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Questions tagged [mathematical-statistics]

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

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If $X_1,\cdots,X_n \sim \mathcal{N}(\mu,1)$, project unbiased estimator $X_1$ on the span of score function and a constant

Question If $X_1,\cdots,X_n \sim \mathcal{N}(\mu,1)$, then project unbiased estimator $X_1$ on the span of the score function and a constant. Partial attempt: It is straightforward to show that ...
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1answer
24 views

What is Factorial Design in the context of Linear regression

I have been trying to understand the concept behind factorial design and its importance in connection to linear regression. I will be glad if somebody can give a clear and tone down explanation in ...
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Posterior mean computation of “Monty Hall Poblem”

Background As I understand that the "Monty Hall Problem" is well studied, e.g., here or here, etc. [I am relatively new to probability theory. So, please help me to learn something from you experts ...
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3answers
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If $X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$ are IID, then compute $\mathbb{E}\left( X_1 \mid T \right)$, where $T = \sum_i X_i$

Question If $X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$ are IID, then compute $\mathbb{E}\left( X_1 \mid T \right)$, where $T = \sum_i X_i$. Attempt: Please check if the below is correct. Let ...
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1answer
54 views

VC dimension of sine family is infinite?

From what I understand, the VC dimension of an hypothesis class is given by the maximum number of points in general position (or random) on the domain space that can be arbitrarily labeled by the ...
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1answer
30 views

Why is the quadratic approximation to the relative likelihood positive?

We can approximate the log likelihood at the real parameter value $l(\theta)$ with the MLE estimate $l(\hat\theta)$ using second order Taylor polynomials, like so: $$l(\theta) - l(\hat\theta) \approx ...
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1answer
44 views

Probability Density Function

x = 2, μ (mu) = 5 and σ (sigma) = 3 I am just wanting to confirm may workings for the second part of the probability distribution function as underlined red in the picture below (see Image_1.). $$e ...
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2answers
135 views

Find Bayes Estimator when Kernel of posterior is not clear

Suppose $x\mid\theta \sim \operatorname{Gamma}(\frac{n}{2},2\theta)$ and $\theta \sim$ inverse Gamma$(\alpha, \beta)$ with loss function $L(\theta, d)=\frac{(\theta-d)^2}{\theta^2}$ We wish to find ...
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Finding the moment generating function

How do I find the moment generating function for the generalized logistic distribution? $$f_X(x) = \lambda \theta (1+\text{e}^{-\theta x})^{-(\lambda+1)}\text{e}^{-\theta x}$$ Where $-\infty < x &...
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How do I measure information loss when converting categorical data to numerical?

Assume that a dataset has a mix of categorical and numerical attributes. The dataset has to undergo numeric processing which necessitates the conversion of the categorical attributes to numeric/...
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2answers
77 views

Why is expected value of random variable equal to mean

While learning about Random variables I came across the mean of random variable X. The definition says that the expected value of random variable E(X) = Mean of Random variable X I am not able to ...
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24 views

Order statistics for log series distribution

I am trying to obtain the probability mass function for various order statistics of a log series distribution for a given n. To do so, I tried modifying the code given in this question: Simulating ...
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1answer
71 views

Proof that random assignment guarantees the same probability for the individual to be selected?

Consider a completely randomized design, where every unit is randomly assigned to a treatment group. Let's say we have 30 observations and 3 treatment groups. When we choose one observation at random, ...
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44 views

Natural Splines and Smoother Matrix

In the context of smoothing splines, one can show that the Reinsch form is given by: $ \hat{y} = N (N^{T}N +\lambda \Omega)^{-1}N^{T} y = (I+ \lambda K)^{-1}y $ where (1) $K = (N^{T})^{-1}\Omega N^{-...
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How to obtain the posterior distribution of a given problem?

Problem: Compute the conditional distribution of a random variable $X$ given $Y$. If a random variable $X$ is Bernoulli distributed with probability $q$ for $X = 0$. The conditional ...
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How to compute sigma in linear discriminant analysis

How to compute sigma in linear discriminant analysis? The formula that I found in Elements of Statistical Learning (https://web.stanford.edu/~hastie/Papers/ESLII.pdf) on page 109 is somewhat ...
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1answer
49 views

How to tell if the “clusters” I see in my pair plots are statistically significant or occurring by chance?

