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Questions tagged [mathematical-statistics]

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

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A question about the definition of sample

Let $(\Omega,\Sigma,\mu)$ be a measure space. The definition of a sample is that it is a sequence of iid random variables $X_1,\ldots,X_n$. Usually random variables are real valued and the ...
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13 views

How to test the validity of Fitt's law in R

I would like to use the Fitt's Law to test some hypothesis. Anyway, it's the first time I use it, and I have some doubts. Then aim of my experiment is to test if in a certain situation the Fitt's law ...
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1answer
30 views

How do you verify independence of a pair of uniform random variables in MATLAB? [closed]

If $A$ is a subset of $R$ and $X$ is a random variable. I have two variables $X_1$ and $X_2$. $I$ being $1$ if $X$ in subset $A$, and $0$ if not in $A$. Let $U$~$U(0;1)$ and determine if this pair ...
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1answer
348 views

How to write diff() mathematically?

I am using R/Python diff() operation. e.g., https://docs.scipy.org/doc/numpy-1.14.0/reference/generated/numpy.diff.html I would like to know if there is a ...
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11 views

Pointwise Mutual Information using spacy or just detailed explaination

So, I have been trying to play around with NLP recently and decided to work on a project involving Emotional Analysis. I have been following this particular research, http://www.cse.yorku.ca/~aan/...
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22 views

How can we call likelihood function a joint PDF when the individual terms do not represent Probabilty

My understanding of the Probabilty Density Function is that they evaluate to 0 at a particular point . So if we have some i.i.d points $x_{n}$ from a Normal distribution and we write : \begin{equation}...
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1answer
98 views

Rao-Blackwell Problem [closed]

Let $X_1, .. X_n$ be iid Bernoulli($\theta$). I want to estimate $\lambda = \theta(1-\theta)$ using $\delta$ (unbiased estimator for $\lambda$) where $\begin{cases} \delta = 1 & X_1 =1 \...
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1answer
33 views

can you help me to do a difference of CDF?

I have 2 CDF's with equal number of points that I want to compare. These are from: Temperature of 1 month from 2012 Mean temperature across months What can I do to obtain this difference, this is ...
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31 views

cumulative distribution functions (CDFs) [duplicate]

i want to know why is important use CDF for this analysis of TMY (typical meteorological year) because i have the data of the month for compare with the long term mean This the example of the manual ...
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1answer
19 views

Creating a custom metric based on diverse feature scalse

I would like to create a custom metric that takes into account multiple features and produces a certain score. However, the issue is that I have data that has many different data types, such as ...
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51 views

Bias Variance Decomposition 2.7 in Elements of Statistical Inference

I try to derive 2.7 from the book. I expose my demonstration $E_\tau[(y_0-\hat{y}_0)^2]=E_\tau[y_0^2]-2E_{\tau}[y_{0}\hat{y_{0}}]+E_{\tau}[\hat{y_{0}}^{2}]$ $= y_{...
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16 views

Encoding quantitative outputs for regression

From Elements of Statistical Learning: For a two-class G, one approach is to denote the binary coded target as Y , and then treat it as a quantitative output. The predictions Yˆ will typically ...
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15 views

How to denote token length in statistics

I have a vocabulary (V) that includes terms. (e.g., V = ['data mining', 'data', 'machine', 'machine learning']) I also have a ...
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32 views

If $X$ is from (canonical) exponential family, do we always know the distribution of $T(X)$?

Assume $X$ are generated by a distribution from exponential family, $$ f(X; \theta) = h(X)\exp\{\eta(\theta)T(X) - b(\theta)\}$$ After solving several exercises with various distribution functions, ...
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1answer
33 views

Asymptotic distribution of median estimator when density doesn't exist

We know that when density(say $f$) exists at the median(say $\theta$) then the median estimator(say $\hat{\theta_n}$) has the following property $$ \sqrt n(\hat{\theta_n}-\theta) \to^d N(0,1/\{4f(\...
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30 views

Why is degree of freedom so important? [duplicate]

As far as I'm concerned, the degree of freedom is simply the number of linear equations need to be satisfied. However, it seems closely related to the statistical deduction. For example Dividing by ...
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49 views

