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

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Average of estimators

Given $X_1,...,X_n$, samples from some unknown distribution, assume that we are given a consistent estimator $\phi_n$ for $\phi$. Is the following true as $n$ tends to infinity? ...
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9 views

Need help establishing a baseline

I'm currently working on a project that involves the computation of a (random) baseline. I want to estimate the mean & variance of a function of a random vector. The function involved doesn't ...
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57 views

Sum of independent binomials is binomial

My question is related to the question Sum of Independent Binomials How is it possible to sum X from 0 to w? I see two potential problems: Since $X\sim$Binomial(n,p), $X$ can maximally be $n$, and ...
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1answer
32 views

How to interpret p values obtained with $\chi^2$-test?

I have the following observed and expected values and I am trying to determine the goodness of fit with $\chi^2$ test. On calculating the p I obtain the value 0.9999742, but how is it possible that ...
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1answer
47 views

Deriving the Ridge Regression $\boldsymbol{\beta}\mid \mathbf{y}$ distribution

Apparently the estimate $\hat{\boldsymbol{\beta}}$ for ridge regression comes up as the mean or mode of the posterior distribution given by $f_{\boldsymbol{\beta}\mid \mathbf{y}}$. This is the ...
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1answer
19 views

Methodology: breaking multi-regression apart

I have to perform a multiple regression where my independent variable is store visits, while the dependents include hour of day, day of week, and others. I need to do this in Excel. Excel limits ...
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18 views

Two-alternative forced choice [on hold]

Suppose that $p[r|+]$ and $p[r|-]$ are both Gaussian functions with means $\langle r \rangle_+$ and $\langle r \rangle_-$ and common variance $\sigma_r^2$. How can I show that $$P[correct] = ...
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34 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 ...
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1answer
17 views

On what basis we are evaluating variance is high or low? [on hold]

I am having a below case where N = 300 , Mean = 67 and SD = 30. With the above data we can say the variance is high because of SD = 30. My question is how we are defining it as high on what basis we ...
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1answer
38 views

What is meant by using a probability distribution to model the outputs for a regression problem?

Often a theoretical text will say something like, 'a probability distribution may be used to model the outputs' or, 'assume a probability distribution such as normal or Lognormal for the outputs'. ...
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21 views

Geometric proof of partical correlation with pythagoras theorem [on hold]

Please help me with my statistics lesson. I need to prove the partical correlation formula with the definition and the pythagoras theorem. The definition (partical correlation): ...
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21 views

R Commander: how to test correlation across all variables? [closed]

If I want to test correlation between two variables I would run the following command in R: ...
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26 views

What does make SVM a “soft computing” method?

Soft computing is defined in [1] by the capability of "operating with uncertain, imprecise and incomplete information in a manner that reflects human thinking". So, based on my limited understanding, ...
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9 views

References for Combine Seasonal Period in Seasonal Time Series

I'm in a middle writing my thesis. I got confused for find the references which said period of seasonal could be chosen from the small period which already containing another periods. Here is my ...
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14 views

Calculate state change probability

I am having telco order management data and need to calculate the probability of each order going through different stages. data is like this: Order No; product_type; time spent in step1; time ...
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37 views

Expectation of a conditional density

I'm trying to figure out why the following equation holds: $$f_{Y}(y) = E(f_{Y|X}(y|X))$$ I have sort of "worked out" the RHS to be: \begin{align} f_{Y}(y) &= E(f_{Y|X}(y|X)) \\[5pt] ...
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1answer
23 views

Algorithm to determine a point in time series data, after which probability of increase in value is very low

I am working with dataset which contains number of movie tickets sold per day. This is basically a count of total number of tickets sold, for a particular movie, for each day after its release date. I ...
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6 views

How to calculate sample size for testing H0: Y is related to X by a sigmoidal function?

