Questions tagged [statistics]

Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory and other branches of mathematics such as linear algebra and analysis.

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How objectively do we evaluate likelihood of two very rare events?

I've been pondering the concept of objectiveness to evaluate likelihood and predictability of extremely rare events. I'd like to demonstrate my questions over trivial hypothethical examples below. ...
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Binomial: normal approximation

I am having the hardest time coming up with the standard deviation that textbook is coming up with. Suppose 14% of the U.S. population is left-handed. If a random sample of 200 people from the U. S. ...
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Probability of someone in a given sized group, given their prevalence in the larger population

I’m sure my title was a terrible description of this question. On twitter, @BobWachter posted this: Asymptomatic test pos. rate @UCSFHospitals now 6.2%. That's 1-in-16 w/ no symptoms of Covid who test ...
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Neyman-Pearson Infimum Integral non rand. Test

*Show for 2 distributions $P_0$ and $P_1$ on $(X,F)$ with densities $f_0$ and $f_1$ with respect to $\mu$ that $inf\{P_0(\phi=1)+P_1(\phi=0) | \phi: X \rightarrow\{0,1\} \text{ non rand. test}\}$ =$\...
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How to use Bayesian Inference in this problem?

I've just learned Bayesian Inference and encountered this math problem: "A box contains 20 of both red anh black balls. Hypothetically, 40% <= the ratio of red balls <= 75%. Pick 5 balls ...
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Can reparameterization make Cramer-Rao bounds tight?

Given a family of distributions parametrized by $\theta$ for which Cramer-Rao bounds on variance for a (biased or unbiased) estimator of $\theta$ exist, these bounds may be unattainable. In the proof ...
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Kalman Filter Time Varying Parameter

I have a simple doubt regarding time varying coefficients Kalman Filter. Lets consider Hamilton's notation for the state equation and transition equation, respectively: $$\xi_t = F_t \xi_{t-1} + v_t $$...
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A problem in ANOVA, I think the result will be useful but I do not know how to prove it.

Consider an ANOVA model: $$y_{ijl}=\mu+\alpha_{i}+\beta_{j}+\epsilon_{ijl}$$ where $i=1,2\dots I, j=1,2\dots J, l=1,2\dots n_{ij}, \epsilon_{ijl}\sim N(0,\sigma^2) $i.i.d, we also have the constrains ...
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Is biased sample variance normally asymptotic?

I need to prove that biased sample variance is normally asymptotic. I tried, but, unfortunately, i couldn't. So, is biased sample variance normally asymptotic?
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What is the variance of the error term for the regression of the sample mean of $x$ on the sample mean of $y$ that is heteroskedastic?

I am having difficulty finding an appropriate expression for the variance of $\bar{e}_{k}$ in $$ \bar{y}_{k} = \tilde{\beta} \bar{x}_{k} + \bar{e}_{k} $$ where $i = 1, \dots, n $ individuals from a ...
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Extrapolate impression values from total impressions.

Im looking to extrapolate impression values for specific page level URLs however I only have the total impressions for a domain, and the ranking of the URLs (where the closer you are to 1 the higher ...
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Strong agreement of the estimator

Could somebody help? I don't have an idea at all. Because, we have $1/2n * [(X_2 - X_1)^2 + (X_4 - X_3)^2 + .. +$ The last element] Let $X_1,. . . , X_2n$ will be a sample of a decomposition with an ...
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Taylor expansion in Hoeffding's Lemma proof

Hoeffding's Lemma proof use taylor expansion with this statement: From Taylor's theorem, for some $ 0\leq \theta \leq 1$ $ L(h) = L(0) + h L'(0) + \frac{1}{2} h^2 L''(h\theta) \leq \frac{1}{8}h^2 $ ...
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need help with this question [closed]

Need help on this question and some explanation
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Can survival Analysis predict lifespan of a company?

Is possible to predict the lifespan of a company based on the company's data by using the survival analysis method? I just followed an expensive course to be a data analyst in my country. One of the ...
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Is the sample path from Monte Carlo Simulation follows CLT?

