Questions tagged [mathematical-statistics]

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

4,378 questions
27 views

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 ...
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 ...
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 ...
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 ...
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/...
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}...
98 views

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 ...
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 ...
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, ...
33 views

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(\... 0answers 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 ... 0answers 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 ... 0answers 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 ... 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), ... 0answers 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 ... 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 ... 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 ... 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 ... 0answers 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 ... 0answers 9 views 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,... 0answers 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}^... 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 ... 0answers 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 ... 0answers 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 ... 0answers 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. ... 2answers 97 views 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? ... 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, ... 0answers 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 ... 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 ... 0answers 17 views 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 ... 4answers 2k views 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 ... 0answers 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ...
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 $c_h$ : ...