition: A statistic $T$ is minimal sufficient iff $T$ is a function of any other sufficient statistic. That ...
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2answers
39 views

Working with expected value of a matrix

If I have two matrices $C$ and $D$ of the same size. If I know that the expected value of $C$, denoted by $E(C)$, is equal to $D$. So $E(C)=D$. In this case, $E(diag(C))$ will be equal to $diag(D)$, ...
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1answer
52 views

Understanding the use of logarithms in the TF-IDF logarithm

I was reading: https://en.wikipedia.org/wiki/Tf%E2%80%93idf#Definition But I cannot seem to understand exactly why the formula was constructed teh way it is. What I do Understand: iDF should at ...
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1answer
28 views

Does the Dickey-Fuller test for a Random Walk?

Is it valid to say that the Dickey-Fuller test, tests for a random walk? Since the AR(1) process $Y_{t} = \rho Y_{t-1} + e_{t}$ with $\rho = 1$ is the same as the random walk. (Next value is maximum ...
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1answer
31 views

Using Pearson's $R^2$ for model selection

I have a question about using $R^2$ as a "best fit" technique for cross-sectional (not time series) type data... Suppose you have a data set, and you're trying to fit a regression model to it. You ...
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0answers
50 views

Distribution of sum of order statistics

The question is from a problem I am trying to solve in Robert Hogg's introduction to Mathematical Statistics 6th version problem 7.2.9 in page 380. The problem is: We consider a random sample ...
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15 views

how to compare the ratings of human rater and ratings of mechine?

Suppose i show some video (rated by some experts) to some number of subjects. They watch those video and rated. Using experts annotation and non-linear regression i get some predicted rate for each ...
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1answer
31 views

UMP test for geometric distribution at $\alpha=0.05$

Let the pmf of a population $X$ be $P$($X=k$)=$p^k(1-p)$ with $0<p<1$. We take a sample of size $n=1$. The hypothesis $H_0$ says $p\leq0.95$ and $H_1$ is $p>0.95$. Find a level $\alpha=0.05$ ...
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19 views

how can I interpret this notation?

I was wonder if anyone could interpret this in words "Under the Gaussian process prior, these functions depend on the model solely through its predictive mean function ...." Thank you in-advance !
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1answer
42 views

Using percentiles and inter-quartile-range for outlier detection in skewed data

I am analyzing the age of a certain group of people and I want to use percentiles and inter-quartile-range in the data to flag possible outliers. I am getting Q1 - 25th percentile, Q3 - 75th ...
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2answers
80 views

Relationship between correlation and linear dependency

I have two variables whose correlation coefficient $\rho=1$. I know that if two random variables are a linear combination of themselves, it implies $|\rho |=1$. Is the converse also true? How to ...
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0answers
121 views

Fastest way to solve Bayes estimator problem

The below problem is from an old PhD qualifying exam in our department. My own solution below is time-consuming and quite possibly wrong. It also relies on recognizing a less common distribution, so I ...
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0answers
15 views

General definition for higher order co-moment matrix

Is there a general equation or procedure for computing higher-order co-moment matrices (i.e., coskewness matrix, cokurtosis matrix, etc) for a vector of random variables? For example, the covariance ...
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29 views

Convergence in distribution

Say I have n boxes and m balls. The balls have distribution of colours given by F. The size of the boxes varies with distribution G. Each box has t balls and the number of balls is given by the ...
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1answer
71 views

$E[\log(X)]$ when $X$ follows binomial

I am looking for solution and some references for the following expectations: $E[\log(X)]$ when $X$ follows binomial with parameters $n$ and $p$, i.e. $E(X)=np$ $E[\log(a+bX)]$ where ...
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0answers
21 views

determining statistical significance of a mean of a mean

I have done an experiment in which I have 8 repetitions per test subject (for each of several measurements). Of this I took the mean. Then I did this for 8 test subjects in total. So basically I took ...
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3answers
127 views

Path to mathematical statistics: Ideal Textbook and approach for Self Study

I'm fairly mathematically inclined — had 6 semesters of Math in my undergrad — though I'm a bit out of practice and slow with say partial differential equations and path integrals my concepts come ...
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4answers
37 views

Variance of linear combinations of correlated random variables

I understand the proof that $$(aX+bY) = a^2Var(X) +b^2Var(Y) + 2abCov(X,Y), $$ but I don't understand how to prove the generalization to arbitrary linear combinations. Let $a_i$ be scalars for ...
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1answer
24 views

Distribution of value closest to 0

Consider $K$ independent Laplace variables $X_i$ ($1 \leq i \leq K$) with mean 0 and scale $\lambda$. Let $X′$ be the variable taking the value of the Laplace variable whose absolute value is the ...
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3answers
72 views

joint pmf of Y1=X1-X2 and Y2=X1+X2

This is not homework. I am just bothered about question 2.2.1 of Introduction to Mathematical Statistics (Sixth or seventh edition) of Hogg,McKean and Craig. Question for ready reference is: If the ...
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1answer
32 views

Distances between successive points - Statistical Test

I have a set of experimental data which comprises distances between successive points: I also have a set of similar data that has been generated by a model that attempts to recapture the ...
2
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1answer
121 views

How much do probabilists need to know about statistics? [closed]

While I was reading this topic, it looked clear to me that statisticians must know some theory of probability (maybe even a lot of probability, depending on the problem). But the inverse is not so ...
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4answers
1k views

Why does the variance of the Random walk increase?

The random walk that is defined as $Y_{t} = Y_{t-1} + e_t$, where $e_t$ is white noise. Denotes that the current position is the sum of the previous position + an unpredicted term. You can prove that ...
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4answers
684 views

Inverse function of variance

For a given constant number $r$ (e.g. 4), is it possible to find a probability distribution for $X$, so that we have $\mathrm{Var}(X)=r$?
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60 views

Mathematics / Statistics [closed]

Today I was thinking how I'm bad at making statements, I know who does statistical cares more about the practical part, but knowing how to do demonstrations in the theoretical part is important? I ...
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1answer
51 views

What is a good test for comparing frequency counts between two groups?

I am looking for statistical methods used to compare frequency of observations between two groups. I have two geographical locations with data on different soil types present. for example, my data ...
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0answers
19 views

Precision and Recall for recommendation system

I am working on movielens 100K movie data for recommendation system. I divide the data into test and training and calculate the precision and recall. In testing there are more than 10K users chosen ...
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22 views

3 correlated questions: Find the roots of an equation, find the inverse of a function and find the c.d.f. of a function of a random variable

I have 3 correlated questions: Find the roots of an equation, find the inverse of a function and find the c.d.f. of a function of a random variable. The questions are in the picture. Sorry for the bad ...
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1answer
46 views

Expected value of a diagonal

If I have $E[A] = B$, where $E$ is the expected value, $A$ and $B$ are square matrices and $\text{diag}$ refers to the vector of coefficients on its diagonal. In this case, what will be the value of ...
2
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0answers
16 views

Randomly Generating Combinations From Variable Weights

The Question I have a list A of n unique objects. Each object Ai has a variable percentage Pi. I want to create an algorithm that generates a new list B of k objects (k < n/2 and in most cases ...