How to derive a ranking function by analysing feature correlations

I am analysing some employee details to find the efficiency of the employees. Ideally I want some rankings to rank them based on these features. My features include; current salary projects ...
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15 views

Hypothesis testing with matrix of measurements

I need to test if a treatment has taken effect in a certain group of patients for which I make a measurement of the relevant variables before doing the treatment and afterwards. Usually, I would do a ...
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0answers
34 views

Proof without words (about statistics or probability)

Are there any well known proofs without words about statistical theorems like for example the CLT, LLN, or the Gauss-Markov theorem? There are many about mathematics in general (e.g. here, or here), ...
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12 views

(Existence part of) Neyman-Pearson via weak-* convergence

I would like a ask whether there is any statistical reference containing the following functional analytic argument for the existence part of Neyman-Pearson: Let $(R, \mathcal{F}, \mu)$ be a measure ...
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1answer
24 views

Why Converting Regression to Ordinal Regression

Intro: Ordinal Regression/Classification is a classification where the labels have orders (https://en.wikipedia.org/wiki/Ordinal_regression) Question: Can you comment what are pros and cons if ...
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1answer
47 views

Regression on Auction Prices, Multiple Prices Randomness

I'm currently building a model to predict internet auction sale prices of products in a marketplace. There are a lot of instances where a product goes for multiple prices but it's basically the same ...
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1answer
57 views

What does the pmf of a discrete random variable look like if it can take on the value $\infty$?

What does the pmf of a discrete random variable look like if it can take on the value $\infty$? Consider a stopping time $\tau$ of a Markov chain, which is a random variable that takes values in the ...
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21 views

Finding Sum of Squares in a multiple regression ANOVA table by hand?

I seem to be having quite a bit of trouble with this problem in an old stats exam. Its basically asking me to fill out an analysis of variance table by hand. My data is given as follows: and we are ...
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covariance for measuring linear relationship, how about the non-linear relationship

I once heard the statement that Covariance is a measure of the linear relationship between two variables, such as X and Y If X and Y has non-linear relationship,...
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87 views

Proof that the mean is a complete sufficient statistic and the sample variance is an ancillary statistic

I have $X_1, X_2, ..., X_n $ that are random samples from the single variate $N(\mu,\sigma^2) $. I want to prove that the mean $\bar{X}$ and the sample variance $s_x ^2 = \frac{1}{(n- 1)} \sum_{i=1}^...
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1answer
26 views

Interpreting the Chi Square for a 2x2 contingency table

Here is my problem. I need to implement an algorithm to find if there is a dependency between 2 categorical variables. My population can vary a lot as it is an input I don't have control over. My ...
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22 views

AICc value How to derive?

initially let me introduce a concept widely used in ARIMA in the following. $AICc = AIC + \frac {2k^2+2k} {n-k-1}$ where n denotes the sample size and k denotes the number of parameters. Thus, AICc is ...
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12 views

What statistical test for multiple groups of paired data

I'm a student performing a research on seeds and their coagulation effectiveness. I have five groups (seeds) and I measured the clarity of water before and after (5 groups, 3 trials total). So as I ...
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11 views

Confidence vs. Count in association rule mining: which one is better?

I am writing a program that mines association rules from a large data set. I have an array of association rules, and I have to decide which ones are more representative of the patterns I am studying. ...
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2answers
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If $F_X(z) > F_Y (z)$ for all $z\in \mathbb{R}$ then $P(X < Y ) > 0$?

I came across this question in a review of an old exam I took. I didn't get the answer correctly then, and I'm struggling to figure the answer out now. Can anyone help me reason through this? ...
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1answer
35 views

How can I prove the following relation between the probabiloity of X and its expectation using Cauchy-Schwarz inequality?

For a random variable $ X \geq 0 $ and $E[X^2] < \infty $, I'm asked to prove the following: $ P(X> 0) \geq \frac{(E[X])^2}{E[X^2]}$ It makes intuitive sense to me that it must be the case, ...
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59 views

Quantile Regression loss function/ check function proof

Can anyone help me to show that this statement is true. I have looked in Koenker's Quantile Regression (2000) and a load of other sources but I cannot find a solution. There seems to be a trick ...
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0answers
26 views

How do I know which GLM family to use?

