# Questions tagged [mathematical-statistics]

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

5,175
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### Why is MSE used in cross validation when selecting optimum number of variables in model?

I'm currently looking through An Introduction to Statistical Learning by Gareth James, more specfically Chapter 6. It discusses ways to select the optimal number of variables in a model using methods ...

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15 views

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### Understanding notation in Bias-Variance decomposition in Elements of Statistical Learning

I'm going through Elements of Statistical Learning and I'm having a bit of trouble understanding this bit of notation from Chapter 2 (this example is (2.27))
$$EPE(x_0) = E_{y_o|x_o}E_T(y_0 - \hat{y}...

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### Multivariate One-Tailed Tests

I know when performing hypothesis tests like the z-test or the t-test for one variable, the one-tailed test (when fitting, usually in the case of verifying claims of superiority) provide stricter ...

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7 views

### Mahalanobis distance between high dimensional arrays

As we know, the Mahalanobis distance (MD) is one of the distance metrics for measuring two points in multivariate space. In practice, I can compute Mahalanobis distance between two 1D arrays using ...

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13 views

### Some questions regarding independence requirements in the conditional likelihood

For iid random variables $Y_1,\dots,Y_n$ from a random sample that are mutually independent with pdf $f(y|\theta)$, my book defines the likelihood as $L(y_1,\dots,y_n|\theta)= f(y_1,\dots,y_n|\theta) =...

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### Time Series Analysis and Forecasting of Astronomical data having a sinusoidal trend

I have data of an orbital parameter of a satellite for 456 days. I am treating this data as univariate time series data and wish to use time series models to forecast future values. However, this data ...

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22 views

### Find corresponding elements in two matrices with many-to-many relationships

I have to matrices, one is the matrix of diagnoses, the other is the matrix of tests. Some tests are always done, some are done to confirm several and some for very few cases. I would like to learn ...

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11 views

### Estimation of beta and sigma for a model truncated from above and below

The following model is truncated from above and below
Y = xā²Ī² + e ,if L < Y* < U
not observed if otherwise
How do we show the estimation of Ī² and Ļ2 using MLE by deriving the Log-...

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6 views

### Can we construct a mapping which transforms a given density to a density which first order stochastic dominates the former? [closed]

Given any continuous density $f$ with a compact support $[a, b]$, is it possible to define a mapping which transforms $f$ into a a density $g$ which first order stochastic dominates $f$ on $[a, b]$?

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### If an event is mutually exclusive find $P(A' \cap B')$

So the question says that you have A and B event they are mutually exclusive so find
$P(A' \cap B')$ I have solved it by :
$$P((A \cup B)') = 1 - P(A \cup B) $$
But what is the effect of being ...

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5 views

### Reference request: Modulus of continuity of the map from parameter space to the space of probability measures

I am thinking of the following condition in my research:
Let $\Theta \subset \mathbb R$ be the parameter space and $\mathbb P_\theta$ be the probability measure on sample space $\mathbb X$ ...

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**1**answer

11 views

### Exponential families and Gibbs measures? Any relation?

The title says it all. The formulas for exponential families and Gibbs measures seems very similar. Is there any relationship, or some kind of translation table?

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612 views

### What distribution does the mean of a random sample from a Uniform distribution follow?

For example, let $X_1,\cdots,X_n$ be a random sample from $f(x|\theta)=1,\theta-1/2 < x < \theta +1/2$.
Clearly, $X_i \sim U(\theta-1/2 , \theta +1/2)$. Some intuition would suggest that $\bar{X}...

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4 views

### Time series split to avoid data leakage when lag variables are created for a univariate forecast model. Please help me in preparing the python code [closed]

I am doing a time series split for a univariate forecast model. Basically I am developing a regression prediction by creating Lag variables. When Lag variables are created then there is a high chance ...

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9 views

### How do the units of the SIR model cancel out?

I was having trouble trying to understand the parameters of the simplest SIR model.
If beta is the effective contact rate and s is the percentage of people who are susceptible, then how do the units ...

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26 views

### PDF of the sum of truncated exponential distributions

The problem is slightly different than finding the sum of classic exponential distributions, this is why I am asking the question.
$p(t|T) = e^{T-t}$ if $t > T$ and $0$ otherwise
The objective is ...

