Questions tagged [mathematical-statistics]

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

Filter by
Sorted by
Tagged with
0
votes
0answers
12 views

When developing the Yule-Walker equations, why can we assume $E(X_{t-k} \cdot Z_t) = 0$

I have a AR(1) process $X_t = 0.4X_{t-1} + Z_t$. I want to derive the Yule-Walker equation. I multiplied by $X_{t-k}$, took the expectation of both sides, and got: $$E(X_{t-k}X_t) = 0.4E(X_{t-k}X_{t-...
0
votes
0answers
10 views

Why does the Yule-Walker equation for AR(1) only hold for $k \ge 1$?

I have a AR(1) process $X_t = 0.4X_{t-1} + Z_t$. I multiplied by $X_{t-k}$, took the expectation of both sides, divided by the variance (which I assume is stationary), and got the Yule-Walker equation:...
3
votes
3answers
54 views

Statistics and machine learning references which require functional analysis background

I am doing a PhD in functional analysis, particularly Banach space theory. Thinking of my future prospect, I would like to venture myself into statistics. However, as I do not want to 'waste' my ...
0
votes
0answers
8 views

How can we apply the rule of stationary distribution to the continuous case of Markov chain?

If the Markov chain converged then $$\pi = Q* \pi$$where $ \pi$ is the posterior distribution and $Q$ is the transition distribution(it's a matrix in the discrete case). I tried to apply that on the ...
0
votes
0answers
16 views

What is the difference of first-order difference and second-order difference in trend analysis?

I have seen that the literature in trend analysis use first-order difference and ...
3
votes
1answer
28 views

Can we calculate the theoretical stationary distribution from a continuous Markov chain?

I have the transition distribution $p(X_{t+1}|X_t=x_t) = \text{N}(\phi x_t,1)$ where $−1<\phi<1$. Can we calculate the stationary distribution and its mean and variance? I know I can do that ...
0
votes
0answers
14 views

Weighted averaging of multi-task (multi-output) regression errors

I am trying to elaborate a multi-task (multi-output) regression metric based on single-task metrics. From my perspective, it should be a weighted average of single-task errors estimated with the same ...
1
vote
0answers
8 views

Granger's representation theorem: Johansen's version

In his book 'Likelihood based inference in cointegrated Var', in order to get the expression for the Granger's representation theorem,, Johansen claims that: (1) $$\beta \bot(\alpha' \bot \beta \bot ...
0
votes
0answers
15 views

What is Prediction Error Decomposition?

In my econometrics material I've come across the term prediction error decomposition. Essentially what I've understood so far is that it relates to maximum likelihood. The net is not very helpful. I ...
0
votes
1answer
19 views
1
vote
2answers
139 views

How can I calculate $\mathbb{P}\left(\sum_{i=1}^5\left[\frac{Y_i}{3}\right]=3\right)$?

Suppose $Y_1,Y_2,\ldots,Y_5$ is a random sample of distribution with Probability function $$\mathbb{P}(Y=y)=\frac{1}{3} , \quad y=1,3,5.$$ How can I calculate $\mathbb{P}\left(\sum\limits_{i=1}^5\left[...
3
votes
2answers
151 views

Calculating $\operatorname{var} \left(\frac{X_1-\bar{X}}{S}\right)$

Suppose $X_1,X_2,\ldots, X_n$ are random variables distributed independently as $N(\theta , \sigma^2)$. define $$S^2=\frac{1}{n-1}\sum_{i=1}^{n} (X_i-\bar{X})^2 ,\qquad \bar{X}=\frac{1}{n}\sum_{i=1}^{...
3
votes
0answers
61 views

Compounding a Gaussian distribution with variance distributed according to the absolute value of another Gaussian distribution

Have there been earlier descriptions of the following compound distribution? Compounding a Gaussian distribution with variance distributed according to the absolute value or square of another ...
0
votes
0answers
17 views

mathematical simulation of svm workings

i cant seem to work my head around the mathematical part of svm i understand the concept and the derivations but the the part that comes after lagrangian formulation is where im stuck..(googling didn'...
0
votes
0answers
16 views

Type 1 and Type 2 Errors with Costs

I know the ex-ante cost of type 1 and type 2 errors in my study. How do I select my alpha, given that I know that alpha governs both type 1 and type 2 errors?
0
votes
0answers
19 views

Cox proportional hazard regression [on hold]