I have the following question: I need to specify the distribution, link function and linear predictor. I know how to do find the link function and linear predictor if I know the first but don't know ...
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Infer statistical model prediction confidence coefficient based on the past predicted and actual results

Currently, I encountered such as a problem: For example, when I built a statistical model based on the past 3 month data and using the model to predict for the next month. But I found that the ...
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4answers
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If I can make up priors, why can't I make up posteriors?

My question is not meant to be a criticism of Bayesian methods; I am simply trying to understand the Bayesian view. Why is it reasonable to believe we know the distribution of our parameters, but not ...
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23 views

Bernstein’s inequality and trimmed mean (truncated mean)

Give that and for <0 How can we calculate upper bound for My best guess is that we can use Bernstein’s inequality but I'm not sure whether this is right way or not... if so, what will be ...
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2answers
33 views

What is the use of knowing the underlying data distribution?

I am trying to understand few basic concepts of data mining, machine learning, etc. and I am new to this field Say that I have a data sample and I have done maximum likelihood estimation or some ...
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1answer
56 views

Best Course of Study for Data-Science/Statistician Interviews [closed]

This is my first question here, so please pardon my gaffes. I am currently working as a Data-Scientist, a position which I worked up from Junior Analyst position.My bachelors is in Computer Science ...
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1answer
29 views

Expectation of matrix X multiply by indicator matrix

Let X1, . . . , Xn, X be i.i.d. R-valued random variables Suppose an indicator matrix I{A} be 1 if A is true and 0 otherwise. Then for τ > 0, and How this could be true....? Can anyone walk ...
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0answers
22 views

High dimensional linear regression inference

I am reading through high-dimensional literature currently but I got confused. Especially about statistical inference with LASSO, anyone can clarify me the main difference among Van De Geer(2014)ON ...
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0answers
13 views

setup time optimization using statistics [closed]

I work at a manufacturer with numerous products. The diversity if variants is very high. For almost every product the machines have to be converted. There are essentially 3 characteristics, which ...
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1answer
19 views

How do I estimate probability of success with no successes? [duplicate]

My $6$ friends and I tried buying tickets to a popular event. Everyone who wanted a ticket got a random number and if your number is less than or equal the number of tickets available, you can buy a ...
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1answer
39 views

Probability integral transforms - Cauchy distribution of 1/x and X

When revising for exams, I recently came across the following question: Suppose that $X$ is Cauchy distributed, ie has a density function $$f_X(x) = \frac{1}{\pi(1+x^2)}$$ Show that $1/X$ is ...
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2answers
122 views

In k-means clustering, why sum of squared errors (SSE) always decrease per iteration?

In k-means clustering, why sum of squared errors (SSE) always decrease per iteration? How can prove it by mathematical derivation of formulas? k : number of clusters m : number of examples $c_h$ : ...
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1answer
37 views

Can I have some help for Statistics/probability theory

Let $\Omega = \{0,1\}^{\mathbb{N}} = \{\alpha=(\alpha_1,\alpha_2,...):\alpha_i \in \{0,1\}\}$ Fact. There exists a $\sigma$-algebra $\mathcal{F}$ such that for every $\beta = (\beta_1,...,\beta_n)\in\...
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0answers
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Mixed Models of this designs

How will I state the Mixed Model of the following designs: 1. Randomized Block Design 2. Split Plot Design Any help will be appreciated, thanks .
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0answers
21 views

Why sigma-finite measure?

I am learning measure theory and the concept $\sigma$-finite measure makes me a bit confused. Why do we need the $\sigma$-finite assumption in many important theorems?
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1answer
54 views

About Mean absolute error and standard deviation

I've calculate the "Mean absolute error" for my predicted data and I want to calculate the standard deviation of an error to find the +- range from mean absolute error. The question is do i need to ...
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2answers
40 views

How to calculate error when $\sigma=0$ in a test sample? [duplicate]

A test has probability $P_f$ of failure where it is expected that $.0001 <P_f < .01$. Given $N$ test samples where all tests succeed and there were no failures (thus $\sigma=0$), what is the ...