Questions tagged [mathematical-statistics]

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

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36 views

Why is the p-value given by p-value${}=P(\chi^2>\Delta D)$ rather than p-value${}=2P(\chi^2>\Delta D)$ as $(*)$?

In the Binomial Regression Models for Binary Data, we have the general Wald statistics result for the Hypothesis $H_0: \beta_j=0, \, H_a: \beta_j \neq 0$ that the p-value is given by $$\text{p-value}=...
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0answers
37 views

Confusion regarding plot of p-value as function of MLE value

I am trying to answer the following question: My work: (a) The power function $\beta \colon \Theta \to [0, 1]$ given by $\theta \mapsto \beta(\theta)$ is the probability of rejecting $H_0$, as a ...
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1answer
60 views

Why is the first canonical direction equal to the left singular vector i.e. why is $w_1 = a = u_1$ in CCA (Canonical Correlation Analysis)?

I want to understand why the canonical direction $a$ is equal to the left singular value of $M = \Sigma^{-1/2}_X \Sigma_{X, Y} \Sigma^{-1/2}_Y$ and not $a = \sigma_1 u_1$. My calculation tell me that ...
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1answer
79 views

Survival analysis: Treatment "parameter" for Weibull

I got following question: We let $T^*_1,...,T^*_n$ be independent survival times for n patients and we let $X_i\in\{0,1\}$ indicate if the i-th patient is treated $(X_i=1)$ or not $(X_i=0)$. We are ...
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27 views

How to calculate the expectation of MLE variance of linear model? [duplicate]

If $$Y_i\sim\mathrm{n}(\beta_0+\beta_1x_i,\sigma^2),\quad i=1,\cdots,n,$$ $$Y_i=\beta_0+\beta_1x_i+\epsilon_i,\quad i=1,\cdots,n,$$ The maximum likelihood estimates of the three parameters: $$\hat\...
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8 views

Posterior distribution derivation from proportionality [duplicate]

Could someone explain to me how the proportionality is derived? What are the guidelines for terms that can/should be omitted when dealing with proportionalities to coil it into a known distribution? ...
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0answers
28 views

Min and Max values of Bayesian Risk Classifier when Posteriori Probability and PDFs are Unknown

I'm struggling to come up with a well reasoned argument for this problem. Let $\tau_1$ be the posteriori probability and let $L(r^*)$ be the risk classifier. For this scenario, assume: $X\in\mathbb{X}=...
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1answer
27 views

Derivation when priori probability is Unknown

I want to derive an expression for the Bayesian classification risk $L(r^*)$ when the priori $\tau_1\in[0,1]$ is unknown. For this problem, let: $X\in\mathbb{X}=[0,1],Y\in\{ 0,1 \}$ $\pi_y=P(Y=y)=1/2$ ...
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20 views

Use Monte Carlo to produce new 'p' correlated data from existing data [duplicate]

As mentioned above, I have a problem where I need to generate new data Y from an existing data X such that Y is p correlated to X. I know their are several ways to do it but I want to know if monte ...
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13 views

What is the orthogonal projection with expectation?

I am reading an advanced econometrics textbook. When it talks about least squares, it says that the orthogonal projection of A onto Z is $P_Z(A)=Z^\prime E[ZZ^\prime]^{-1}E[ZA_k]$ and when A is a ...
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2answers
100 views

The role of bias terms in binary recommender systems

I realize that a recommender system applied to, for example, the Movielens dataset needs to account for bias. That is, one needs to adjust for the varying popularity of movies, and that users have ...
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1answer
17 views

Distinction between plug-in classifiers and Empirical Risk Minimization classifiers?

I am currently reading the paper Fast learning rates for plug-in classifiers under the margin condition by Audibert and Tsybakov (2014), in which the authors prove that, for binary classification ...
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21 views

How to approximate the expression to $\sum x_i$

How to approximate the expression on the left hand side to $\sum_{i=1}^Nx_i$ as $n\to \infty$ $$ \frac{\sum\limits_{i=1}^{N}x_i^2}{n-2\frac{\sum\limits_{i=1}^{N}x_i}{N}} \left(\sqrt{1+\frac{Nn\left(...
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28 views

Model the correlation between random variables with events

I have a set of correlated random variables (eg. $[x_0, x_1, ..., x_N]$) and I am trying to find a way to represent the correlation between them with a set of independent events (eg. $[E_0,E_1,...,E_M]...
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1answer
54 views

Likelihood of a predicted value falling within a range of values

I have a model that produces the expected points of every football (soccer) team for the following season, given the results from the previous season (the model considers the number of goals scored/...
1
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1answer
56 views

Does averaging affect statistics?

