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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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I ran an ANN model and got an extremely low R2 but a pretty good MSE, what does this mean?

I ran an artificial neuron network on data with about 2,000 rows and 3 features. I got a R2 of .06 which is really low, but a good MSE of .41. Why are these performance evaluators of this model ...
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Is this Forex trading algorithm flawed?

I'll try to explain this the best I can, I'm sorry for my lack of economic concepts and terminology. There's a Foreign Exchange Market where a trader gets to make predictions about the currency ...
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1answer
10 views

Variance of a fitted model

Show that for any linear model, $\sum_{i=1}^{n}\frac{\text{Var}(\widehat{Y_{i}})}{n} = \frac{p\sigma^2}{n}$. Wasn't too sure where to start here. I know that Bias($\widehat{\sigma^2}$)=$-\frac{p\...
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Writing $\beta$ and X for a qualitative model

Express the following model in matrix form, ie: specify $\beta$ and $X$ so that the model can be written as $Y = X \beta + \epsilon$. The model $Y_{ij} = \mu_i + \epsilon_i$ where $Y_{ij}$ represents ...
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Calculate prediction interval

I am trying to use the formula $\tilde{x}^n_{n+m}\pm c_{\alpha/2}\sqrt{P^n_{n+m}}$ to calculate a prediction interval, $\tilde{x}^{100}_{101}\pm c_{\alpha/2}\sqrt{P^{100}_{101}}$ where $\tilde{x}^{100}...
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12 views

Mixed two-way ANOVA

How can you tell from some ANOVA table was the constrained or unconstrained version of interaction terms used? We know that a mixed two-way ANOVA model with interaction has been fit to a set of data ...
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14 views

Relation between Maximum Likelihood Estimator (MLE) and Cramér-Rao Lower Bound

Suppose that X1,X2, ... are i.i.d Bernoulli random variables with success probability equal to an unknown probability θ∈[0,1]. Show that the MLE of θ attains the Cramér-Rao lower bound and is ...
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Hypothesis Testing with Two-Variable Hypotheses, Arbitrary Significance (Neyman-Pearson Lemma)

I'm trying to catch up in my statistics class before finals and I am posed with this "try at home" question in our notes. It asks me to find the best critical region for testing a null hypothesis vs. ...
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12 views

Bayesian Posterior Mode VS MLE [on hold]

Suppose that $X_1, X_2, \dots $ are i.i.d Bernoulli random variables with success probability equal to an unknown probability $θ∈[0,1]$. Considering the example of flipping a coin with $n=1$, provide ...
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15 views

Independence of sample mean and sample variance

It is well known that under normality assumption, the sample mean and sample variance are independent, by Basu's Theorem. My question is that, is the normal distribution the only distribution whose ...
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1answer
10 views

Variance of a sum of variables with coefficients too

Can someone help me with the proof here? How do I start the proof? How do I simplify $(\sum_{i=1}^k a_i(Y_i - E(Y_i))^2$? I'd end up getting a large multiplication between each $a_i(Y_i - E(Y_i)$ ...
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Find Pr[$H_A$ is true given we reject $H_0$]?

Suppose that $$H_A: \theta = \theta_A$$ $$H_0: \theta = \theta_0$$ Let's suppose that Pr[Reject $H_0$ if $H_0$ = true] = 0.05. Pr[Reject $H_0$ if $H_0$ = false] = $\gamma $ What is: Pr[$H_A$ = ...
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1answer
44 views

MLE of simultaneous exponential distributions

Given the $X_i\sim \text{exp}({\theta})$ and $Y_i\sim \text{exp}(\frac{1}{\theta})$, where $X_i$ and $Y_i$ are indpendent, with the same $\theta>0$. I have to find the MLE and its distribution. I ...
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1answer
24 views

Simple probability and statistics problem

If there are 12 football teams in a league, how many different bets can you make if you bet on 3 first teams and 2 that will get kicked out of the league? The solutions says its (12*11*10*9*8)/2. ...
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1answer
19 views

How can I calculate if my data is diverse

I conducted a survey in which the first question I asked students what year they were in. I want to check that I have a diverse amount of students (from many year groups) so I can say the survey data ...
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Simple question about dummy variables

I have a regression where I include several groups of dummy variables: race “white” and “black” in my regression and "other" if both "black" and "white" are equal to 0. Where can I find the ...
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Hi guys. I am abit confused as to how the percentages are calculated of busting when hitting on specific Blackjack hand values. [on hold]

Blackjack probability Not sure how the percentages are derived and what probability formula is being used. According to the table below: Hand value Percentage of busting 21. ...
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1answer
18 views

Testing order statistics and finding if it is MP test using Neyman Pearson lemma.

