Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [mathematical-statistics]

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

2
votes
1answer
49 views

Finding expression of $n$-th derivative, when $n$ is large

For completeness, assume $C$ is an Archimedean copula with some generator function $\varphi$, which is usually assumed to have nice properties. It is known that $$ C(u_1, u_2, \ldots, u_n)=\varphi^{-1}...
0
votes
0answers
33 views

Likelihood of Gamma Distribution

I'm studying bayesian stats on my own and came across the following problem on Coursera. Can anyone help me understand how to work through this? $x_{i}\stackrel {iid}{\sim} Beta(\alpha,\beta), i=1.......
0
votes
0answers
9 views

R^square for a pre-determined linear regression

I would like to produce the R^square goodness-of-fit statistics for a predictive model. I have the base data (10, 000 number of x-values) which are the true values given by an analytic/deterministic ...
0
votes
0answers
19 views

A/B test : Is there a way interpreting factor “A” group is greater than factor “B” group?

Plz.. understand my lack of English skills.. T.T I have a problem dealing with statistical hypothesis testing. To be specific, we have two special delivery method like (new) : method "A", (old) : ...
0
votes
0answers
17 views

About the power spectrum and confidence upper limit

For now, I have a coupled system with 5 variables and use the Runge-Kutta method to integrate. ...
2
votes
2answers
63 views

Is “geometric mean” the same as “the first moment of the lognormal distribution”?

I would like to compare the results of two studies, one reporting "geometric mean diameter" and the other one reporting "the first moment of the lognormal size distribution". I am not sure whether the ...
0
votes
1answer
37 views

How to justify using Beta distribution as a prior distribution in the following problem

Let $\theta$ be the proportion of people who are ready to quit smoking within 6 months. Let's say we perform a survey in $2017$ with a $n$ volunteers who ask people this question until they obtain yes ...
1
vote
1answer
63 views

Finding the MVUE from two independent random samples

Suppose we have a random sample $X_1, X_2, \ldots, X_n$ from exponential$~(β >0)$ $\text{i.e. }f(x\mid β) = {1/β} ~e^{−x/β}$ and a random sample$~Y_1, Y_2, \ldots, Y_n$ from exponential$~(⍺ >0)...
0
votes
2answers
59 views

What would p(a,b|a) be equal to?

What is p(a,b|a) equal to in conditional probability? Any sort of breakdown as to why it might be equal to p(b|a) would be helpful (if it is true in all cases). My reasoning for asking this question ...
2
votes
0answers
29 views

Finding the limit of a quotient of hazard functions

Let $\lambda_i(t),S_i(t)$ be the hazard and survival functions of two populations for $i=1,2$ and satisfy that: $\frac{S_2(t)}{1-S_2(t)}=\phi\frac{S_1(t)}{1-S_1(t)}$ (1) I want to proof that $\lim_{t\...
0
votes
1answer
32 views

Seeking origin of variance equation

In my data science textbook, it says that the variance of a variable $Y$ can be written as: $v_y = \frac{1}{n-1} \sum_{k=1}^{n} y_k^{T} y_k$, I have never seen variance defined like this before. ...
1
vote
1answer
66 views

How to prove that the robust F statistic is asymptotically chi squared distributed?

The linear model is $$y_{i}= x_{i}'\beta+u_{i}$$ When written in vector notation such that $y_{i}$ is a $1$ x $1$ matrix of outcomes, $x_{i}'$ is a $1$ x $k$ matrix of control variables, $\beta$ ...
0
votes
0answers
16 views

What is the preferred way to show that a 3-dimensional function is a copula?

I am currently working with 3 dimensions, and have some functions $C(u_1, u_2, u_3)$ for which I need to check whether or not they are a copula. What are all the requirements that I need to check? In ...
0
votes
1answer
50 views

Canonical link of Gamma Distribution [duplicate]

I wonder why my professor said that Gamma's canonical link is $\frac{1}{\mu}$. My thoughts are: EDIT: $\theta$ is the canonical parameter. Since $$\mathbb{E}_\theta(Y)=b^{'}(\theta)=-\frac{1}{\theta}...
2
votes
0answers
32 views

Consistency of Adaptive LASSO

I'm reading the paper on Adaptive LASSO estimator (Zou, 2006). In one of the presented numerical simulation examples (Model 0 (Inconsistent lasso path), page 6 (1423)) they claim the following: To ...
2
votes
1answer
38 views

