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Questions tagged [mathematical-statistics]

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

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27 views

Is there stationarity?

I'm trying to analyze a financial time series, these are the ACF and PACF returns graph. What could I say? Lag–9 and lag–15 are significant? I would say they are not, and there is not weak ...
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23 views

when do the principal components of PCA form a basis for the dataset?

Suppose I do a PCA on a data set and get $k$ principal components that explain 100% of the total variance of the data set. We can say any observation from the data set can be reconstructed by the ...
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12 views

Difference between several/Multivariate Random Variables (RVs) & a Sequence of RVs?

What is the difference between several/Multivariate Random Variables (RVs) & a Sequence of RVs? Example: Picking a student from class and noting his/her height and weight is several RVs (...
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1answer
23 views

Is joint conditionally independent equivalent to marginally conditionally independent?

Heading ##I am wondering whether these two properties are equivalent: $X$ is conditionally independent of $Y$ given $Z$ $X$ is conditionally independent of $Y$ given $a^T Z$, $\forall a \in R^p$ ...
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30 views

How to understand Jacobian Matrix from the geometric perspective?

I found a good lecture about Jacobian Matrix which was part of a statistics course. However, it was published 20 years ago and lack of explanation. As a beginner of statistics, I'm not able to find ...
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7 views

Proving the convergence of k-means algorithm using fixed point theorem

I'm learning machine learning and functional analysis this semester. When I learn the k-means algorithm , it came to me that the stopping criterion is very similar to the fixed point theorem thought. ...
4
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1answer
60 views

Which of the Gauss-Markov assumptions does error-in-variables violate?

The Gauss-Markov theorem states that for a linear model $$y = X \beta + \epsilon $$ if both of the conditions are true $$\operatorname E[\epsilon \mid X] = 0$$ $$\operatorname{Var}(\epsilon) = \...
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0answers
58 views

Proving an inequality for CDF's

I am working on a proof to show that given $x_1, x_2,\ldots,x_k$ random variables with a joint pdf and joint CDF, show that $$ 1-\sum_{i=1}^k \overline{F_i(x_i)} \leq F(x_1,x_2,\ldots,x_k) \leq \min_i ...
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32 views

Joint probability distribution of correlated data points

I have a query with respect to joint distributions. Here, each output data point in $\mathbf{y}$ is conditionally independent given the inputs $\mathbf{x}$ and the mapping $f:\mathbf{x}\rightarrow \...
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1answer
49 views

Variance of OLS Coefficients with Omitted Variables

I've read in several sources, for example here page 51, that if you omit a relevant variable from an OLS regression the resulting standard errors are smaller. Their arguments make sense but I'm trying ...
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17 views

Trying to find a posterior distribution from non-normal prior and squared loss function

I've been reading an economics paper regarding rational inattention by Sims (link: https://www.sciencedirect.com/science/article/abs/pii/S0304393203000291) and have been trying to implement his ...
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5 views

Statistical meaning of $\mathbb E_P[score_\theta(Z)]^T\operatorname{Cov}_P[score_\theta(Z)]^{-1}\mathbb E_P[score_\theta(Z)]$

Consider a random vector $Z$ with distribution $P$ having mean $\mu$ and covarance matrix $\Sigma$. Question Statistically what is the meaning of the quadratic quantity $\mu^T\Sigma^{-1}\mu$ ? More ...
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22 views

Interval estimation - sample problem

A random sample of 81 credit sales in a department store showed an average sale of $75.00$. From past data, it is known that the standard deviation of the population is $27.00$. ...
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0answers
8 views

Calculate association trends

I have time series dataset for 10 consecutive periods (i.e. T, T+1, T+2, ..., T+9). Moreover, I also have 100 term triplets in each time period. Each triplet contains 3 objects namely x, y and z. I ...
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17 views

Variance estimates of Grand Mean and Treatment Effect in ANOVA

Suppose a random sample of size $n_i$ is taken from $i = 1,2,...k$ and $k$ is the number of populations to be compared, the ANOVA Model is given by $Y_{ij} = \mu +\tau_i + \epsilon_{ij}$. I would like ...
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1answer
22 views

Exponential family form for product of Gumbel distributions

Consider the Gumbel distributions $(P_\vartheta)_{\vartheta\in\theta}=(G(\beta,\mu))_{(\beta,\mu)\in(0,\infty)\times\mathbb{R}}$ with distribution functions $$F_{\beta,\mu}(x)=e^{-e^{-\frac{1}{\beta}(...
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1answer
25 views

prerequisites to algorithms and then to python and Machine Learning

I spent the last 5 months studying full-stack web development using the MERN stack, that is (MongoDB, Express, React and NodeJs). Now that I have seen the full picture of what I would be doing ...
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13 views