I'm pricing some derivative securities with Monte Carlo Simulation. I don't know population follows which distribution. But When I some calculation with my samples(all the each path) ,for example $X = ...
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Why is this independent random variable cannot deduce $\mathbb{E}(Y^2)=\mathbb{E}(Y)\mathbb{E}(Y)$

Here is the proof of biased and unbiased sample variance. Since they are random sample they are i.i.d. From the process of $$\sum^n_{i=1}\mathbb{E}(Y^2_i)-n\mathbb{E}(\bar{Y}^2)$$ I was thinking about ...
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Measure for bias in a confusion matrix

To measure the quality of classifiers I want to have a 'comparable' way to express a detection bias. I guess I'm looking for a function e.g. $ F: R^{n,n} \rightarrow \vec{x}, \text{ where } \vec{x} \...
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2 votes
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Variance-stabilizing transformation on a simple linear regression

I am currently working with variance-stabilizer method and readed something about it from my textbook. I want to understand it better so I would like to consider a case where I for instance have a ...
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Find the value $\hat{\beta}$ which minimizes $\sum_{i=1}^{4}|i||i- \beta|$ for $\beta \in \mathbb{R}$

I am looking for the value $\hat{\beta}$ which satisfies the following condition: for each $\beta \in \mathbb{R}$, $$\sum_{i=1}^{4}|i||i - \hat{\beta}| \leq \sum_{i=1}^{4}|i||i - \beta|\text{.}$$ ...
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Is recycling samples better than drawing fresh ones?

At a high level, I am wondering if in a sequential process it is better to reutilize samples even if these samples have been used to make past decisions. Let me formalize my doubts in a toy example ...
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Question about how to construct Brownian Motion. [duplicate]

My first question was answered within minutes by Sangchul Lee, but after a week of asking people I am stuck again, and let me describe. I will try to write out the standard definitions carefully. Let $...
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Deriving estimators for functions of an unknown parameter $\theta$

I am wondering if there is an easier way to find an estimator of a function of an unknown parameter $\theta$ for a particular statistical model. For example, if I derive the Maximum Likelihood ...
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Consistent estimator of function of moments up to given order

Asume we the moments of a distribution exist up to order $n$. Consider the estimator $$t (x)=f(m_1(x),..., m_n(x)) $$ where $m_k(x):=1/r \sum_{i=1}^r x^k$. We assume we have $r$ samples of the ...
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preventing rare extreme values in linear regression prediction

I am trying to train a model with a lot of input variables using linear regression. For technical reasons, my training data is obtained from a simulation that closely but not perfectly mirrors the ...
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TF2 crithack detection via statistical analysis

This is a idea that I've been thinking about for quite some time, but I'm not good enough at statistics to work out how to do it. Sorry in advance. I don't plan to implement this, but feel free to ...
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Causal discovery for pairwise independent joint dependent variables

Consider the standard example for variables that are pairwise independent but joint dependent. $$ (x,y,z)= \begin{cases} (0,0,0) & \text{probability 1/4} \\ (1,1,0) & \text{probability 1/4} \\ ...
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Rao Score test and Neyman-Rao test

I've got the Laplace location/scale model where $X_1,...,X_n$ are iid random variables with common density \begin{equation} p_{\theta,\eta}(x)=\frac{1}{2\eta}e^{-|\frac{x-\theta}{\eta}|}. \end{...
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Recalculate average based on subset of poisson distribution

Assuming a series of independent events, probability of the event occurring on a given day is a poisson distribution with λ=0.8. This includes probability of event ...
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Expectation of the largest order statistic from uniform random variables

If $X_1, ..., X_n$ are iid random variables from the Uniform[$0,\theta$] distribution, where $\theta >0$, compute the expectation of the largest order statistic denoted $X_{(n)}$. I am looking to ...
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Bayesian Approach to Hypothesis Testing: Posterior Probability that Two Distributions Differ

I'm trying to take a Baysian approach to Hypotheisis testing but I need a bit of help formalizing what claims I can actually make. Let's assume I have two datasets $X$ and $Y$ that each consist of $N$ ...
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Deviation matrix of mixed number data sets

As an example, let's suppose you have a list of student scores in $3$ classes: Math: $90, 90, 60, 30, 30$ Physics: $80, 60, 50, 40, 20$ English: $40, 80, 70, 70, 90.$ Let's denote a matrix of these ...
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3 votes
1 answer
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Is it possible that linear transform changes interval data to ratio?

I'm going through a stats intro and got puzzled by the concept of interval and ratio data. Celsius temps is an example of interval data that can not be multiplied and divided, because there is no true ...
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Time series: ARMA characteristic polynomials have common roots

I have a question regarding the idea that if the roots of the characteristic polynomials of a time series (say some ARMA process) lie outside the unit circle, then the series will be invertible/causal ...
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How to solve the least squares for A with both X and Y as matrices?

