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Questions tagged [mathematical-statistics]

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

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1answer
33 views

What if errors (residuals) follow other distribution rather than linear regression?

At first, sorry for the bad english. I'm not good at english because I'm a Korean. Recently, I'm studying linear regression. I've heard that errors always follow normal distribution because they are ...
2
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1answer
98 views

Should we trust statistics? [closed]

My mathematical statistics theory professor, an expert in the field, said today he doesn't trust statistics enough to use it in practice. He just likes the beauty of the math. He said he would never ...
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0answers
78 views

Conditional expectation of an exponential variable

I have trouble with conditional probabilities, therefore I am wondering if the following derivation is correct: Both X and Y are exponential random variables, with $\lambda_x$ and $\lambda_y$ ...
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0answers
20 views

compare the outputs of two different ML models

Let's imagine I have two completely different multi-class ML models, let's call them ML1 and ML2. The models were trained on completely different data with different target classes. As an output, the ...
4
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1answer
36 views

Poisson distribution and completeness, what happens when one point removed from parameter space?

Long time ago (early 1980's) my professor showed me a paper (I think it was Teachers' Corner or something similar) about the Poisson distribution and completeness. Showed that if only one point was ...
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0answers
56 views

Derive the Likelihood Ratio Test for multivariate normal and specific covariance matrix

Let $X_1,\ldots, X_n$ be i.i.d. $N(µ, C)$ random $p$-vectors. Derive the Likelihood Ratio Test for $H_0: C = σ^2(1 − ρ)I_p + ρ1_p1_p^T$, where $1_p = \begin{pmatrix} 1 \\ 1 \\ \vdots \\ 1 \end{...
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0answers
19 views

Proof that the pooled sample variance is biased when paired comparison is not used when it should be

Given the model for what is meant to be a paired comparison design: $y_{ij} = \mu_i + \beta_j + \epsilon_{ij}$ $i = 1,2; j= 1, 2, ..., 10$ Eg. An experiment comparing the different in mean hardness ...
3
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1answer
38 views

Sum of square of gaussian random variable and exponential random variable

I have 2 independent RVs $s$ and $N$ with distribution as below: $\begin{array} { c } { f _ { s } ( s ) = \frac { 1 } { \sqrt { 2 \pi \sigma ^ { 2 } } } e ^ { - s ^ { 2 } / 2 \sigma ^ { 2 } } } \\ { \...
2
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1answer
86 views

Define statistical potential energy [closed]

I am looking for a statistical method that closely relates to the idea of potential energy. Here is a quick google definition for potential energy "...the energy possessed by a body by virtue of ...
0
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1answer
31 views

standard deviation with mean for final score [closed]

I have an array of numbers The goal is for every number to be as close to 1 as possible The way I have interpreted this is that I want a mean as close to 1 as possible for the whole array, and a ...
1
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2answers
51 views

Significance of eigenvector components in PCA

Long time reader, first time poster. Hopefully I won't screw this up... In the context of Principal Component Analysis, I have the sense that the components of an eigenvector are a measure of the ...
1
vote
1answer
36 views

How to find the average given that you have the mean, standard deviation and amount?

I have an assignment where I'm asked to find the the average shoe number for men and women from given data: Sum of men and women: 312 Amount of men: 239 Standard deviation of men: 1.63 Mean of ...
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2answers
51 views

Symmetrical nature of binomial distribution [closed]

If $X$ is a $binomial(n,p)$ random variable, and $P(X<15)<0.5$ Find $n$. Less than $30$ More than $30$ Equal to $30$ None of the above. I know that binomial distribution is symmetric when ...
4
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1answer
42 views

KL divergence of models with both continuous and discrete variables

When $x$ is discrete, KL divergence is $D_{KL}(P||Q)=\sum\limits_{x}P(x)\log \frac{P(x)}{Q(x)}$, when $x$ is continuous, $D_{KL}(P||Q)=\int\limits_{x}p(x)\log \frac{p(x)}{q(x)}dx$. However, when the ...
0
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1answer
52 views

Interpreting sample space of a unknown random variable

After looking for a dozen of questions on cross validated, I've decided to write my own. We know the following mapping is called a random variable $$X:\Omega\to \mathbb{R}$$ where $\Omega$ is set of ...
2
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1answer
61 views

Invariance property

I am a bit confused regarding what exactly is the invariance property of sufficient estimators, consistent estimators and maximum likelihood estimators. As far as I know, Invariance property of ...
1
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1answer
37 views