When to update a model

This is an open question that I have - when should a model be updated? In practice, the modelling process could take a number of weeks/months to update and I would just like to know what some of your ...
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1answer
50 views

Calculation of relative distance

I have 100 term triplets as shown in the below mentioned figure. Each triplet contains 3 objects namely x, y and z. I want to rank the triplets according to the following two properties. y should be ...
2
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1answer
51 views

How to calculate divergence from standard deviation

I have 20 people I wish to give performance score every 6 months. I want to identify the weakest performers in the bottom 10 and remove them from the group but only those from the point of the most ...
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1answer
34 views

Finding UMVUE of a family of continuous random variables

Let $X$ has probability density function $f_X(x;\theta) = a(\theta)b(x)I_{(0, \theta)}(x)$ (where $a(\theta)$ and $b(x)$ are nonnegative). I have to find the UMVUE of $\theta$ or show that one doesn't ...
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14 views

Estimating excess count

I have values of sales made for 20 months over two months period. The values are recorded as count per month and we have a hunch that the values were under-reported. I am trying to build a ...
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0answers
23 views

Expected Value of Cross Validation approximates Predictive Square Error

In the context of Smoothing Splines, Im trying to show that the expected value of the cross-validation can approximate the predictive square error. More specifically, I want to show that $$E[(y_i - \...
3
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1answer
179 views

Some Questions about reference measures and maximum entropy priors (from The Bayesian Choice)

I am relatively new to statistics and Bayesian theory but I am trying to understand it by working through a few books. There are some things I am confused about. ( As I believe my analysis and such is ...
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30 views

Where is my misunderstanding/mistake in calculation about maximum entropy

I am looking through the book Philosophy of Science : Entropy and Uncertainty by Teddy Seidenfeld and have some questions. He talks about the basic dice example, where by you have a fair six sided ...
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0answers
23 views

R2 - contribution of explanatory variables x1, x2 and x3 to the variance of the explained variable y

I wish to look at the contribution of explanatory variables x1, x2 and x3 to the variance of the explained variable y. To summarize the contribution of the explanatory variables alone to the ...
1
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1answer
31 views

Notation for nested multilevel model

I am fitting the following logit model in lme4 (R): dummy ~ ind_var1 + state_var2 + country_var3 + (1 | country/state) This specifies varying intercepts for ...
2
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0answers
46 views

Chi-square origin of the name

What are the origins of the names and letters in these distributions: What is the origin of the name in chi-square distribution $\chi_k^2$? And the origin of $t$ in student's $t$-test? And naming $...
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1answer
119 views

Uniqueness of Reproducing Kernel Hilbert Spaces

Digging in the definition of Reproducing Kernel Hilbert Spaces (RKHS) I came across the following example taken from pages 49-51 of [1]: Given the kernel $k(x,y) = \langle x,y\rangle^2$, with $x,y\in ...
3
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1answer
38 views

Minimal sufficient statistics of increasing dimensionality (not equal to the number of observations)

Restricting the attention to the case of fixed parameters support, it's my understanding that (minimal) sufficient statistics of fixed dimensionality, i.e. a fixed number of of them, exists in, and ...
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0answers
8 views

Quantifying separation of groups of bivariate data

Simple question really; is there a statistical test for significance between groups using bivariate data? If I have two groups on a scatterplot that clearly separate, how can I quantify that. For ...
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1answer
25 views

Which coeffcients are used for a F-statistic

I have an F-statistic, $F(4,10)$, my constant and 4 Coefficients $\beta_2 , \beta_3 , \beta_4$ and $\beta_5$ I already know that the (in this case) 10 reflects the number of obsverations. But what ...
2
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1answer
29 views

Lower incomplete gamma function format in series representation and R [closed]

As known that the lower incomplete gamma function can be written as $\gamma(a,x) = x^{a}e^{-x}\sum_k^\infty{{x^{k}}\over a^{k+1}}.$ What is the format for $\sum_j^\infty{\gamma(v/p-j,rx^{p})} $ in ...
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0answers
16 views

Determine confidence Interval of the model fit with Small Number of Observations

There are two parts to this question: I have two observations of a star. I would like to determine its temperature so I use a model to fit its spectrum. Each fit gives me one value of the star's ...
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0answers
43 views

Using Leibniz integral rule when minimizing Expected Absolute Loss [duplicate]