I have an array that represents client's bills for a certain period of time. Each client has several values. I need to group data by week or month and calculate mean and variance. However, I have a ...
2
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1answer
61 views

Derivations of Bayesian Risk Classifier when Posteriori Probability is Unknown

I found two expressions for a Bayesian risk classifier when the posteriori probability is unknown, but I don't understand how and why the derivations were made. For this scenario, assume: $X\in\mathbb{...
2
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1answer
58 views

Does adding interaction terms affect the power of main effects not included in the interaction?

Say we fit a regression model between the binary variables, $\mathbf{X} = (X_1, X_2, X_3) $ and a continuous response variable, $Y$ with a $$E(Y| \mathbf{X}) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \...
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1answer
64 views

Wilcoxon signed-rank test vs Mann Whitney U Test [closed]

Background Qualitative measurements are taken of a subject before and after some external stimuli Aim Determine if measurements taken before and after stimuli are different (p<0.05) Question 1.) As ...
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20 views

A random walk question about optimal outcome?

My mom and I were walking my dog. But we forgot to bring a poop bag, so we went home and came to the place where she pooped to pick up, which is a perceived small mount at the center of the field(can ...
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2answers
40 views

What is the theoretical justification of testing for equal mean/distribution via resampling?

If we have two data sets $X_1,\ldots,X_m$ and $Y_1,\ldots,Y_n$, each i.i.d., and wanted to determine whether $\mathbb{E}[X_1] = \mathbb{E}[Y_1]$ or not using $\bar{X}_m - \bar{Y}_n = \hat{\Delta}_{m,n}...
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0answers
42 views

Reference prior of normal distribution with unknow mean and variance

Problem: Assume that $X|\theta \sim N(\theta, \sigma^2)$ for unknow $\theta$, and unknown $\sigma$. a. Find the reference priors of $(\theta, \sigma)$, when $\sigma$ is of interest. b. Find the ...
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23 views

Mixture random variable density

I have $X = B_1X_1 + B_2X_2 + ... + B_nX_n$ where the $X_i$ are independent random variables and the $B_i$ are independent $Bern(p_i)$ such that $\sum_{i=1}^{n} p_i =1$. I want to find the PDF and CDF ...
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0answers
10 views

Create score from various features/variables

I have 4 measurements for different sensors and I would like to somehow create a score for each measurement ranging from 1-10. I have tried performing PCA with 1 component on these 4 columns and then ...
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0answers
31 views

Fisher Information Matrix for Non Canonical link function for Poisson regression

I'm trying to solve this problem and I don't know how to do it since is not using the canonical link function. When I was researching online I couldn't find an example when the canonical function is ...
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0answers
16 views

Prediction interval from regression, variance question

I'm struggling a bit with the notation to get the prediction interval from a maximum likelihood regression. Having 2 random variables $X, Y$, where they relate as $y = f(x; \beta) + \epsilon$, where $\...
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0answers
26 views

Whether the correlation structure of random field $Z(u) + Y(u)$ is equal to the correlation structure of $Z(u)$ plus that of $Y(u)$?

I want to simulate a Gaussian random field (RF) with correlation structure (represented by the geostatistic tool 'semivariogram' $\gamma (h) \: +\: pure \: nugget \: effect$). I want to know whether ...
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0answers
42 views

Confusion surrounding notation in proof of central limit theorem

In john rice Mathematical Statistics and Data Analysis we find a proof about the central limit theorem. Let $X_{1} , X_{2}...$ be a sequence of independent random variables having variance $\sigma^{2}$...
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1answer
26 views

Beta regression for ratio of counts

I have a question about whether it is appropriate to use beta regression when the proportion is formed as a ratio of counts. I have seen information suggesting it is fine and others pointing out that ...
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0answers
38 views

Probability of independent transmission attempts (independent Bernoulli trials)

I am trying to figure out the independent transmission attempts (independent Bernoulli trials) probability $q$ for a sender node to remain in the same (waiting) state if receiving node is in sleep ...
2
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0answers
22 views

What is cohort effect in general?

I have run into cohort effect in longitudinal data analysis book for quite a while. However, I still could not quite grasp what is cohort effect. What is the meaning of cohort effect and how it will ...
1
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2answers
55 views

How to select the level of confidence when using confidence intervals?

Saw the expression of "95% confidence interval is commonly used" in a lot of literature. I want to understand more about why 95% (or 99%) is a good level to use. Or is there a general ...
2
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1answer
45 views

Definition of Corrected Contingency Coefficient

I quote RACHEV, HÖCHSTÖTTER, FABOZZI, FOCARDI (2010). Starting from the bivariate variable $(x,y)$, with the component variable $x$ taking values in $(v_i, i=1,\dots,r)$ and the component variable $y$ ...
3
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3answers
538 views

Doubt around Covariance definition

Given the general definition of Covariance between two random variables $x$ and $y$: \begin{equation*} \text{Cov}(x,y)=\frac{1}{n}\sum_{i=1}^n(x_i-\bar{x})(y_i-\bar{y}) \end{equation*} Does the above ...
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0answers
20 views

How to correlate different variable types and test for the correlations significance?