Covariance of Constrained Maximum Likelihood Estimators

I plan to numerically estimate the parameters of a GLM but with constraints imposed on some of the parameters. In this case, does the general approach of estimating the covariance matrix of my MLE ...
1
vote
0answers
21 views

Help analyzing data ranked on a scale

So, I thought it would be interesting to try an experiment with my family. I baked 13 different kinds cookies this year, and then told my 8 family members to rank from each cookie from 1-13. Is there ...
1
vote
0answers
34 views

Intuition behind the no convergence of the variance of sum of random variables

$$Var[\bar{X}] = \sigma^2/n $$ $$Var [\sum{X}_i] = n\sigma^2$$ $$lim_{n \to \infty} Var[\bar{X}] = 0 $$ wich means at $\infty$ we will always get the same $\bar{X}$ after every simulation. I ...
0
votes
1answer
23 views

Bernoulli distribution/ SOME probability/conjugate prior

I would like to know what "SOME probability of seeing tail" means in the second answer here. I.e. how much is it? EDIT: I do not understand how can I see that there is SOME probability of seeing Tail ...
1
vote
0answers
37 views

how should replications be done for ANOVA? [closed]

I generate my own data (an order list) and with that data I have multiple scenarios which can be performed which give a resulting value. Now I was wondering when I want to do a ANOVA, should I ...
5
votes
2answers
110 views

How to justify that $(Y_1,Y_2)$ is not bivariate normal without finding its exact distribution?

Suppose $X_1$ and $X_2$ are independent $N(0,1)$ variables. Define $$Y_1=X_1\,\text{sign}(X_2)\quad,\quad Y_2=X_2\,\text{sign}(X_1)$$ I have to show that $(Y_1,Y_2)$ is not bivariate normal ...
2
votes
1answer
64 views

Math questions in Kalman filter equation derivation

I am interested in data analysis. While my working data (actually it's shopping mall's daily sale) is accumlating, I wish to find some statistical laws underlying business phenomena. I left school for ...
0
votes
2answers
22 views

Beta distribution and normalization [duplicate]

Here in the 4 pictures in the last answer, is the vertical axe the probability? I.e. it seems to me that it is somewhat unnormalized : it has the value 2 in the 2nd picture and 3 in the 3rd picture. ...
1
vote
1answer
46 views

Linear regression formula with sum of residuals

The relationship in our model between X and Y is linear $Y = X^T\beta + \epsilon$ For arbitrary test point $x_0$ we have prediction $\hat{y_0} = x_0^{T}\hat{\beta}$. Alternatively this can be ...
-1
votes
2answers
38 views
2
votes
0answers
34 views

Observational Data and Bias - A real problem

I'm hoping you all can provide some guidance. I'm working a problem with the following objectives and data set. I would like to be able to predict, for each unit, at each sampled moment, the expected ...
0
votes
0answers
21 views

How to derive the distribution of OLS starting from the sample moments?

I know I am supposed to start from $N^{1/2}[N^{-1}\sum x_{i}u_{i}]$ Then by central limit theorem that that it is asymptotically $ N(E(x_{i}u_{i}),var(x_{i}u_{i})) $ and $E(x_{i}u_{i})=0$ so $ ...
0
votes
0answers
38 views

Unbiased Estimate of a squared difference between sample of random matrices

In the description below, IER stands for "Inhomogenous Erdos Renyi" random matrix, which is basically saying that the entries in the matrix are Bernoulli distributed and iid. I didn't get the part ...
2
votes
0answers
28 views

Simulation - problem of maximization inside a circle

I am doing some projects related to statistics simulation using R based on "Introduction to Scientific Programming and Simulation Using R". In the Students projects session (chapter 24), I am doing ...
1
vote
1answer
49 views

Non Linear transformation of Random Variable

I have a pdf say $p(x)$. Now, I apply some transformation (may be linear or non-linear) to the variable $x$ say $g(x)$. Let the new pdf be called $p(y)$. For, a small change in $x$ say $dx$, there ...
0
votes
0answers
24 views

Why do we use geometric distribution here in place of the multiplication rule?

For the question: "Only 4% of people have type AB blood. What is the probability we won't find a type AB donor before the 10th person?" the textbook says to use a geometric distribution. But for the ...
0
votes
0answers
23 views

Writing the matrix form of a linear regression model?