PCA with SVD give me differents results compare to manually computed

I am trying to do PCA using the SVD method manually and compare the results with the results computed by PCA on sklearn. Let's say that our matrix is A = [[3 0],[0 -2]] Following the SVD method, I ...
1
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0answers
25 views

How are these two expected values equal? [closed]

From understanding machine learning: How are the two expected values equal in the proof of the theorem in the red box below? Notation: $D$ is a distribution on the set $Z$, $E_{z \text{ ~ }D}$ ...
0
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1answer
14 views

Consumer Surplus (Williams 1977 and Rosen 1981)

$U_{nj} = V_{nj} + \epsilon_{nj}$ for all $j$ and $\epsilon_{nj}$ captures the factors that affect utility but are not included in $V_{nj}$. The decision maker chooses the alternative that provides ...
3
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0answers
36 views

Require understanding regarding the concept of restricted estimators

I was reading "The Elements of Statistical Learning Book by Jerome H. Friedman, Robert Tibshirani, and Trevor Hastie" where I encountered the following: The part tells us that the RSS criterion will ...
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0answers
13 views

how to incorporate individual-level classification uncertainty into population-level uncertainty (via Poisson distribution)?

How do I incorporate uncertainty in an individual-level predictive model into the prediction of a population-level variable? To give a specific example: Suppose I have a binary classification model ...
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0answers
18 views

$R^2$ associated with a restricted LS estimator is never larger that that of the unrestricted LS estimator

Prove that the $R^2$ associated with a restricted least squares estimator is never larger than that associated with the unrestricted least square estimator. So, I tried doing this question but I can'...
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0answers
20 views

How to approximate continuous function from discrete uneven interval value?

To clarify, I have a discrete value of an unknown distribution. The value of this distribution domain is from 0 to 1. But the value which the discrete value is sampled has an unknown interval. For ...
11
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5answers
644 views

Are confidence intervals useful?

In frequentist statistics, a 95% confidence interval is an interval-producing procedure that, if repeated an infinite number of times, would contain the true parameter 95% of the time. Why is this ...
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0answers
21 views

Methods to find relationships between counts

I am looking for if there is any statistical method(s) to view if there exists any relationship between counts. I have a large astronomical type of dataset but for the sake of breaking it down for an ...
1
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1answer
43 views

Conditional logit or multinomial logit?

I have the following problem I need to use a discrete choice model to determine the time values (VOT) of 3 transport alternatives (car, bus and train). Looking at the literature of the discrete choice ...
4
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3answers
74 views

Moment Generating Function for Lognormal Random Variable

I'm working through the proof of a lognormal random variable and am having some difficulty in moving through it. I understand the following: Our CDF is $\Phi(\frac{logx - \mu}{\sigma})$, and thus our ...
0
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0answers
8 views

how to perform Wald test when we don't have replicate

I was reading this manuscript https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3287564/ They wrote that they have done statistical test and found that Wald test is best when there is no replicate I want ...
2
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2answers
44 views

The Kelly Criterion for interest bearing accounts

How is the Kelly criteria formula modified when the unused capital accrues interest ? Let us assume you have $\$1000$ and this generates interest compounded at $e^{(rt)}$ For example, you can bet \$...
3
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2answers
42 views

Estimating sample size en poussant

If we have sample mean of $n$ observations as $\bar{x}$. It is given that maximum value of the whole set is $M$ and minimum value is $0$. How many more observations we need so that the estimate for ...
0
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0answers
51 views

Viewing PMF as an instance of a PDF

I'm having difficulties in thinking about the probability mass function (PMF) as a special case of the probability density function (PDF). I understand that PMF's are used in discrete examples, but ...
0
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1answer
18 views

The MLR increasing property (for Karlin-Rubin) with indicator functions

Let $\theta_0<\theta_1$, and let the ratio of likelihood functions for the random variables $X_1,...,X_n$ be $$\frac{L(x ; \theta_1)}{L(x; \theta_0)}=\frac{I(x_{(n)}<\theta_1}{I(x_{(n)}<\...
5
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1answer
89 views

Distribution of min(X+Y,Y+Z,X+Z,Z+V,X+V,Y+V)?