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### What are the disadvantages of using mean for missing values?

I have an assignment (Data Mining course) and there is a part which asks: "What are the disadvantages of using mean for missing values?" in Missing Value section.
...

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32 views

### How can I compare the lengths between these two confidence intervals?

Let $X\sim Beta(\theta,1)$. I was asked to find two confidence intervals and compare their lengths.
The first confidence interval is for $Y=-(logX)^{-1}$ over the set $[y/2,y]$.
The second ...

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**1**answer

47 views

### How can I show that $\prod^n_{i=1}X_i$ has a monotone likelihood ratio?

We have a random sample $X_1,\cdots,X_n \sim Beta(\theta,1), \theta > 0$ is unknown. My ultimate goal is to find a UMP size $\alpha$ test for $H_0: \theta \le \theta_0$ v. $H_1: \theta > \...

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13 views

### Is there a minimum number of entries in a bin to consider it for computing $\chi^2$ ? and for gaussian behaviour?

Let's consider a set of data points of some observable in several bins, and a theoretical model.
The agreement between the two is using the $\chi^2/ndf$ formula (where we divide $\chi^2$ by the number ...

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10 views

### Writing covariance with a scale factor, why?

Why are covariance matrices often written as
$$\Sigma = \sigma^2 \Omega,$$
where the variance scale $\sigma^2$ is written explicitly. What is it that motivates this kind of notation?

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25 views

### Some of my data is not normally distributed, what test should i use?

I am testing dating profile information and whether wanting children (yes, no, maybe) has an effect on how many attributes the seekers ask for. When testing for normality and homogeneity of variance ...

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8 views

### How can I score a y in a loop regression model with multiple x? [closed]

Iām currently trying to implement a linear regression model on different target with a column list loop start. (I'm using the software KNIME for large data analysis).
Basically in each column ...

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8 views

### Existence of an optimal learning algorithm for a given distribution

I am working on Exercise 3.8 from Understanding Machine Learning by Shai Shalev-Shwartz and Shai Ben-David. In it we consider the binary classification problem where
$\mathcal{X}$ is a set of data ...

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15 views

### Is the estimator consistent for sigma squared? [closed]

Suppose that X1,X2,...,Xn and Y1,Y2,...,Yn are independent random samples from normal populations with means mu1 and mu2 and variances sigma^2 and 2(sigma)^2 respectively, n>2. the x's and y's are ...

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### Sample selection SI

I have to solve this exercise: We want to know about households income. We have N individuals of all households and we select n individuals of them, then we identify which home they belong to and the ...

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11 views

### Why I obtain different variances in a same exercise? [closed]

I'm solving a sampling problem, in this I have to compute the variance of $\pi$ estimator using two differents ways: first, using a definition and second: using a "direct formula" , well, the ways are ...

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42 views

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### Probability distributions associated to the logarithm numeration system

The most elementary logarithmic numeration system is defined as follow. Any random number $X \in [0, 1]$ can be represented uniquely as
$$X=\log_3(A_1 + \log_3(A_2+\log_3 (A_3 + \cdots)))$$
with $A_k \...

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### How to follow-up a repeated measures ANOVA with 1 between and 4 within-subject factors?

I'm having trouble determining the best way to follow-up the results of my repeated measures ANOVA with 1 between (group) and 4 within (time x IV A x IV B x IV C) factors. Time is pre-test, post-test ...

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17 views

### Low P value and Low R square

I am developing a model which can predict the amount of funding which a research institute can get from the Government. I have used Linear Regression for it.
Response variable is the amount of ...

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8 views

### What is the point of using bootstrapping in logistic regression of sentiment analysis?

Apologies if the title is a bit ambiguous. Let's say I am trying to use logistic regression to predict whether a book review is positive or negative using the bag of words approach. My homework asks ...

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### How to find the bias of $\hat{\sigma}^2=\frac{\sum^n_{i=1}(X_i-\bar{X})^2}{n}$ for the population variance $\sigma^2$

My question is
The bias of an estimator $\hat{\theta}$ for parameter $\theta$ is defined as $E(\hat{\theta})-\theta$
please find the bias of $\hat{\sigma}^2=\frac{\sum^n_{i=1}(X_i-\bar{X})^2}{n}$ ...