What is the importance and role in adjusting the Cox proportional hazard regression survival curve for covariates of interest? The study is about the relationship between lycopene levels and the ...
0
votes
1answer
159 views

finding probability density function $U=Y-X$ [on hold]

suppose joint probability density function of $X,Y$ is $$f(x,y)=\frac{1}{2}\\ x>0 ,y>0,x+y<2$$how can i find probability density function $U=Y-X$
0
votes
0answers
11 views

Normal probability of default vs conditional forward default PD

Hi everyone, is the diagram correct in calculating foward PD(conditional default) ? Or should the formula be Probability of default = probability of survival x forward PD Whatcha of this is equal ...
0
votes
1answer
23 views

Correct test for measuring a difference in Gender and Performance

I'm often sharing data around performance and typically cut the data by gender to show differences around male vs female. So for example, I'll say 19% of males (which represents 550 out of 2800 total ...
-1
votes
1answer
37 views

Deriving a no closed-form expression for a density

Suppose U~N(0, Ip) and X|U~N(0, Im), then what will be the distribution of X? Will it be normal? If yes, what will be the mean and variance? If not, is there any easy way to derive it?
6
votes
3answers
11k views

How to create ROC curve to assess the performance of regression models?

I knew that, ROC curve are use to assess the performance of classifiers. But is it possible to generate ROC curve for the regression model? If yes, How?
3
votes
2answers
540 views

Inequality on variance of sum

I want to prove that $$\operatorname{Var}\left(\sum\limits_{i=1}^m{X_i}\right) \leq m\sum\limits_{i=1}^m{\operatorname{Var}(X_i)} \,. \>$$ A too complicated proof is to write $$ a_{ij}=\sqrt {Cov(...
0
votes
0answers
27 views

How to calculate High Order Moments?

I have a raw data set for Radio Signals (type: complex) and I want to calculate the High Order Moments and Cumulants (see this paper PART: II-A) In the Paper a ...
0
votes
0answers
29 views

Samples from a Gaussian distribution in theory [on hold]

I am pretty confused about the term sample from Gaussian distribution. Let $X=[X_1, X_2,\cdots, X_n]$ be a standard Gaussian $N(0,1)$ random sample. So, in theory every random variable $X_i \in (- \...
1
vote
1answer
19 views

Linear Quadratic Regulator

from http://people.eecs.berkeley.edu/~somil/Papers/lqrlecture.pdf Why must the matrices be positive semidefinite? What is the input authority cost? What is the purpose of multiplying the transpose ...
0
votes
1answer
16 views

Expectation over complex multiplication is multiplication of expectation

Suppose I have $\mathbb{E}\left[{\bf x} {\bf x}^H \right]$, where ${\bf x} \in \mathbb{C}^{N \times 1}$ is a random vector which has a uniform distribution, then can I say, $\mathbb{E}\left[{\bf x} {\...
1
vote
0answers
26 views

What are the differences between probabilistic models, stochastic models, and statistical models? [on hold]

What are the differences between probabilistic models, stochastic models, and statistical models? Do statistical models deal with data sets, and model them mathematically to capture the summary ...
5
votes
1answer
220 views

Unbiased estimator of $\lambda(1 - e^\lambda)$ when $x_1,\ldots,x_n$ are i.i.d Poisson($\lambda$)

Suppose $x_1, x_2, x_3,\ldots, x_n$ are i.i.d. random variables with a common Poisson$(\lambda)$ distribution. I was trying to find an unbiased estimator for $\lambda(1 - e^\lambda)$, but I could not ...
3
votes
1answer
82 views

Correlations between two sequences of irrational numbers

If $x$ is an irrational number and $b$ an integer, let's define $g(x,k) = \mbox{Correl}(\{nx\},\{nb^kx\})$. Here $k=1,2,\cdots$ is an integer. The brackets represent the fractional part function. ...
1
vote
0answers
13 views

U~U(0,1) and Xt = sin(2*pi*u*t), t = 1,2,3,… Does Xt (t is a suffix of X) is strictly stationary, why or why not? [closed]

The time series X_t = sin (2*piUt), where t=1,2,... and U follows uniform (0,1). Does X_t is strictly stationary?
0
votes
0answers
11 views

Reconstruct data of the past of a time serie

I'm trying to construct a time series model in which the water flow (caudal) is a function of temperature and precipitation. I want to forecast the results of the water flow. My problem is that I don'...
1
vote
1answer
30 views