Problem: Suppose X1,X2,...Xn follows exponential (mean=1). Only the largest values of Xi are recorded and denoted by T. To test H0:n=5 vs H1:n=10, show that the most powerful test of size 0.05 rejects ...
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12 views

Correspondence of smoothing parameters and variance components in generalized additive models

I would like to better understand the correspondence between smoothing parameters and variance components in generalized additive models (GAM), particularly GAMs with multiple smoothing parameters (e....
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24 views

How to calculate Poisson probability with float $k$?

Ok, so I see the equation is $\frac{λ^k * e^{-λ}}{k!}$ but what if my $k$ is not an integer but a float possibility? for example, $k = 2.5$ and $λ = 5$. I have tried to multiply $k$ and $l$ with the ...
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Proof for normal posterior distribution for a data for linear regression.

Let $D=(x_1,x_2,....X_N)$ be the data and $x_i$ is D-dimensional data. For linear regression the likelihood is as follows $$p(y|X,w, u, \sigma^2) = N(y|u + Xw, \sigma^2 I_N) \propto exp(-\frac{1}{2\...
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1answer
45 views

Where he used the information that $\mu \leq \mu_0$?

This example is from casella_statistical inference book. Where he used the information that $\mu \leq \mu_0$? What if $\mu \geq \mu_0$? Thanks.
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1answer
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(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 ...
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1answer
26 views

Conditional Variances: show that $Var(X-E(X|Y)) = Var(X|Y)$

How do I show that for two random variables $X$ and $Y$ $$ Var(X-E(X|Y)) = Var(X|Y) $$
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Why does independence between v and z guarentee (statement) when finding LATE in IV [on hold]

I'm trying to decipher why this expectation can be written. I just don't get the "Furthermore, joint independence of idiosyncratic gains and v from z also guarantees" part also the article goes ...
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1answer
20 views

Under what conditions $g(X_n,Y_n)\overset{d}{\rightarrow}g(X,Y)$?

Under what conditions, given $X_n\overset{d}{\rightarrow}X$ and $Y_n\overset{d}{\rightarrow}Y$ $\Rightarrow$ $g(X_n,Y_n)\overset{d}{\rightarrow}g(X,Y)$ I know that we can't apply the continuous ...
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21 views

Help me identify this decay function

I'm interested in identifying a formula/equation that i used as a decay function in order to learn more about it - the equation is formatted in Python code as that is where it was originally expressed....
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24 views

Likelihood interval for binomial counts

I have an assignment question regarding a "likelihood interval" that I don't really understand. The question asks to consider counts of $X_i$, with $i\in \{1,...,N\}$, modelled as independent ...
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24 views

estimate Markov chain mean transition time

Assume a continuous time Markov chain which is run through in one direction and finally absorbed at the last state $1 \rightarrow 2 \rightarrow 3 \rightarrow ... \rightarrow n $ The transition ...
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1answer
19 views

Asymptotic distribution of median estimator when density doesn't exist

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(\...
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1answer
21 views

Proving that (simple) regression can estimate average treatment effect if a covariate is binary

I'm trying to prove we can use regression to estimate average causal effect if a covariate is binary and a randomized treatment. In other words, I want to prove the following. \begin{align} \frac{\...
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1answer
29 views

Maximum Likelihood Estimator for Non Absolutely Continuous Distributions

Let $\theta\in[0,1]$ be a parameter. Suppose that $Y$ is a random variable that takes value $\theta$ with probability $1/2$ and is uniformly distributed on $[0,1]$ with probability $1/2$. What is ...
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1answer
19 views

Unbiased estimator for normal distibution [on hold]

Let $X_1,X_2,...,X_n$ be iid observations from a normal distribution with mean $\mu$ and variance $\sigma^2$, $\sigma^2>0$ is known and $\mu$ is an unknown real number. Let $g(\mu)=2\mu$ be the ...
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1answer
19 views

