Questions tagged [mathematical-statistics]

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

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Efficiency of an estimator using Cramer-Rao bound

I want to compute the efficiency $\displaystyle e(Z)={\frac {I(Z )^{-1}}{\operatorname {Var} (Z)}}$, where $I(Z)$ is the Fisher information of $Z$, for $Z=\alpha\sum_{i=1}^n |X_i|,\ X_i\sim\mathcal{...
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Existence of UMP test of poisson distribution

Let $X_1, \dots, X_n \sim \mathrm{Poi}(\lambda)$. I know that (a) There exists a UMP test of $H_0: \lambda = 0.5 $ vs. $H_1: \lambda = 1.5$ (b) There exists a UMP test of $H_0: \lambda \leq 1$ vs. $...
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Measuring standard/formula when we know length, mean, highest and lowest values of a set

This is probably a basic statistics question, sorry about that. I have sets of data that I wanna compare. Each set has a list of values from which I know the length, mean, highest and lowest values ...
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Confusion on when you can apply CRLB

I am somewhat confused on when you can apply CRLB when estimating parameters which are not of densities directly. For example a classic problem is estimating the magnitude, phase and frequency of a ...
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how does realization of random variable work?

Reading different resources about statistical inference, I have hard to time to build good intuition about how some fundamental tasks stated. Many resources say something like this: Given random ...
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P-value range for 3 samle anova [closed]

I am working on a question where I am asked to find a p-value range based off of a test statistic. I have calculated a number that ,with proper df's, falls in the range of p = 0.01-0.025. However, the ...
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Q-Exponential distribution in Numpy

I am studying a phenomena, which I know is characterised by the q-exponential distribution, with parameters $c=1$, $q = 1.355$ and $b = 0.524$ in the equation $$e_{q,b,c}(t) = c(1 + b(q-1)t)^{\frac{1}{...
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1answer
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Subset of the event {$i$ aces in $i$ cards} and its probability

As far as I'm aware, $A$ is a subset of $B$ if $x\in A$, then $x\in B$ for all $x\in A$. If so, how could this statement (please see the image below) say the event {4 aces in 4 cards} was a subset of ...
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Approaches to Processing exam results [closed]

I am trying to process around 8000 exam results I have put these in a datatable. What I would like to know is how you would go about processing this data from a statistical point of view. I have ...
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Is $E(T) \ge \sigma$ or is $E(T)\le\sigma$?

Let $X_1,\cdots X_n \sim N(0,\sigma^2)$ be i.i.d., and let $T=\sqrt{\frac{1}{n}\Sigma^n_{i=1}(X_i^2)}$. Is $E(T) \ge \sigma$ or is $E(T)\le\sigma$? My work: $\frac{X_i^2}{\sigma^2}\sim \chi^2_1$ ...
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Asking for some hints

Can anyone help me or give me some hints that how can I answer the following questions? An experiment was carried out to study the effect of temperature (x) on the yield of a chemical reaction (y). ...
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R - basic vectors' statistics questions [closed]

I am just starting my steep learning curve in R, its' statistical aspects specifically, and currently want to tackle the following. I created the following random vectors: ...
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Minimal Sample Size needed to have at least one element when there are 3 possible outcomes?

This is my first question here, I am really glad this community already answered so many from other users that have greatly helped me. So here is my problem, I have a pretty "large" sample (N>200) ...
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Doubt in Continuity Correction

In this question - Let us say the mean number of Gems per packet is 20, and the variance is 2. Assuming that the distribution of Gems in the packet is Gaussian, what is the probability of finding a ...
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1answer
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Correlation matrix: decline in correlation

I performed a correlation matrix with Python. The following variables were included total_equity,net_income,...
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Approximate posterior distribution

Let $\mathbf{a_k} \sim \mathcal{GP}(\mathbf{m}_k,\mathbf{C_k})$ where $\mathbf{a} \in \mathbb{R}^T$ is modeled as Gaussian process with mean $\mathbf{m} \in \mathbb{R}^T$ and prior covariance $\mathbf{...
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Bayesian normal posterior distribution

I am new to the Bayesian and here I have a question about deriving the posterior distribution function. Suppose my prior in main study is based on the posterior distribution of a pilot study. The ...
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1answer
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Simplifying modified Bessel function of the first kind

