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3.823 mathematical-statistics questions.

Given the following likelihood function $$f(y|x,\tau) = \prod_{i=0}^Nf_T(u_i-x_i-\tau) \tag{1}$$ where, $f_T(t)$ is the probability density function of an Inverse Gaussian distribution given by ...

As a former mathematics student, when reading any math-related materials I tend to care about their mathematical rigour very much. Such high attention to mathematical details might be a good habitat ...

I have the a time series data, the acf and pacf for which have been displayed below: I get that MA term is 1. But I'm confused about AR term since it is geometrically decaying from 7th lag. Do I need ...

I am being asked to apply a statistical technique that I do not think is optimal. I have asked before, but was told my question was vague. I have re-worded it appropriately here. I am being asked to ...

This is from a very cool paper about horse racing, I tried it at Excel much time, but all failed, I mean cannot estimate the likelihood. https://imgur.com/WsC95aC "formula" My Excel is like this: ...

Lyapunov condition $$\lim_{n\to\infty} \frac{E|V_n-E(V_n)|^{2+\Delta}}{n^{\Delta/2}\sigma^{2+\Delta}V_n}$$ is to be prove for $\Delta=0$, where $$V_n=\frac{1}{h}\exp\left\{-\left(\frac{j-x}{h} +\...

Given $X_1, \dots, X_n, \dots \sim \mathscr{N}(0,1)$, consider the random variables $$ Z_n := \max_{1 \le i \le n} X_i\,. $$ Question: What is the most "important" result about these random ...

1. For a sequence of random variables $V_n$, and a deterministic sequence $b_n$, does $$ \frac{V_n}{b_n} \overset{a.s.}{\to} c \quad \left( \implies \frac{V_n}{b_n} \overset{P}{\to} c \right) $$ ...

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 ...

Find the r th moment ? Then find the expected and variance Note : the answer use the digamma

I have a random variable and wondering if there's a mathematical test to see if it has an oscillation pattern like a sine wave. Data is a time series collected every day and follows a random walk. ...

In Elements of Statistical Learning, a problem is introduced to highlight issues with k-nn in high dimensional spaces. There are $N$ data points that are uniformly distributed in a $p$-dimensional ...

I am working on a project where my predecesor has been analyzing a table of rows by columns of count data. Brands represent the columns, and statements about those brands represent the rows. The cells ...

Let us assume that we have two random samples, $X$ and $Y$, where $Y=X+Z$, so $X$ and $Y$ are dependent and have different number of observations. What is more, suppose that $X$ and $Y$ do not have ...

In the text they say they used a Mann Whitney U test. In one section of the results they report this: Association of PBP 2 with antimicrobial susceptibility. Overall, considerable variation ...

I've run up against a wall in reconciling two different definitions of the Ornstein-Uhlenbeck process, and would appreciate some help. On the one hand, as discussed here, we can define an Ornstein-...

In classical statistics, there is a definition that a statistic $T$ of a set of data $y_1, \ldots, y_n$ is defined to be complete for a parameter $\theta$ it is impossible to form an unbiased ...

I tried to obtain the score vector (1st derivative of density function w.r.t. parameters) of multivariate normal distribution. Density function of multivariate normal distribution: \begin{align*} f(x)...

I have several confusions regarding estimating sample size. Consider the situation where I have data from a pilot, from which I estimated effect size and want to do power analysis (using software such ...

(edited version) I have a dataset of UK general election results and want to compare two groups on some criteria (education, health etc based on 2011 Census). These are the groups: the seats gained ...

Let $V$ be an $n$ dimensional space with sets of positive class vectors $P$ and negative class vectors $N$. The task is to find a vector $x$ such that AUC is maximized, based on ranking generated by ...

I was reading the original paper on ADAM (Adam: A Method for Stochastic Optimization), which mentions: [...] invariant to diagonal rescaling of the gradients, [...] What does it mean? Also, ...

I have various products and for each product I have 5 types of cost(not just monetary cost) variables associated with it, the value of each variable for any product is a positive integer. I want to ...

I'm trying to forecast out sales in 2019 using significant independent variables, however these are mostly, if not all, unknown. The method I'm currently using is to use excel "forecast" function to ...

I want to measure the improvement of students using exam marks of semester 1 to semester 3. The following detail should be considered when calculating the measure; If Student A get 30 (sem 1) and 50 (...

I have a signal (Y) with 200000 samples. I plotted probability density function (PDF) of the ...

Let $X$ be an $n \times p$ matrix with rank$(X)=k$. Let $\epsilon \sim N(0,I_p)$ be a vector with the i.i.d. Gaussian components. (a) Show that $$E\|\ X\epsilon \|_2^2 = \|\ X \|_F^2 = \sum_{...

