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3.970 mathematical-statistics questions.

Let $P_t$ denote a stock price distributed as $\operatorname{lognormal}(\mu , \sigma^2 )$. Suppose we construct simple returns $R_t=\frac{P_t-P_{t-1}}{P_{t-1}}$. My question is: What is the ...

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 ...

$X_i \stackrel{iid}{\sim} N(\mu,\sigma^2)$ i=1,2,...,n $Z_n=\sqrt{n}(\bar{X} - \mu)$ I believe the asymptotic distribution of $Z_n$ is $N(0,\sigma^2)$. So what would the asymptotic distribution of ...

I would like to know the name of a statistics textbook in which the notion of $p$-value is treated as detailed as possible.An Amazon link will help.Thank you.

Here on page 7, example 2.7. The claim is that sufficient statistics for $d$ dimensional multivariate normal $\mathbf{x}_i \sim N(\vec{\mu}, \Sigma)$ is $$\left(n^{-1}\sum_{i=1}^n \mathbf{x}_i, \hat{\...

A basketball player succeeds in making a basket three tries out of four. How many times must he try for a basket in order to have greater than 0.99 probability of making at least one basket? In this ...

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 ...

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 ...

In one of assignments in a study programme I got this statement: Hypothesis: University towns have their mean housing prices less effected by recessions. Run a t-test to compare the ratio of the ...

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 (...

I was thinking about the meaning of location-scale family. My understanding is that for every $X$ member of a location scale family with parameters $a$ location and $b$ scale, then the distribution of ...

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 $a^T Z$, $\forall a \in R^p$ ...

I am still confused about Hypothesis testing. How does one set up the null hypothesis $H_0$ and the alternative hypothesis $H_a$? I have read a post here that doesn't give it much credit, as far as ...

Basically, let's assume I have a simple rules-based function/model (if weight >= 150) -> return true. Simple binary answer (true or false) from a single feature input. If I have a range of samples/...

In a certain university, the students were informed that they need a grade in the top 8% of the computer science students to get scholarship for the next semester. In the standardisation of the test , ...

In my hypothesis test, under H0, if I get the p-value for the test statistic as exactly 5%(please assume), then should I reject H0 or fail to reject H0. Test carried out at 5% level significance. ...

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 ...

I was doing some work in scipy and a conversation came up w/a member of the core scipy group whether a non-negative discrete random variable can have a undefined moment. I think he is correct but do ...

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) = \...

I have a sample $X=(X_1, ...,X_n)\sim N(\mu,\sigma^2)$ with $\sigma^2$ known. The hypotheses are $H_0: \mu=\mu_0, H_1:\mu \neq \mu_0$. I know that in such a case an UMP test does not exist and so ...

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. ...

Tried but failed to prove the following assertion: if $ (X_n)_{n \ge 0} $ is a process adapted to a filtration $ ( {\mathcal A}_n )_{n \ge 0} $ such that the family $ \{ X_n : n \ge 0 \} $ is ...

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 ...

Suppose that an urn contains $6$ balls in all, of $\theta$ different colors, with an equal number of balls of each color, where $\theta\in\Theta ={\{2,3}\}$. (a) Letting $X$ be the number of ...

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 \...

I've read in several sources, for example http://lipas.uwasa.fi/~sjp/Teaching/ecm/lectures/ecmc3.pdf page 51, that if you omit a relevant variable from an OLS regression the resulting standard errors ...

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 ...

I am doing bayesian analysis. Exploratory analysis shows the parameter might has a left skewed shape. So what kind of distribution should I used as prior distribution for this parameter? Any kind of ...

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 ...

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 $27.00$. ...

I need to show that F test is equal to T test squared, when the T test is for 2 independent groups and assuming variances are equal. I know that $F=\frac{MSB}{MSW}=\frac{SSB/k-1}{SSW/N-K}$ and I know ...

The following question I found on an old exam: Given $n$ i.i.d. random variables $X_k$, $1 \leq k \leq n$, with uniform distribution on $[-1,1]$, it is easy to compute the characteristic function of ...

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 ...

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}(...

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 ...

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 ...

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 ...

I have recently come across a testing design that Adobe uses in their "Adobe Target" tool that they call "Auto-Allocation". Here is their article on the test design. https://marketing.adobe.com/...

I have a serious doubt about one characteristic of the two-sided test: Suppose we have a random variable: $ X \sim \mathcal{N}(\mu,\,\sigma^{2}) $ where $\sigma = 0.4$ but $\mu$ is unknown. We ...

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 ...

I'm trying to work out why one-hot encoding is required, rather than, for example, assigning a different integer to each of the classes. Eg, in an machine learning image classification task, why can'...

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 ...

I'm reviewing the proof the estimator of the trace of a matrix and am having trouble reconciling a jump in the proof of the variance of the estimator. The paper with the proof is found here. The ...

I have been struggling computing the Fisher's information of the Wishart distribution. I'll write what I have gone through. Let's $\Omega$ denote a $p\times p$ Wishart random variate denoted by $\...

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 ...

Let $X_1,...,X_n$~ $n(\theta,\sigma_0^2)$, where $\sigma_0^2$ is known. Given the hypothesis $H_0: \theta = \theta_0$ vs $H_1: \theta \neq \theta_0$, I know that a LRT has rejection region \begin{...

I am trying to understand the asymptotic distribution of the Wald test statistic, specifically under the alternative hypothesis which I've found little reference to. For clarity, the binary ...

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 ...

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 ...

I understand that a stable distribution is a distribution whose linear combination of two independent random variables with this distribution has the same distribution (ignoring location and scale ...

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