Let $X,Y,Z,V$ be i.i.d continuous random variables in an interval $[a,b]$. What will be the distribution of $\min(X+Y,Y+Z,X+Z,Z+V,X+V,Y+V)$? Assume the distribution of the random variables to be $F(.)$...
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0answers
18 views

inverse Wishart

Suppose $X\sim IW_{p}(n,\Sigma)$ has an inverse Wishart distribution, which probability density function is $$f(X\mid n,\Sigma)=C(n,\Sigma)|X|^{-\frac{n+p+1}{2}} \exp\Big(-\frac{1}{2} \operatorname{...
3
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1answer
55 views

How to call this frequentist interval estimate that is neither a prediction interval nor a confidence interval

This question is inspired by Confidence Interval on a random quantity?. That question introduces an interesting concept for a type of interval that is neither a prediction nor a confidence interval (...
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0answers
9 views

2-Stage Least Absolute Deviations (2SLAD) Estimator with Quantile Regression in R

I was reading a paper, in which they introduce the following analysis, but without much further explanation as to what is being done: Since our key variables of interest are endogenous, we follow ...
1
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2answers
42 views

Understand a statement about likelihood function

I'm reading Agresti - Categorical Data Analysis and it says Consider two models, $M_0$ with fitted values $\hat{\mu}_0$ and $M_1$ with fitted values $\hat{\mu}_1$ with $M_0$ a special case of $M_1$....
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0answers
6 views

What methods exist to determine importance/weights for variables?

I have a variable which I decomposed into 5 components, such that $$Variable_i = c_{1,i} + c_{2,i} + c_{3,i} + c_{4,i} + c_{5,i}$$. I am trying to determine the similarity measure between two ...
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0answers
19 views

Build customer ratings from subscription datetime data for a recommendation system

I want to build a recommendation system with only some customer's subscription and unsubscription date. I have a database that looks like: ...
1
vote
1answer
73 views

Machine learning & mathematics - [closed]

I am not from a statistics (or mathematics) major background. Just bachelors in computer science but had learnt algebra, probability, statistics, Integration & Differentiation, Permutation & ...
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0answers
35 views

Why my proof of unbiased estimator seem fine without using expectation?

https://math.stackexchange.com/questions/787939/show-that-the-least-squares-estimator-of-the-slope-is-an-unbiased-estimator-of-t The above link is what I used as the related reference that I got the ...
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0answers
18 views

plot two groups of wind direction [closed]

I have two groups of wind direction and I want to use spoke plot in R peogramme I search about it in R programme using ?spoke plot but i cant found any information . for more detials the first ...
0
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0answers
13 views

Why the ratio of (distance from expectation)^2 / expectation in a goodness fit test follow a chi-square distribution?

I know that the sum of square of normal random variables follow a chi-square distribution. But when I learn how to do a goodness-fit test I don't know why the ratio of (O-E)^2/E follows a chi-square. ...
0
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1answer
48 views

How to prove theoretically that Treatment Sum of Squares + Error Sum of Squares = Total Sum of Squares using the formula below?

How to prove theoretically that Treatment Sum of Squares + Error Sum of Squares = Total Sum of Squares using the formula in the picture?
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1answer
66 views

Can lower inner fence be a negative number? [duplicate]

I checked my calculations and I am getting a negative number. Can LIFs be negative?
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0answers
31 views

Incorrect computation in Knight and Fu (2000)?

I'm currently reading Knight and Fu's 2000 paper on the asymptotics of "Bridge" estimators with a particular focus on LASSO as a special case. In the proof of theorem 2, they make the claim that under ...
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0answers
18 views

What is Moment generating function in layman terms [duplicate]

What is Moment generating function in layman terms. I am not able to understand what MGF is.
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0answers
24 views

how to differentiate mean residual life function

For survival analysis, with 2.5(CDF) and 2.7(Mean residual life function) how can i derive 2.9 step by step?
0
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1answer
40 views

Joint conditional density of two iid exponential random variables conditioned on their sum

I have the following question: Suppose $X_1, X_2$ iid $\sim f_X(x)=\theta e^{-\theta x}1\{x\ge 0\}$ and $S=X_1+X_2$. What is $f_{X_1,X_2}(x_1,x_2|S=s)$? The solution from our exercise class to this ...
0
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1answer
12 views

Confusion about the derivation of the TD-Learning update rule

I am currently trying to understand the paper "Learning to Predict by the Methods of Temporal Differences" by Sutton. I am stuck with the following step: (From "Learning to Predict by the Methods of ...