I have a dataset that is structured as follows: Lot Number Lot Group Booking stage Date mID Measurement 1 Measurement 2 Measurement 3 19208.000 1 PROBE 2013-08-28 10650 34.7 44.89 67.00 48209.000 2 ...
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3answers
396 views

Hypothesis testing: why $\mu > 0$ (or even $\mu > \epsilon$) "seems easier” to substantiate than $\mu \neq 0$?

Consider a random variable $X$ following a normal distribution $N(\mu,\sigma^2)$. Suppose that we have drawn iid samples of $X$, obtaining a data set with a sample mean $\bar{x}>0$. We want to test ...
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25 views

If you take the maximum value from two random draws from a normal distribution, what is the mean and standard deviation of the resulting distribution? [duplicate]

Let's say I draw two numbers from a normal distribution with mean 50 and sd 10, and take the maximum of those two numbers. If I generate a distribution of that maximum, what are the resulting ...
1
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1answer
23 views

sample size for continuous skewed data

I have an outcome which takes values from 0 - 100. I have added a figure of the distribution (below). However, 30% or more will have a value of 0 (so zero inflated). Then the next lowest value it can ...
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0answers
30 views

Correlation of Financial Returns using Period-End vs. Period-Average Values

I have two time series of financial returns for assets $A$ and $B$ defined below for $n$ periods. The return $a_i$ is the percent growth in the asset price of $A$ using period-end values for $i-1$ and ...
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0answers
21 views

Implications of expectation of score function being zero

I understand why mathematically, the expectation of the score function (evaluated at the true parameter) is zero. But what are the implications of this fact? Can you give one or more examples on where ...
1
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1answer
26 views

Experimental Design: Choose Data Points to Minimize Quadratic Term Variance in Multiple Regression

$\newcommand{\eps}{\varepsilon}\newcommand{\szdp}[1]{\!\left(#1\right)} \newcommand{\szdb}[1]{\!\left[#1\right]}$ Problem Statement: Suppose that you wish to fit a model $$Y=\beta_0+\beta_1x+\beta_2x^...
2
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1answer
13 views

While deriving Least Squares Estimators, how to find the derivate of a summation operate?

I'm calculating the Least Squares Estimators. There was one step here: $\frac{d}{d\hat\alpha}{\sum(y_i-\hat\alpha-\hat\beta x_i)}^2=0$ --> $-2{\sum(y_i-\hat\alpha-\hat\beta x_i)}=0$ I know it is ...
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0answers
29 views

How to prove that the two equations for the slope of the line of best fit are equivalent? [duplicate]

I was reading this great article on deriving the equation for the line of best fit (https://www.neelocean.com/simple-ols-estimators/), and got confused when I came across: Rearranging: $$\hat \beta = \...
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0answers
17 views

Comparing two time series using trend analysis

I wanted to know if there is a way to compare two time series or some statistical measurement. I was completing some trend analysis and couldn't find a good statistical measurement to use. I tried ...
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0answers
38 views

Easy to follow video lectures or online help on measure-theoretic mathematical statistics

Due to some unexpected events, I was not able to follow my measure-theoretic mathematical statistics classes for a while and now I have to cover these materials (chapter 1 of Jun Shao's book and ...
3
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1answer
98 views

Hazard function of a simple system of components

I have been asked to complete these tasks, and a few more, but I am stuck somehow: Figure 6 is the simple system above. For task number 1: [DONE] I started by recognizing that the total survival time ...
5
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1answer
316 views

Sklar’s Extension Theorem and support restrictions

This question is about an application of the Sklar's Extension Theorem, whose proof can be found in Sklar, A. (1996), "Random variables, distribution functions, and copulas: A personal look ...
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0answers
69 views

Normalization of $M_{n} = \max(U_{1}, ... , U_{n})$

Let $M_{n} = \max(U_{1}... , U_{n})$ be the maximum of a sample size $n$ from $U(0 , 1)$ distribution. In my statistics textbook it says that $M_{n}$ normalized is equal to $n(1 - M_{n})$ but I'm not ...
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0answers
27 views

How is Mean Squared Error calculated for Point Estimates?

I am currently going through the book "All of Statistics" by Larry Wasserman, and when talking about inference, with regards to a particular parameter "theta", he states the ...
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0answers
20 views

Statistics on an event happening in a series of records

I'm at a loss to calculate the theoretical probability of an event happening. I'll appreciate any kind of guidance. Say i have a few thousand records with a simple series of -1 or 1 with something ...