I don't know how to write a simple linear regression model in a matrix form.. in our book we are given a table having values of $ x,y,x2,y2,xy.$ . I created a very small example and I attached it as ...
0
votes
0answers
8 views

Mini Batch Gradient Descent Backpropapagation

I am a beginner to machine learning. I have derived the equations for backpropagation, and for the weight update for hidden layers, the update rule uses the output vector of the layer to multiply with ...
0
votes
1answer
26 views

Type-I and Type-II errors example explanation

In the following example, can someone help understand how the value for $k$ (highlighted in yellow) is derived below? How is $z_\alpha$ introduced in there?
4
votes
3answers
120 views

Does $\mathbb{P}(X < a) = \mathbb{P}(f(X) < f(a))$?

If $f(x)$ is a monotonic increasing function, then does $\mathbb{P}(X < a) = \mathbb{P}(f(X) < f(a))$? My intuition says it's true but I cannot prove the case nor find the name of the theorem.
0
votes
0answers
12 views

What is the impact of seed weights in raking (iterative proportional fitting)?

In survey weighting, I've seen that first a design weight adjustment is made (to reflect over- or under-sampling), and then a non-response adjustment is made (to reflect groups that are less likely to ...
-1
votes
0answers
15 views

Frequentist properties of Bayesian posterior probabilities under cut-off or classification rules

Consider a two-group randomized control trial for a new hypertension drug with a single primary metric of blood pressure. Suppose we obtain the estimated treatment effect (viz,, mean difference ...
1
vote
0answers
20 views

Computing the p-value of a test statistic converging to a standard normal

We are asked to do a hypothesis test for two random variables, X~Bernoulli(p1), Y~Bernoulli(p2), Ho: p2<=p1, H1: p2>p1. We are given the data for a realisation (x,y) and a test statistic T(X,Y) ...
0
votes
1answer
43 views

Serial correlation structures with two nested random intercepts

This is a Gaussian model of spatial correlation: \begin{align} \boldsymbol y &\sim N(\mu, \boldsymbol V), \text{ where} \nonumber \\ \mu_i &= \ldots \text{ (depends on fixed effects of the ...
1
vote
0answers
21 views

Simulated estimate from a linear mixed model with random intercepts

I wanted to see if there was a quick way to get a bootstrapped parameter estimate and CI using confint() from merTools? ...
0
votes
0answers
13 views

Taking into account sample size

Let's assume that we have: 5 different conditions with an increasing concentration of a stress-causing agent. in each condition, 4 Growth curves and their area [under the curve] were generated from ...
0
votes
0answers
20 views

Degrees of Freedom in 2k factorial design

I am working on a 2k factorial design experiment, and got stuck at one point. If the suppose there are only 2 factors A and B (K=2) each with only one replicate each. This means n=2. Then according ...
0
votes
0answers
28 views

Does conditioning on a random variable yield a random variable?

Let $X$ and $Y$ be random variables and $y\in Im(Y)$ a possible value of $Y$. Is $X|Y=y$ a random variable in the mathematical sense? Or is that just abusing notation?
-1
votes
1answer
45 views

Is Maximum Likelihood Estimation the median? [closed]

I asked what maximum likelihood estimation to a friend of mine. He told me that it is the median which I don't understand.
1
vote
1answer
74 views

How can I calculate $\int^{b}_{a}\Phi\left(\frac{x-\mu}{\sigma}\right)\phi(x)\,\mathrm dx$

Suppose $\phi(\cdot)$ and $\Phi(\cdot)$ are the density function and cumulative distribution function of the standard normal distribution. $a<b$ are finite real numbers. How can I calculate the ...
3
votes
1answer
50 views

Sufficient Conditions for the Central Limit Theorem

My understanding is that the central limit theorem applies as long as the variance of the random variable is less than infinity. Is this equivalent to saying that all moments are finite? If not, what ...
0
votes
0answers
20 views

Bayesian hierarchical design

Hope someone can give me a hint!!!! Bayesian hierarchical design Some people think that in the Bayesian statistical model, the design can be distributed to the unknowns in an endless design. For ...
0
votes
0answers
25 views

P-value on statistical tests [duplicate]

If I have a p-value of 0.01 on a statistical test, does that mean the probability of my hypothesis being wrong is 1%? If not, why not?
0
votes
1answer
36 views

Equivalence of gamma and inverse gamma:Does choosing a prior lead to possibly different evaluations?

Suppose we had an expressions that was proportional to $$\frac{1}{\sigma^{3}}\exp(\frac{-\beta}{\sigma^{2}})$$ My question is, can you choose different possible parametrization for a prior ...
0
votes
1answer
54 views

Type I and Type II errors in Hypothesis Testing

I am confused about the last highlighted sentence regarding finding a subset S, for which BOTH Type I and Type II error probabilities are 0. For $P_\theta S$, which is the probability with which we ...