The modified Bessel function of the first kind shows up in the normalizing constant of a lot of random variables (e.g. the normal product distribution, the noncentral chi-square distribution, the ...
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Finite Mean and Finite Variance [duplicate]

Let $X$ be a random variable. Suppose there exists a constant $c ∈ R$ such that $E(|X − c|^2) < ∞$. Show that the random variable $X$ has finite mean and variance. And I'm quite confused about the ...
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Proving an alternative form for Mallow's $C_p$ statistic

I want to show that Mallow's $C_p$ statistic, $$C{_p}{_j} = \frac{RSS_j}{S_E^2} + 2s_j-n$$ can be written as $$C{_p}{_j} = (k+1-s_j)(F_j-1)+s_j$$ where $RSS_j$ is the residual sum of squares for model ...
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How do I calculate Confidence Interval for Gamma Distributed Pivotal Quantity?

I'm studying confidence intervals and then I came across the following problem: It's said that a random variable X has Skewed Exponencial Distribution with parameters $\alpha >0$ and $v \in \...
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How do I interpret the paired t-test results?

How do I interpret these results? Which food does the Corvid like best? What does the p-value and t-value mean? Is this test significant? My hypothesis is: H0 Corvids have no preference on food ...
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Beta as a limit distribution of a proportion of Poisson process [closed]

Problem Assume $\{X_i(t) : t\ge0 \}$ for $i=1,2$ : two independent Poisson processes with the same parameter $\lambda$. We set $X_i(0) = n_i$. Prove that $$ \mathrm{lim}_{t \to \infty} Pr \...
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Why can't ridge regression be used when there is only a single predictor variable? [closed]

Trying to understand why there are problems when ridge regression is used when there is only a single predictor variable, but can't understand if it is just because there are better things for a ...
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Calculating p-value for testing alternative hypothesis [duplicate]

this is a recent stats question I came across: A random sample of 847 13- to 17-year-olds found that 396 had responded better to a new drug therapy for autism. Let p be the proportion of all teens ...
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25 views

Conditional expected value variable change

Assume that we have $X_1,X_2,X_3$ that are jointly Gaussian and with non-zero covariances. We also have: $$Y_1=X_1$$ $$Y_2=X_2−X_1$$ $$Y_3=X_3−X_2$$ With $0$ covariances among all $Y_i$ and $E(Y_i^...
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1answer
16 views

How to rank data based on cross-validation

I had this problem from a long time. I have small dataset with about 1000 data points. The data is labeled as 1 or 0 (i.e. ...
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11 views

How to know if your data obeys a functional central limit theorem?

How may I know that my dataset obeys the FCLT. There is a test that can be conducted if the FCLT holds. However, in practice, I am not sure if my data follows the FCLT. Is there a way to test for this?...
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Are the forecasting methods like mean, naive, drift, weighted average applicable to non stationary time series?

Like AR, MA models essentially need the series to be stationary, do the other forecast methods mentioned above also follow stationary?
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1answer
22 views

Understanding t bounds on MGF

I'm having trouble understanding what the bounds of the $t$ variable are for an mgf. My questions are bolded. Here's an example from a textbook: Suppose X is a random variable for which the pdf is: ...
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31 views

How to change the distribution of classes?

I have two classes [0,1] and I want to evaluate algorithm on different distributions of classes. I did label flipping incrementally such as 0%, 10%,20%,....,90%,100%. Does label flipping change the ...
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Example: probability-integral transform. Normal distribution [closed]

CONTEXT This question is related to [1], in that I'm seeking a deeper understanding of probability integral transforms. This question is related to an example in [2], which concerns the probability ...
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1answer
26 views

What is the rejection region for this test?

A sample of size 16 was randomly drawn from a normal population with known standard deviation σ = 5. The sample mean is equal to 20.5. The researcher wants to run the followingz-test(α=5%): H0 :μ=...
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1answer
22 views

What are the broadest class of distributions for which the range statistic is ancillary to the expectation of the random variable?