I'm assuming I would need to specify a given functional form and conditional standard deviation for Y at each X. That would give me some parameters to be estimated and I would need to set a threshold ...
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1answer
41 views

Interchanging limit and derivative for CDFs

Let $F_{\theta}(x)$ denote a cumulative distribution function indexed by the parameter vector $\theta$. Given this definition is the following equation correct (and if so under which conditions)? ...
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129 views

For least squares estimation, what is the difference between using the estimator $\hat{\beta} = X^{T}Y$ vs $\hat{\beta} = (X^{T}X)^{-1}X^{T}Y$

For least squares estimation, the estimator $\hat{\beta} = X^{T}Y$ is an unbiased estimator while $\hat{\beta} = (X^{T}X)^{-1}X^{T}Y$ is also an unbiased estimator given that $X$ is well-defined. Is ...
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1answer
39 views

Is it possible to determine how many effects I can estimate in a least squares problem just by looking at the correlation matrix?

I currently have a model matrix $X$ with $6$ columns, which is being used for a factorial design problem, with each column associated with an effect. The ultimate goal is to be able to estimate as ...
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3answers
68 views

Express correlation matrix of $X$ in terms of $X^{T}X$ (in the OLS context)

In least squares estimation where $Y = \beta X$, how can we find the correlation matrix of $X$ in terms of $X^{T}X$? It seems that $X^{T}X$ is very close in structure to the correlation matrix, but ...
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Convergence in distribution and degenerate random variable

Let $\{Y_n\}$ be a sequence of random variables with an associated sequence of CDFs $\{F_n\}$ given by : $$F_n(y) = \begin{cases} 0 & \textsf{for}&y <0 \\ (\frac{y}{\theta})^n & ...
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12 views

Distribution of samples from a uniform distribution [duplicate]

Let's say we are taking $n$ samples from a uniform distribution, that spans from $0$ to $1$. According to the central limit theorem, the mean of the $n$ samples will follow a normal distribution with ...
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1answer
51 views

Percentage interpretation of negative values when you can't use log transformation

I have a data set of 5 indicators of the stock market. 2 of the indicators have negative values: e.g. they range from say -50 to 100. After running a regression I would like to be able to compare the ...
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1answer
49 views

data normalization after dimension reduction for classification

The classifier is KNN or RBF-SVM. After doing dimension reduction (e.g., PCA, LDA or KPCA, KLDA), does it need to do normalization before classification? In LIBSVM ...
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15 views

What mean to use in odometer calculation based on model?

Im trying to figure out what would be considered a valid calculation. Imagine I have a dataset that looks like that: ...
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1answer
32 views

On One-to-One Functions of Complete Statistics

Why is a one-to-one function of complete statistic also complete? How might you go about proving this?
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1answer
24 views

On the proof of admissibility of constant estimators under squared loss

The question concerns the discussion in Wasserman, All of Statistics, Section 13.6. He defines: An estimator $\hat{\theta}$ is inadmissible if there exists another rule $\hat{\theta}'$ such that ...
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1answer
54 views

Limiting distribution of $\frac{\sqrt{n}\left(\bar{X_n}-\mu\right)}{\sqrt{\bar{X_n}}}$ from mean of Gamma$\left(\mu,1\right)$?

Given $\bar{X_n}$ is mean of random sample with size $n$ from Gamma distribution with parameter $\alpha=\mu$ and $\beta=1$. I wanna find the limiting distribution of ...
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18 views

What statistical test should I use to compare the binary outcome of two groups (with pre- and post- test)?

I have two groups of people: experiment group and control group. People in each group accepted pre-test and post-test. The outcome of the tests is binary: "good" or "bad". In other words, each people ...
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27 views

Show that weighted least squares estimator for a specific model is not consistent

Here is the background for this problem: $\qquad\qquad\qquad$ $Y_{1},...,Y_{n}$ iid $N(\mu,c^2\mu^2)$, $\,\,$ $c^2$ known. $\,$ The problem is as following: Consider the above model. Define ...
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1answer
10 views

Is there a framework for reinforcement learning with states and actions in the same domain?

In reinforcement learning, there are states, actions, initial states, terminal states, a progress function and a reward function. Is there a theoretical framework or setting where states and actions ...
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26 views

Does the UMVUE have to be a minimal sufficient statistic?