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### One tailed test

I was reading about an article on One tailed test and Two tailed test. I know that using one tailed test we focus only in one direction and are not concerned about direction.
I came across the below ...

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8 views

### Question about the confidence band definition

Can anybody explain where the formula for en in this confidence band definition comes from?

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### Inv covariance matrix for Mahalanobis distance between vectrors in 2D space

I want to determine Mahalanobis distance between vectors in 2D kinematic space,
sample data:
K.E. P.E.
110 , 510
38 , 504
44 , 172
71 , 146
91 , 300
How to do it in python ...

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### How can I use this pivotal quantity to find the shortest length confidence interval for $\theta$?

Let $X_1,\cdots,X_n \sim f(x|\theta)=\frac{\theta}{x^2}, x> \theta$ be a random sample where $\theta>0$ is unknown. I want to use $\frac{\theta}{X_{(1)}}$ as a pivotal quantity. How can I use ...

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21 views

### Proving sufficiency [closed]

I am taking a very math heavy math stats course (graduate level). This is the definition of suffiency given in class notes:
Definition $1.0 .1 .^{2}$ sufficiency
$Y=\phi(X)$ is sufficient over $H \...

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18 views

### LLN and Consistency are equivalent expressions?

Is it true to claim that LLN produces the property of Consistency? How do we intuitively differentiate between the terms as they seem equivalent?

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### How do you calculate standard deviation for a mean change, given mean 1, SD 1, mean 2, SD 2, and total n?

How do you calculate standard deviation for a mean change, given mean 1, SD 1, mean 2, SD 2, and total n?
For example: n=590. Initial mean is 26 with a standard deviation of 8. The mean at the end ...

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### Have incidence for full population and an odds ratio for a sub population. How to determine an adjusted risk with that sub population removed?

I have a population, and for that population I know the incidence of a disease, and for this population there is both high risk and low risk groups. The size of the high risk group varies over time.
...

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### Why can information gain be used in Hoeffding Inequality as mean?

The Hoeffding inequality
$$P(|\mu-\bar{X}|>\epsilon) \leq 2exp(-2n\epsilon^{2}/(b-a)^2)$$ with real mean $\bar{X}$ and sample mean $\mu$ is used in Hoeffding trees to determine the best feature to ...

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37 views

### Question about calculating confidence intervals

I am reading about confidence intervals and got stuck with this example from L. Wasserman's book titled "All of Statistics". Could anybody explain why PQ(Īø ā C) = 3/4 in this example? Below is the ...

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### Is normalizing/standardizing features and target separately a good method?

Suppose I scaled the features and target by creating separate objects, like this
...

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30 views

### Bayes Classification vs Naive Bayes Classification

Generally, known that Bayes Classifier is optimal for the probability of error. But when I did some experiments:
First Case:
I have 2 classes data and their covariance matrices correlated in this ...

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28 views

### Probability of failure set

I want to prove the following theorem, which in general allows us to compute the probability of a certain set even if it contains no observations (which is all too common in extreme value analysis of ...

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### Finding quantiles of scalar non-decreasing function of two independent variables

Let $X_1 \sim exp \circ N(\mu_1, \sigma_1)$ and $X_2 \sim exp \circ N(\mu_2, \sigma_2)$ be two independent lognormal distributions, and $f:R^2\to R$ such that $f$ is non-decreasing on both arguments. ...

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### Risk of a learner as random variable

Assume the learner $h: X \to Y$ where $X \times Y$ with joint probability distribution $P_{X,Y}$ and assume a loss function $L$. Then the risk of $h$ associated to $L$ (and $P_{X,Y}$) is defined by ...

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### Would a z test be ok in this situation

I would like to know if I am doing this question correctly.
The data I am using is from http://www.kaggle.com/mohansacharya/graduate-admissions.
What I would like to do is, estimate the population ...

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### Deriving the bias of the self-normalizing importance sampling estimator

Suppose we are interested in the expectation of a test function $f(X)$ with respect to target distribution $\pi(X) \propto \gamma(X)$ using importance sampling with proposal distribution $q(X)$ with $...