Transforming sum of exponential variables to chi-squared distribution

Assume $X_i$ are generated with the following distribution: $$ f(x; \theta, c) = \theta^{-c}cx^{c-1}e^{-(x/\theta)^c}$$ $\theta>0$ and $c>0$ is known. Further, assume $T(X)=\sum^{n}_{i=1} X_i^...
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1answer
40 views

Prove $\lim_{x\to\infty} x . [1 - F(x)] = 0$ [duplicate]

I have no idea how to proceed with proving this. If $X$ is a continuous random variable, $P(X > 0) = 1$, $E(X)$ is defined and $F(x)$ is the CDF, then prove $\lim_{x\to\infty} x . [1 - F(x)] = 0$
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1answer
26 views

Why does $ \mathbb{P}\left(X < -z\right) = \alpha \Rightarrow -z = \chi^2_{1 - \alpha}(2n) $ hold?

Assume $X_i$ are generated by $\Gamma(\theta_0,n)$ distribution, and $S_n = \sum X_i$. Further, it is known that $2 \theta_0 S_n$ follows a $\chi^2(2n)$ distribution, $\theta_0$ is known, $\theta_1 &...
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How do I show that the MLE for all sides of a die is the same as MLE for 5/6 sides of a die

I have two sets of hypotheses: $H_0$: each of the six sides of a die has probability 1/6 of being rolled, versus $H_a$ being not $H_0$ $H_0$: 5 sides of a six-sided die each has probability 1/6 of ...
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1answer
23 views

UMVUE of $g(\mu)=2\mu$ for a normal distribution

Let $X_1,X_2,...,X_n$ be iid observations from a normal distribution with mean $\mu$ and variance $\sigma^2$, $\sigma^2>0$ is known and $\mu$ is an unknown real number. Let $g(\mu)=2\mu$ be the ...
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0answers
11 views

Perturbation Bound for Left Singular Matrix

Let $A$ and $\hat{A}$ be two $N\times N$ matrix with rank r, let $A=\hat{A}+H$ where $H$ is some perturbation. Suppose $A$ and $\hat{A}$ have following SVD: \begin{align} A&=U\Sigma V^T\\ \hat{A}&...
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1answer
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Does it make sense to compare coefficient of variance between samples with different sample size?

I have two samples with different sample sizes. The difference is quite large: one has sample size of 10 and one has sample size of 200. Two samples are same type of data but are collected from two ...
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How to perform hypothesis testing on samples formed by matrices 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 ...
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how to measure the peak of a distribution?

As mentioned and explained in detail in this Math Exchange here, particularly by Peter Westfall, Kurtosis only measures "extremity of the tails", not the "peak" of the distribution which can even ...
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1answer
30 views

Statistics question regarding instrumental variables

Can someone tell me how in equation (40) the two equal signs are attained here? I know pictures and specific questions are not cool but I am so lost
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2answers
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Why is $ce^λ=1$ equal to $c=e^{-λ}$? [on hold]

Why is $ce^λ=1$ equal to $c=e^{-λ}$?
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10 views

Tree Pruning: SSE

Can someone show me an instance of a data set where a no split causes a reduction of SSE, however can be modeled by a tree with 3 splits? Basically, an example that shows that it's sometimes needed to ...
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1answer
18 views

Probability that a die game ends on an even throw (Linear Solution)

I've researched solutions for this problem seeing as I couldn't seem to solve it on my own, and there's one in this very forum that albeit looking the most simple, I can't fully understand. Here is ...
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12 views

Sufficiency in the exponential distribution

I am trying to show that given a random sample $\{X_i\}_{i=1}^n$ where $X_i\sim exp(\lambda^{-1})$, the statistic $T(\mathbf{X})=\sum_{i=1}^n X_i$ is sufficient by using only the definition. I have ...
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Expressing quadratic form of normal variables in terms of chi-squared variables

On Wikpedia and in the references therein [1,2], it is stated that any quadratic form of normally distributed random variables can be expressed as the sum of many independent non-central chi-squared ...
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1answer
21 views

Questions of Variance, mean and interpretation

Let X be a continuous random variable with the density function: $f_x(x) = \begin{cases} x+1, & \text{if}\ -1\leq x \leq 0 \\ -x +1, & \text{if}\ 0 \leq x < 1\\ 0 &...
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What is the Q-matrix in fixed effects estimation?

Sorry to bring you pictures but I stumbled upon this piece of evidence for the fixed effect estimator that I can't seem to unhinge. I have two questions 1) what is the 'plim'-term? 2) what is the Q ...