I'm studying point estimation and I have found this question that seems pretty tricky to me. If $T$ is a minimal sufficient statistic for $\theta$ with $E(T) = \tau(\theta)$, can you say that $T$ ...
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7 views

Significant change over time between categories

I am trying to establish whether change over time is significant at an archaeological site. Basically, there are seven different time periods (lets call it A-G) and in each time period there are six ...
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1answer
54 views

How should I calculate the variance of a circular random variable?

Consider the following function being the PDF of a circular random variable (orientation angle from the zenith) ...
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31 views

How to compare PCA with KPCA for dimension reduction?

Both linear principal component analysis (PCA) and kernel principal component analysis (KPCA) are unsupervised dimension reduction methods. I have a dataset with $4000$ training samples and $40000$ ...
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113 views

Pooled Variance for Dependent $t$-test statistic?

I'm looking into finding a way to calculate Cohen's $d$ for correlated samples. Assuming pooled variances, we end up getting $$\text{SE}\left( \Delta \text{ of means}\right) = ...
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1answer
33 views

Random unit vectors in $\mathbb{C}^n$ and $\mathbb{R}^{2n}$

Suppose $\mathbf{u}\in\mathbb{C}^n$ is a complex random vector with circular symmetry, uniformly distributed on the unit complex $n$-sphere, so we have $\|\mathbf{u}\|=1$. In other words, $\mathbf{u}$ ...
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125 views

Marginal normality and joint normality

Let $X$ and $Y$ be two independent standard normally distributed random variables $N(0,1)$ .If we define a new random variable $Z$ such that : $$Z = \begin{cases}X & \text{if} &XY > ...
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1answer
62 views

Compute $E\left[ \Phi \left(X \right) \Phi \left(Y \right) \right]$ for a bivariate normal distribution

Assume that $X$ and $Y$ follow the bivariate normal distribution with correlation coeffcient $\rho > 0$, zero means and scale parameters equal to one. I am looking for an elegant way to compute ...
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1answer
17 views

Derivation of formula for sample size of finite population

I found here the formula for computing the sample size $n$ of a finite population $N$ $$ n = \frac{n_\infty}{1 + \frac{n_\infty - 1}{N}} $$ where the sample size for an infinite population $n_\infty$ ...
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28 views

calculating PMI for co-occurrences of words

I am in the process of building a question answering system. I am interested in calculating the PMI for words $x$ and $y$ occurring within 5 words of each other in a document. I have the formula and ...
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8 views

Count of the biggest bin in histogram, C#, sharp [migrated]

I want to make histogram of my data so, I use histogram class at c# using MathNet.Numerics.Statistics. ...
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0answers
23 views

What are the correct ways to check if a variable is significant?

I've conducted a series of computational experiments varying a set of discrete parameters. I see on the figures that some of the parameters don't affect the result in any consistent way and look like ...
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49 views

$\mathbb{P}(A_1)≤\mathbb{P}(A_1)$ in Boole's inequality ($n=1$) proof?

Why does this proof use $≤$ in the $n=1$ (induction base case) case for Boole's inequality, when in fact it's an equality? That is, why claim, $\mathbb{P}(A_1)≤\mathbb{P}(A_1)$, when it should be a ...
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1answer
20 views

Question on Integrating Over Joint Probability

Let us say that we have a joint probability density denoted as $$P(x_1,x_2,...,x_n)$$ If we are trying to find the probability that every $x_i$ is greater than $0$, is it correct to say that the ...
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1answer
37 views

How to do this transformation $Y= g(x) \sim \mathcal U(0,2)$?

If $X$ is a continuous random variable with probability density function $f_X(x)=2(1-x)$ for $0 < x < 1$, find the transformation $Y=g(X)$ such that the random variable $Y\sim \mathcal U(0,2)$.
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1answer
44 views

Hypothesis testing for the (Pearson) correlation coefficient

I don't understand why we have to assume ρ=0 in to get the probability density function? If I say null hypothesis p is something like 0.3, I can still use the probability density function, can't I?
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1answer
29 views

Boosted Trees: Objective Function clarification

Reading through this overview of boosted trees, I'm having trouble understanding how the second line was derived. $$ Obj(t)=\sum_1^n{loss(y_{i} - \hat{y}_i^{(t)})} + \sum_1^t{\Omega(f_i)} \\ = ...