I have a data set with one row per subject. Some variables include laboratory parameters for blood chemistry, hematology, etc. I also have some flag variables: any = 1 if the subject experienced an ...
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1answer
31 views

Prove that $\sum \hat y_i(y_i-\hat y_i)=0$ for linear regression model

Prove that $\sum \hat y_i(y_i-\hat y_i)=0$ for linear regression model. Attempt We have that $\sum \hat y_i(y_i-\hat y_i)=\sum x_i\hat\beta(x_i\beta-x_i\hat\beta)=(X\hat\beta)'(X\beta)-(X\hat\beta)'...
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2answers
116 views

Find the UMVUE of $\frac{\mu^2}{\sigma}$ where $X_i\sim\mathsf N(\mu,\sigma^2)$

Suppose $X_1, ..., X_4$ are i.i.d $\mathsf N(\mu, \sigma^2)$ random variables. Give the UMVUE of $\frac{\mu^2}{\sigma}$ expressed in terms of $\bar{X}$, $S$, integers, and $\pi$. Here is a ...
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59 views

Expectation of the minimum of dependent random variables

How do we compute the expectation of the minimum of dependent random variables? In other words, what is the value of $\mathbb{E}[Y]$ in the following case: $$ \mathbb{E}[Y]= \mathbb{E}\big[\min(X_1,\ ...
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2answers
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Self-Study Plan Help (no undergrad math or stats experience)

This has definetely been answered in part in a lot of places but alas, here I am asking again. Anyways background I did calculus and vectors and advanced functions in high school. Then I did a ...
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1answer
58 views

Establishing convergence in probability from a related convergence in distribution

Is it true that $\sqrt n (\hat{\theta}-\theta) \ \rightarrow_d \ N(0,\sigma^2)$ implies $\text{plim} \ \hat{\theta} = \theta $? If so, how can I prove this? Attempted proof: My proof is like ...
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If a statistic can be written as a function of a minimal sufficient statistic almost everywhere, is it minimal sufficient?

I know that if $T(X) = f(W(X))$ for one-to-one $f$, where $W(X)$ is minimal sufficient, then $T(X)$ is also minimal sufficient. But my textbook does not include "almost everywhere" or "almost surely" ...
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Finding MSE when parameter space is restricted.

Let $X_1,.. X_n$~ Exp ($\lambda$) , where $\lambda \ge 2$ . I need to find the Mean squared error (MSE) of the maximum likelihood estimator (MLE) of $\lambda$. The MLE of this is $ \hat{\lambda_{...
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How do I adjust action outputs when accounting for volume at which said actions are played?

I apologize for the vague and potentially misleading title, I am very new to statistics and do not yet have a handle on the jargon. Essentially, I have the table below:   ...
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3answers
97 views

Likelihood function when $X\sim U(0,\theta)$

Let $X_1, ..., X_n$ be $i.i.d$ random variables, uniformly distributed over $(0,\theta)$. Derive the likelihood function given the sample $x_1, ..., x_n$. Answer The likelihood function is: \begin{...
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1answer
28 views

Control for over-dispersion. Why do this: take natural log of metric, exponentiate, rank, remove top and bottom 10%

I'm looking at some NHS healthcare data on the number of deaths in England The measure i'm looking at is called the SHMI - it's simply: The number of observed deaths at a hospital / The expected ...
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3answers
226 views

Consistent unbiased estimator for the location parameter of Cauchy (theta, 1)

Given Cauchy distribution with pdf $p(x) = \frac{1}{\pi ((x - \theta)^2 + 1)}$ how can I find a consistent unbiased estimator for $\theta$? My reasoning so far Tried MLE, but there seems to be no ...
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23 views

Binary dataset correlation

There is following table ; ...
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1answer
58 views

Sufficient statistic when $X\sim U(\theta,2 \theta)$

Let $X_1, ..., X_n$ be $i.i.d$ random variables, uniformly distributed over $(\theta,2 \theta)$. Find a sufficient statistic for $\theta$, and compute $\widehat{\theta}_{MLE}$. Answer The joint ...
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1answer
29 views

Why is the ML ratio test of two normal means with different variances impossible?

Say I have IID samples from possibly different means and exclusive variances, $X_{1i} \sim N(\mu_1,\sigma_1^2), X_{2j} \sim N(\mu_2,\sigma_2^2)$, where $(i = 1, \cdots, m, j = 1, \cdots, n)$, and $\...
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1answer
22 views

Cumulative Probability Distribution of Maximum and 2nd from Maximum of 4 Variables

I understand that the cumulative probability distribution cum(x) of the maximum of 2 variables x1 and x2 with probability distribution p1(x1) and p2(x2) is the product of the two cumulative ...
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1answer
80 views

Biasedness of Uniform Distribution MLE

How do I show that the maximum likelihood estimator for uniform distribution on $[0, \theta]$ for a random sample of size $n$ is biased? I've calculated the MLE as $\max_i\{X_i\}$. Intuitively, we ...
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26 views

Does a data-dependent sampling rule induce correlation?