Consider choosing $\theta^*$ that minimizes the expected absolute loss: \begin{align} \tag{1} \int_{\Theta}|\theta-\theta^*|\pi(\theta|\mathbf{x})d\theta= \int_{-\infty}^{\theta^*}(\theta^*-\theta)\...
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0answers
12 views

Restricting Properties of Point Estimators in Discrete, Unordered Parameter Spaces

A common principle used to deal with the fact that there are typically no uniformly best estimators (in the sense that they uniformly have least frequentist risk) seems to be to restrict the space of ...
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3answers
224 views

OLS effect of X squared ond dependent variable

I have my OLS regression: $y = \beta_1 + \beta_2 X_2 + \beta_3 X_3 +\beta_4 (X_3)^2$ Could anybody please explain to me the effect of a change in $X_3$ on the dependent variable?(Is the effect ...
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0answers
8 views

Applicability of Trimmed Estimators in estimating population parameters

I have been recently focused on trimmed estimators. I read a couple of articles but they seem to be giving a somewhat conflicting output. Below I have outlined the two different conclusions I have ...
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1answer
38 views

Confusion with covariance

For distributions of random variables X and Y, their covariance can be defined as the difference between the multiplication of X and Y, normalized by their joint probability and the multiplication of ...
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0answers
31 views

Aikaike Information Criterion: derivation in original paper

I have been reading AIC paper 'Information theory and an extension of the maximum likelihood principle' by Akaike (1974). I have been able to understand up to the third section of the paper, but I am ...
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0answers
42 views

Establishing an upper bound for the tail probability $P(X-\lambda \geq z)$ for any $z>0$, where $X$ is Poisson r.v. w/ parameter $\lambda$

Poisson random variable $X$ with the parameter $\lambda$ has, respectively, the pmf and the moment generating function of the forms $$P(X = k) = \dfrac{e^{-\lambda}\lambda^k}{k!}, \quad k=0,1,2,\...
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1answer
136 views

Finding the distribution of sample range for a Beta population

Let $X_1,X_2,\ldots,X_n$ be i.i.d random variables having density $$f(x)=2(1-x)\mathbf1_{0<x<1}$$ I am trying to derive the distribution of the sample range $R=X_{(n)}-X_{(1)}$. The ...
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0answers
54 views

How to detect relationship between two variables over time?

Considering there are two series Sa, Sb over time T.Series Sa doesn't have a defined range and can range within any values , Sb ranges within 0-1. There are already pattern in the data such as when Sa ...
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0answers
14 views

Proving new features theoretically help learning

I've been recently working with feature construction. I was wondering, whether how one would go about proving, that given a standard classification setting, a new feature can improve a classifier's ...
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1answer
59 views

Deriving marginal pdf from joint pdf

Problem setup: $X \sim \Gamma(\alpha,\beta)$, $f_{Y|X}(y|x)= \tau xy^{\tau-1}e^{-xy^\tau}$ for $y>0$ and $f_{Y|X}(y|x)=0$ for $y\leq0$, where $\tau\geq1$ is a constant. I am asked to derive the ...
2
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1answer
63 views

Admissibility does not imply minimax

The answer to minimax estimator explains why minimax does not imply admissibility. The relevant statement is from https://www.stat.berkeley.edu/~yuekai/201b/lec6.pdf which says, minimaxity does not ...
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0answers
29 views

Finding complete statistic from a single sample

How do I find a complete statistic for the sample of size 1 of the following random variable: $f(x; \theta) = \frac{2x}{\theta}I_{[0, \theta]}(x) + \frac{2(1-x)}{1-\theta}I_{[\theta, 1]}(x)$ ? ...
3
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1answer
51 views

Upper bound on $P(n^{-1}\sum_{i=1}^n (X_i - \lambda_i)>t)$ for independent $X_i\sim\operatorname{Poisson}(\lambda_i)$

Let $X_1,\dots,X_n$ be independent random variables, $X_i \sim \operatorname{Poisson}(\lambda_i),$ $i=1,\dots,n.$ Let $$S=n^{-1}\sum_{i=1}^n X_i, \quad\quad \lambda=n^{-1}\sum_{i=1}^n \lambda_i.$$ ...
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0answers
33 views

Bias variance tradeoff when the estimated target function is also random

I'm interested to understand "bias variance tradeoff" notion in a different setting than usually presented. In a setting where target $f$ (see the map $f$ below) is a random map rather than ...
4
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2answers
142 views

question about MSE mean square error

The following is taken literally from Wikipedia's mean squared error in the mean subheading: "Suppose we have a random sample of size $n$ from a population, $X_1$, ... ,$X_n$. Suppose the sample ...