I’m trying to solve the following problem: I have a series of n-dimensional vectors x and another of m-dimensional vectors y, so that y = Ax, with A being a m,n ...
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I have to determine $d_{\alpha}$ in terms of critical values of a chi squared disdtribution, significance level $\alpha$ and sample size n

So far I have the following: Assuming the Null hypothesis $X_{i} \thicksim$ EXP$(1)$ $Y: = \sum_{i = 1}^{n} X_{i}$ $Y \thicksim$ GAM$(1,n)$ $\bar{X} = Y/n$ $P(\bar{X} \leq$ $d_\alpha) = 1 - \alpha$ $P(...
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What is the best way to interpret this "delta" growth metric?

I am trying to parse a spreadsheet with the following table. Penguin Type Jan '21 Count Feb '21 Count Jan '22 Count Feb '22 Count Jan Growth Feb Growth Delta Emperor 200 220 210 250 5% 13.6% 7.1% ...
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Easy problem involving joint probability distributions

Let $X_1$ denote time time (hr) it takes to perform a first tast and $X_2$ denote the time it takes to perform a second one. The second task always takes at least as long to perform as the first task. ...
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Probabilty question - multiple choice question with two correct answers

There is a multiple choice question with 4 options and 2 right answers. One has to choose 2 options while answering. What is the probability of getting this question right? My first instinct is to ...
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-1 votes
1 answer
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Why is "More Data Better than Less Data"?

According to Wikipedia: The Fisher Information is a way of measuring the amount of information that an observable random variable X carries about an unknown parameter θ of a distribution that models ...
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Are we able to Compare the Fisher Information of Two Variables?

According to Wikipedia (https://en.wikipedia.org/wiki/Fisher_information), "the Fisher Information is a way of measuring the amount of information that an observable random variable X carries ...
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Find $\mathbb{E}[\hat{\alpha}]$. Suggest a new estimate $\check{\alpha}$ of $\alpha$ which is unbiased, such that $\mathbb{E}[\check{\alpha}]=a$

I'm currently working on the following question: $$Define \: m=\sum_{i=1}^n log(\frac{\theta+X_i}{\theta}) \sim Pareto(n,\theta)\: and \: i.i.d$$ $$Let \: \hat{\alpha}=\frac{n}{\sum_{i=1}^n log(\frac{\...
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Can we calculate the opponent's hidden values in this statistical battle?

First-order statistical battle Imagine there is a game in which the user should guess what values the opponent is hiding from the user. In the first battle, the opponent has two hidden values ...
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Calculating a sample size

I can't figure out how to crack this problem. It's as if I don't have enough information. Walk me through please Jogging is a popular sport in Brilliantia. According to a recent survey, between 80% ...
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Calculate Treatment Effect for the Treated

This is from the paper 1991 Identification and Estimation of Local Average Treatment Effects by Angrist and Imbens. On p5, example 1 which says: Consider the following model: $Y_0 = \epsilon$ $Y_1 = ...
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Find estimates for $\alpha_0$ and $\alpha_1$ and covariancematrix

I have that $$f(x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}x^2}$$ I have the conditional distribution: $$f_{\beta}(y|x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}(y-\beta_0-\beta_1x-\beta_2x^2)^2}$$ and we ...
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Datasets for optimal experimental design.

I am looking for some real-world datasets/benchmarks for optimal experimental design problems. Nomatter which criteria (A, D, E-optimal) we use for the optimal design, the information matrix is the ...
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MSE of the Phase Estimation of a Noisy Complex Quantity

Presentation of the Problem I am making a measurement which yields me the complex quantity: \begin{equation} S_k = e^{i \cdot \phi} + X_k + i \cdot Y_k \quad \text{with} \quad X_k, Y_k \stackrel{iid}{...
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Estimates of parameters of interest $θ_1$ & $θ_2$ ($\pm$ standard errors) were $25\pm10$ & $10\pm3$. Find estimate & Standard Error of $δ=θ_1/5−θ_2$

Problem: In two independent studies, the estimates of the parameters of interest $θ_1$ and $θ_2$ ($\pm$ their standard errors) were computed to be $25\pm 10$ and $10\pm 3$, respectively. If one is ...
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Particle concentration dependency by particle volume for a given mass. [closed]

I have a mixture with a lot of nano-particles (suspended in water) with diameter ranging from 15nm to 40nm with a skewed distribution. I know the count of each particle by size. I know the mass of the ...
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