I'm fitting a Cox model with one predictor, X. That is, $$h(t) = h_0(t) exp(X_i \beta).$$ I'm an interested in getting a confidence interval for the hazard ratio of a 10 unit change in X instead of a ...

I have a data set containing sales data per day for Jan-April 2017 and Jan-April 2018, I need to forecast the sales values for Jan-April 2019. What would be the best method for this? Example of the ...

For ridge regression I learned before, $\hat{w} = argmin_{\theta}||y-Xw||_2^2 + \lambda||w||_2^2$. Thinking about if the bias is added, so the new $X$ become $[1,X]$, and we have a new weight $\theta$...

I've been in a debate with my graduate-level statistics professor about "normal distributions". I contend that to truly get a normal distribution one must have mean=median=mode, all the data must be ...

Let the task be classification and the neural network under discussion to have Sigmoid activation functions and be trained by Backpropagation and SGD. How can I force the networks hidden activations ...

For any random variable $X$ whose density is $\mathbb{P}(X=x)=p(x;\theta),$ where $\theta$ is a parameter, its deterministic function representation is $X=f(\theta, \omega)$ where $\omega$ is a random ...

I have a data set which consists of > 500 hedge funds, their historical monthly returns, and their benchmark (index) monthly returns. The number of data points (# of monthly returns) differs from a ...

Introduction: I'm studying "Statistics-Business-Economics-Paul-Newbold", chapter 6, topic: "Acceptance intervals" (page: 260). (https://www.amazon.it/Statistics-Business-Economics-Paul-Newbold-ebook/...

I wanted to prove the equation, but I'm somehow stuck in the middle of the process. To solve the right side, I have $[{E[(Y|X)^2}] + E[Y^2] - 2E[Y*E[Y|X]] + E[{E[Y|X]}^2]$ ==> $2 E[{E[(Y|X)^2}] + E[...

A random variable is a function from the space of outcomes to real numbers (there are extensions to this, but it's not important for the purposes of this question: see wikipedia). The question is: ...

After studying James-Stein estimators for a few weeks and looking at many different sources I am stuck at trying to understand how Efron and Morris calculated the Toxoplasmosis example in their 1975 ...

Given a sample $x_1, x_2, \cdots x_n$ from the pdf: $$ f(x ; \theta) = (\theta + 1) x^\theta $$ where $0 < x < 1$ and $\theta > -1$ is unknown. What is the bias of the MLE of $\theta$? I'...

When we perform multiple linear regression with centered predictors (that is, $x_{ij}^c = x_{ij} - \bar{x}_j$) we get the same coefficients as with the original predictors but a different intercept. I'...

How do I work out the p value if the variance of my control and experimental group are different? I can't do the t-test anymore. How do I work the p -value out, what method? Also, my two groups are ...

Today, I read a post Computing and Sustainability: What Can Be Done?. And I found that the author of this post can easily find statistical problems in other fields, such as computer science. Since as ...

So I am working with the time people have been living for in urban renewed place, so looking at the influx of people over the years, during three periods: before renewal, whilst renewal and after ...

I'm trying to make sense of the following sentence from introduction "Multiple discoveries: Distribution of roots of determinantal equations" http://statweb.stanford.edu/~ckirby/ted/papers/...

suppose $X_1,X_2,\ldots,X_n$ be random sample of $N(0,\sigma^2)$. how can I calculate UMVUE of parameter $(1-\sigma^2)^{-\frac{n}{2}}$

This maybe a naive question, but I found two ways to calculate the relative error, one is common and another is from the papers I read. Assume that the exact number is $n$, and the estimated number ...

Suppose I have $X_1|\theta_1$ and $X_2|\theta_2$ such that $X_i \sim D(\theta_i)$ and $D$ is some known distribution with parameter $\theta$. $X_1|\theta_1$ and $X_2|\theta_2$ are assumed independent. ...

Suppose there is a sample of varying size $n\in \mathbb N$, each sample point taking values in $\mathbb R$, and a statistic $T$. If I am correct, a statistic can accept arbitrary sample size. What are ...

I don't quite understand "by the properties of the Fourier transform, $it^r\psi(t)$ is the Fourier transform of $(-1)^rD^r\Psi(x)$" from Wikipedia about Gram–Charlier A series and Edgeworth expansion:...

On page 161 of Lehmann and Casella's Theory of Point Estimation, they introduce Functional Equivariance and Formal Invariance. $\bar{h}$ appears in two places: in "Functional Equivariance": $\...

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