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Questions tagged [mathematical-statistics]

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

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Questions about confidence interval and presenting metrics for classification problem

A friend of mine and I are working on our bachelor's thesis and we're getting close to its completion. We have a few disagreements about how to present the result metrics for the experiment that we're ...
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1answer
28 views

How to calculated a probability for a sum of integers to be equal to a given value if probabilities of addends are known?

Let's assume that we have a process generating some integer numbers with different probabilities. The set of possible integers is small. For example: -2, -1, 0, 1 and 2. We also know probabilities for ...
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How to compare two small sample proportions using R?

All the data is in the attachment below& The question is: Do these data provide convincing evidence that there is a difference in how good people are at recognising the backs and the palms of ...
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Verify that data have property $X$ with hypothesis testing

In data analysis, one usually need to verify that data have property $X$ before applying method $Y$, which takes $X$ as a prerequisite. To illustrate, possible values of $(X, Y)$ include $(\text{...
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Estimate frequency based on given sample

I have a list of users (~25 million) and a dataset containing events created by the users in a specific time window (eg all tweets of all Twitter users that happened in a month - on avg I have 100K ...
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13 views

Time Series: ACF and PACF plot, how to tell what's the best model by looking at the plot?

The question ask me "The equation of the model you think is most appropriate, given the plots. Justify your choice of model. " the ACF plot looks like cut off after the first lag. And I'm not too sure ...
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11 views

T-test with conditional data

I have some data giving the vital signs of people before and after one experiment. Also for each single measurement I have some informations about the subject that could affect the results and need to ...
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1answer
56 views

“A property holds if it cannot be rejected”

Most examples in my Time Series Analysis slides and Multivariate Analysis textbook (Applied Multivariate Statistical Analysis, 6th Edition) conduct a hypothesis testing on data's normality, or zero ...
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Why does this pattern in powers happen? [migrated]

Hopefully this not to frivolous to ask, I have wondered it for years. There is a strange pattern in the differences between various power. For example if you square the numbers from one to ten the ...
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Question about a new type of confidence interval [duplicate]

I came up with the following result, tested on many data sets, but I do not have a formal proof yet: Theorem: The width $L$ of any confidence interval is asymptotically equal (as n tends to infinity) ...
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2answers
132 views

What is the distribution of the difference of two iid noncentral Student t variates

Let $X_1$ and $X_2$ be iid non-central t random variables. I'm interested in the question: what is the distribution of $X_1 - X_2$? i.e. what is the distribution of the difference of two iid ...
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2answers
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Score function of poisson distribution

I have a stupid question I haven't figured out. So when counting the score for poisson distribution I get the log likelihood $$S(\mu ) = \frac{\partial \ell(\lambda )}{\partial \lambda } = \sum_1^n \...
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Conflicting unilateral and bilateral tests. Which one should I base myself on?

Suppose I have $H_0: \beta = 0$ vs $H_1: \beta \neq 0$. My data tells me that I don't reject the null, with a p-value of approximately 0.06. Then out of curiosity, since by the scientific theory one ...
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1answer
31 views

Confused about conditional counterparts to traditional probability laws

I'm self-studying probability and have seen the following in various readings. The "conditional counterpart" $$P(x,y|\theta) = P(x|y,\theta)P(y|\theta)$$ to the traditional conditional probability ...
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2answers
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What's an example of an estimator that is $o_p(n^{-1/2})$?

For $X_i$ i.i.d. normal with mean $\mu < \infty$ and variance 1, by the law of large numbers, we have $\bar{X} \overset{\mathcal{P}}{\rightarrow}\mu$, i.e. $\bar{X} - \mu = o_p(1)$. Similarly, by ...
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Mysteries of the Normal Distribution [duplicate]

I have been studying ML for over a year and am actually a Bachelors of Statistics myself and am sick of not knowing the beauty of the Gaussian distribution and why it is so prevalent in nature. I've ...
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Prior distribution on a transform parameter [duplicate]

Let say that my likelihood has the following form: $[logit^{-1}\{\theta_1 + \theta_2\log(x_i)\}]^{y_i}[1-logit^{-1}\{\theta_1 + \theta_2\log(x_i)\}]^{n_i-y_i}$ where $\theta_1$ and $\theta_2$ are my ...
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1answer
48 views

Moment Generating Function of Beta ( Taylor series)