How to calculate tf-idf for a single term

I am following the tf-idf method described in this paper: Measuring, Predicting and Visualizing Short-Term Change in Word Representation and Usage in VKontakte ...
2
votes
1answer
149 views

Normalization constant for uniform distribution over categorical distributions

Suppose we have a uniform distribution over all categorical distributions p for m categories, where the pdf has the form $$ f(x) = \left\{\begin{aligned} &c, && 0 \le p_i \le 1, i = 1, ......
2
votes
0answers
31 views

Intuitive explanation of scaled inverse chi-distribution

I am having a hard time understanding the scaled inverse chi-distribution. I looked for Wiki and other resources which are pretty math heavy without an intuitive ...
1
vote
0answers
113 views

Non-negativity of interaction information for special trivariate case

Consider a discrete trivariate distribution $P(X_1, X_2, Y)$, which satisfies $$ p(x_1, x_2, y) = \min( p(x_1,y), p(x_2,y) ), $$ for all $x_1$ and $x_2$ for which $p(x_1, x_2) > 0$ and for all ...
0
votes
0answers
7 views

How do I apply the difference between two simplexes to a third simplex and have the result still sum to 1?

My question concerns simplexes, but I'll give an applied example instead. Imagine we are polling a national election. Three political parties compete. At the last election, the overall result was as ...
8
votes
1answer
149 views

Do random variables follow the same algebraic rules as ordinary numbers?

In the comments on my answer to a recent question about the sum of random variables, I came across a link to the Wikipedia article on the ratio distribution, and noticed the following peculiar claim ...
0
votes
1answer
38 views

what is statistics uncertainty on a quizz score?

Remark : I have created the question from scratch : it is not my teachers who asked me that. I'm wondering myself about a general problem of uncertainty. Let's consider a 'school' quizz with D ...
0
votes
0answers
63 views

What does it mean if the sobol main and total effects indices are the same?

What does it mean when the total and main effects ANOVA indices are the same? Does it mean there is zero interaction of the different inputs? Is there some other way to quantify or understand that? I ...
2
votes
2answers
80 views

Sufficient statistics - how can the conditional pdf not depend on $\theta$ when $\theta$ is in the equation?

I understand the heuristic definition: say you know a statistic, $T$, of some sample that you want to use to estimate the corresponding population parameter - but you don't know the data points of the ...
2
votes
1answer
91 views

How to measure the consistency of improvement on different conditions?

I want to measure whether the speed improvement of method 1 over method 2 is consistent on different conditions. Below are two examples of the speedup values of method 1 over method 2 on 5 conditions. ...
2
votes
2answers
158 views

Regression model when the dependent and independent variables show exponential distribution

As the Title suggests i am trying to figure out what would be the regression model to use when both the dependent and independent variables show an exponential distribution. Do I have to perform a ...
0
votes
1answer
9 views

Calculation of critical values

Are the critical values for the two-sample t-test with unequal variances calculated the same as for the two-sample z-test?
2
votes
2answers
331 views

Treating continuous variables

I attended a conference on ML and Data Science and I have a general question that was not answered in the conference. If we have a continuous variable, let's say age. What is the best way to handle ...
1
vote
0answers
32 views

Problem on sufficient statistics

Let the distribution of $X_1,X_2,...X_n$ depend on two parameters $a, b$ such that there exists a single sufficient statistic, for either parameter when the other is fixed/known. Show that there is ...
5
votes
3answers
177 views

Which likelihood function is used in linear regression?

When trying to derive the maximum likelihood estimation for a linear regression, We start by a likelihood function. Does it matter if we use either of these 2 forms? $P(y|x,w)$ $P(y,x|w)$ All pages ...
0
votes
0answers
12 views

A pharmacologist is interested in determining whether or not three different psychoactive drugs differ in [closed]

A pharmacologist is interested in determining whether or not three different psychoactive drugs differ in how long they remain in the body before they are excreted or metabolized. The pharmacologist ...
5
votes
1answer
1k views

Basu's Theorem Proof

I am having trouble with the proof of Basu's theorem... specifically, I'm not sure about the $\theta$s in the expectations below: Let $T$ be a complete sufficient statistic. Let $V$ be an ancillary ...
3
votes
1answer
38 views

A basic question about a randomized test involving the error type I

I have a basic question in the context of testing statistical hypotheses, more specifically, about randomized tests. Suposse that I have two actions (altenatives) about a certain unknown parameter $\...