[This question is cross-posted on math SE here ] Suppose I have two iid streams of data that are independent of each other: $X = (X_1, X_2, \ldots)$ and $Y = (Y_1, Y_2, \ldots)$. I want to estimate ...
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1answer
75 views

Gauss' proof that “the best estimate for a random variable is the average”

I have read that "Gauss proved that the best estimate for a random variable is the average." Can someone provide Gauss' publication containing that proof? To be clear, I am not looking for you to ...
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1answer
20 views

How to prove this Corollary regarding ratios of densities being sufficient

The following Corollary is used in "Theory of Point Estimation" by Lehmann to prove a theorem. However I'm unsure how to prove this Corollary (it's left as a problem, so proof is omitted). The ...
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Non-Significant Total effect but significant direct effect in mediation?

I am doing a mediation in R. I have 2 mediators, 1 IV and 1 DV, and my results are showing something weird for one of my mediators. I have a significant direct effect (c) but a non-significant (c') ...
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1answer
40 views

How to regress ELO scores back to the mean in R? [closed]

I am using the elo package in R to calculate college football scores over the course of several decades. But I am having trouble understanding the regression function in the package: This is the ...
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1answer
32 views

Power of test and error Type I [closed]

I need some help with these multiple choice questions: 1) Considering a statistical test for the mean. the error type I is a=0.027. How at least is the power of the test? The possible answers are 0....
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1answer
54 views

Elements of Statistical Learning training set [closed]

I am trying to read the Elements of Statistical Learning Tibshirani, Hastie and Friedman, however I have a problem with understanding the expected (squared) prediction error ($EPE$) formula that they ...
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3answers
111 views

When are correlated Normal random variables multivariate Normal?

I know that there are many example of correlated normal random variables which are not jointly (multivariate) normal. However, are there conditions which state when correlated normal random variables ...
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0answers
28 views

reA function of sufficient statistic

I'm following notes at onlinecourses and I got confused on transformation of sufficient statistics. For example, if $X$ is a sufficient statistic for $\mu$, why $Y=X^2$ is not a sufficient statistic ...
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1answer
52 views

Rules based Model (Function) - Derive Probability & Ensembling

Basically, let's assume I have a simple rules-based function/model (if weight >= 150) -> return true. Simple binary answer (true or false) from a single feature input. If I have a range of samples/...
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1answer
62 views

How to model and estimate interference and subsequent shocks on panel data?

I have the following setting: In my factory, we have mutliple assembly lines(>10). Each line produces an amout of itmes every day, with some weekly and mothly production peeks. Thus its basically a ...
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What is the correct approach to know if something exists at any time, with given mean/SD persistence time and start date (but exact time is unknown)?

I am doing research on food resource competition between two predator species, and I am not exactly an expert on statistics so I'll thought I'll come here and ask. I have information on mean and SD ...
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Any sub-vector of an isotropic sub-gaussian random vector is an isotropic sub-gaussian random vector.

Any sub-vector of an isotropic sub-gaussian random vector is an isotropic sub-gaussian random vector. Note that a random vector $x\in\mathbb{R}^n$ is sub-gaussian if for any $a\in\mathbb{R}^n$, $a\...
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Does Hoeffding or Bernstein inequality provide a more accurate bound for this problem?

Let $X_i$, $i=1,\dots, n$ be independent random variables with $EX_i = \mu_i$, $\operatorname{Var}X_i=\sigma^2$ and $|X_i - \mu_i| \leq b$, $i=1,\dots,n$. Let $X=\sum X_i$ and $\mu = \sum \mu_i$. Let $...
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1answer
46 views

How many sample needed for 95% confidence of a rule

I've got an electronical component measuring some voltage. The documentation of this component say the maximum error of measurement of this component is 3%. Assuming a chosen component always do the ...
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2answers
41 views

Does the random sample from the population always have the same distribution as the population?

Let $Y_1,Y_2,...Y_n$ be a random sample of size $n$ from a population with distribution X. From this information, can I also always conclude that $Y_1,Y_2,...Y_n$ will have distribution X? Or is this ...
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Are there occasions when both the two-sample t-test and the Wilcoxon rank-sum test are appropriate?

As the title says, Are there occasions when both the two-sample t-test and the Wilcoxon rank-sum test are appropriate? What kind of problem would be correct to use both on ?