Suppose X is a random variable with a Beta ( a =$\frac{1}{2}$ , b=1) distribution and x in (0,1) Then the moment generating function is calculated as below $ M_X(t) $ = $\mathbb{E}[e^{tX}]$ =$ \...
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An elementary proof of the equivalence of measure theoretic and density expected values

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $X\colon \Omega\to\mathbb{R}$ be real-valued and measurable. Suppose there exists $f\colon \mathbb{R}\to [0,\infty]$ such that $P(X\in A)=\...
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2answers
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What's $E[X - E[X | Y] |Y]$

I'm trying to figure out what $E[X - E[X | Y] |Y]$ equals to. Assuming that $X$ and $Y$ are continuous random variables, I thought that $E[X - E[X | Y] |Y] = E[X |Y] - E[E[X|Y]|Y] = E[X|Y] - YE[X|Y]$ ...
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Standard error for difference between two independent gains

I've calculated an effect size for the difference between two standardized pre-post differences of two independent groups of subjects. That is: $[Mean_{dif_{group~1}}/SD_{dif_{group~1}}]-[Mean_{dif_{...
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1answer
33 views

Understanding p value

Hi i have seen few videos and read some articles as well but i am still confused, i'll trying quoting the example i saw. Helen sells chocolate nutties which claims to have 70gm or more peanuts in a ...
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15 views

How to find the limit in probability of certain distribution?

I was working on finding a limit in probability of Normal, Exponential, and Uniform distribution. 1st Problem I figured the answer is And here's Exponential distribution problem. I figured the ...
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Bernouilli variables - bias and variance of estimator

Reading through this I work on Example 1 in 3. Consistency. $X_{1},... , X_{n} ∼ Bernoulli(p)$. The mle $\hat{p}$ has bias 0 and variance $p(1−p)/n \rightarrow 0$. Here $\hat{p} = \sum_{i} Xi/n$. ...
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1answer
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Bayes Minimax Estimation

Let $S\sim B(n,\theta), l(\theta, a) = (\theta - a)^2,\delta=\bar{X} = S/n,$ and $$\delta^*(S)=\left(S+\frac{1}{2}\sqrt{n}\right)/\left(n+\sqrt{n}\right)$$ where $B(n,\theta)$ is Bernoulli ...
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How to know which value in a feature is significant

this question might be silly, but how do I know which category (value) within a feature is significant? Say I have some exam result data of students from four different countries: How do I know ...
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Need help to formulate correctly a statistical problem [closed]

Assume that we want to test whether a meal contains olive oil, spices and cheese. We assume that a meal can just contain one of them. So, we do some tests each day. Consider the following statement ...
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What's the difference between loadings from partial least squares (PLS) regression and beta coefficients from multiple linear regression?

I have a set of independent variables (X1, X2, ..., X10) and I have run a PLS to find a combination of the X1, X2, ..., X10 that best predicts an outcome Y (a single-variable outcome). As a result, I ...
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Linear Discriminant Analysis coefficients - ex. 4.6 Elements Statistical Learning

From https://web.stanford.edu/~hastie/Papers/ESLII.pdf exercice 4.6. In (c) we are to show that least-square regression coefficient is identical to the LDA coefficient, up to a scalar multiple. What ...
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Logistic regression - residual sum of square

In Elements of Statistical Learning https://web.stanford.edu/~hastie/Papers/ESLII.pdf Section 4.4.3 about Logistic Regression infers from the Newton-Raphson procedure for fitting linear logistic ...
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How to calculate the percentage of values that lie within some range of Standard Deviation around the Mean in a Normal Distribution Curve) [duplicate]

This wiki article on 68–95–99.7 rule states - In a Bell Curve (i.e Normal Distribution Curve), 68.27%, 95.45% and 99.73% of the values lie within one, two and three standard deviations of the mean, ...
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How to compare or evaluate different ratings

First of all, I would like to state that English is not my first language so apologize me if I use the wrong terminologies. In our company we collect reviews for our branches, so after each client's ...
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1answer
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How do you find the most probable state path given an observed sequence with infinite emissions values in MATLAB?