Let $X_1,X_2,X_3$ be iid random variables such that $E(X_1)=\mu$ Define $X_{(3)}$ and $X_{(1)}$ as the maximum and minimum order statistics respectively. I know that if $X$ is normal, $R=X_{(3)}-X_{...
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1answer
44 views

Distribution of sum of independent random variables using MGF

Assume you have $x_i \sim \operatorname{Bernoulli}(p_i)$ with $p_i \sim \operatorname{Beta}(\alpha,\beta)$. and let $Z=X_1+ \dots +X_n$ and I wanted to show that $Z$, $Z \sim \operatorname{...
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1answer
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Invariance of maximum likelihood estimates to rearrangements of parameters/constants in the model?

I know that maximum likelihood estimates are invariant to re-parametrization (https://stats.stackexchange.com/a/335368/267430). Is the MLE also invariant to rearrangements of the constants and ...
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What is constraint and effectivesize in structural holes?

Today, I spent the whole day trying to understand what structural holes in network means. More specifically I want to clarify what the two measurements constraint and effective size means? Is it good ...
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Determining the degree of confidence of a Prediction

I'm handling a text multi class classification problem using RNN LSTM and model is pretty accurate (approximately 95%). But I'm interested in finding the degree of accuracy of a Prediction. Can I make ...
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31 views

What is the purpose of the generalization error bound?

I could not understand what is the purpose of the generalization error bound, why do we need to calculate it?!. How does the generalization error bound work with 1-nearest neighbour algorithm ?. Does ...
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How do you prove that samples are equally distributed even when taken without replacement? [closed]

I was reading Casellar and Berger's book and came across this well-known property. Although, in the book, the proof for any Xi is not provided. Does anyone know how to follow up on it? I'm curious, is ...
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Linear regression - Bayesian Predictive distribution

I am trying to answer a question about linear regression but i am stuck: $y=w \cdot x + \epsilon, \epsilon \sim N(0,\alpha)$ i am also given a prior: $w\sim N(0,\beta)$ from which i was able to ...
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1answer
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What does “the denominator does not contain any theta dependence” mean in Bayes' Rule? [duplicate]

Every lecture and book says that the denominator in Bayes' Rule does not depend on the parameter $\theta$. However, the denominator also includes $\theta$ in the formula of Bayes' Rule. I just cannot ...
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2answers
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Do Z-test and T-test assume normality?

I’ve been digging into the internet trying to find an answer for this question. Notwithstanding, all that I’ve found were empty statements like: “z-tests depends on the population to be normally ...
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Under what condition(s) running a simple regression between two time series variable is valid?

Suppose we have two stationary time series variable: gdp growth rate and S&P 500 returns. And, we fit a simple linear regression model for S&P 500 returns with gdp growth rate as the sole ...
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21 views

Rao-Cramer lower bound for bernouli distribution [closed]

The question asks for Rao-Cramer lower bound but I don't understand why the answer assumes a binomial distribution with 2 trials. Does it mean that we need to at least have two trial to come up with a ...
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23 views

Maximum likelihood function questions [closed]

Your friend throws an unknown number, v, of standard dice and reports that she got x sixes. (i) Assuming all throws of the dice are independent, write down the likelihood of given x. (ii) By ...
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1answer
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MCMC Metropolis-Hastings Understanding [closed]

I am having trouble understanding a key aspect of MCMC - does the collection of states the Markov chain passes through represent the target distribution, or for a single sample, are we running $n$ ...
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1answer
66 views

Putnam problem - guess an integer in an odd number of attempts

I have no clue how to answer this question. This is 2002 Putnam examination question B-4. B4 An integer $n$, unknown to you, has been randomly chosen in the interval $[1,2002]$ with ...
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1answer
110 views

“Associative” Correlation

For 3 random variables, $X,Y,Z$ all with zero mean If $E[XY]\ne0$,$E[YZ]\ne0$ then can we say $$E[XZ]\ne0$$ Alternatively $E[XY]=0$,$E[YZ]=0$ then can we say $$E[XZ]=0$$ Or even $E[XY]=0$,$E[YZ]...
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Is there a significant relationship between two data sets?

I am interested in image processing in python. I worked on detecting and tracking particles in an image. I obtained velocity data of particles in different regions. I believe that the difference ...