I have created code to output observed values of a Hidden Markov Model. I see that 'hmmviterbi' would give you the most probable state sequence given an observed sequence, transmission matrix, and ...
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1answer
23 views

How close to the critical value to be accsepted? [closed]

What if this number 12.32056265 is so close to the 0.05 critical value which is 12.59. Can it still be a significant value with a positive correlation? Or is it then with the critical value of 0.10 ...
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1answer
35 views

Relation between two statistical measures

Consider a random variable $X$ with the associated density function $f_X(x)$ and "zero" mean. Define the following quantities: (1) $E[X^2] := \int_{-\infty}^{+\infty} x^2 f_X(x) dx$ (2) $E[ |X| ] :...
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How to derive a formula based on hierarchy

I have a hierarchy as follows. I want to rank the leaf nodes (i.e. 2, 3, 4) by comparing its position relativly to the two nodes 1 and 5. For instance, node 4 should be ranked highest as it is ...
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What is the most common number appearing at the units digit of the scores in NBA game?

In the lottery market of NBA, the boss will divide the numbers(0-9)to 5 groups: 0and5, 1and6, 2and7, 3and8, 4and9, and the five groups will be given the same odds. How and why does the host of the ...
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1answer
49 views

MLE of Weibull hazard and lognormal frailty in R

Consider a Gaussian random variable log(U), with mean mu and variance sigma^2. How can the parameter estimates of the corresponding log-normal frailty model (i.e the frailty random variable is U which ...
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26 views

Population vs Sampling Frame vs Sample

Could someone please explain how the sampling frame is different from population and sample? I understand that the population is all the sampling unit that match our criteria for the study. And the ...
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1answer
14 views

On some confusion regarding the autoregressive model and the definition of a statistical model

Citing Wikipedia the stationary AR(1) model (without constant trend parameter) is defined as $$ \begin{aligned} y_{t} &= + \beta y_{t-1} + \epsilon_{t}, \\ \epsilon_{t} &\stackrel{iid}{\sim}...
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What is the expected value of overlapped genes fetched from one dataset? [duplicate]

I have a simple question about calculating the expected count of overlapped genes from one dataset. I have one dataset containing 7000 genes, and I randomly fetch two subsets from it. The gene counts ...
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1answer
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Finding optimal subspace for Linear Discriminant Analysis - Elements of Statistical Learning 4.3.3

Linear Discriminant Analysis (LDA) possibly operates a dimension reduction. Section 4.3.3 in Elements of Statistical Learning explicits this notion as well as a method for computing the "optimal ...
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1answer
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If $X\sim\mathcal{N}(\mu = 1,\sigma = 4)$ find $\textbf{P}(X^2 - 2X \leq 9)$

If $X\sim\mathcal{N}(\mu = 1,\sigma = 4)$ find $\textbf{P}(X^2 - 2X \leq 9)$. I understand how to find the pdf of $X$, but I'm not sure how that would work for a function of $X$ like $X^2 - 2X \leq 9$...
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is there any other use cases where correlation is applicable besides bivariate normal distribution?

per this professor, correlation is applicable for bivariate normal distribution. besides bivariate normal distribution, is there any other use cases such as other bivariate distribution where ...
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1answer
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Computation of LDA in Elements of Statistical Learning 4.3.2

Elements of Statistical Learning 4.3.2 elaborates on computation for Linear Discriminant Analysis. https://web.stanford.edu/~hastie/Papers/ESLII.pdf Procedure is said to be • Sphere the data with ...
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1answer
30 views

Rao-Blackwell for Minimum-Variance Unbiased Estimator

Let $X$ be an observation from a distribution with probability mass function:$f(x;\theta) = \left(\frac{\theta}{2}\right)^{|x|}(1-\theta)^{1-|x|}I_{\{-1,0,1\}}(x), \, \theta \in (0,1).$ Use Rao-...
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EM Algorithm for Poisson Gamma

I would like to check if I have done this question right. I am trying to derive the EM algorithm for $\mu$ in the following distributions: $$f(Y|Z) = \frac{z^y}{y!} e^{-z} $$ $$f(Z) = \frac{\theta^\...
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1answer
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Linear Discriminant Analysis - deriving classifier expression for multivariate normal distribution

From Elements of Statistical Learning chap 4 - https://web.stanford.edu/~hastie/Papers/ESLII.pdf We have K classes and we are modeling the posterior probability : $P(G=k|X=x)=\frac{f_{k}(x)\pi_{k}}{\...
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1answer
67 views
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Difficulty with problem 3.11(d) in An Introduction to Statistical Learning (ISLR) (simplifying an equation)

I'm working through An Introduction to Statistical Learning and am getting stuck on question 3.11(d) from the third chapter. In short: For the regression of Y onto X without an intercept, the t-...