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0
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6 views

log multivariate normal differentiation with VAR process

I am trying to estimate a regime switching model with an autoregressive component using the EM algorithm. The process itself can be presented this way: $$ r_{t}= A_{n \times (n+1)} \boldsymbol ...
0
votes
1answer
16 views

Identification of asymptotic distribution of a function of the MLE

Let $X_1, X_2, \dots, X_n$ be Bernoulli$(\theta)$ and let $\hat{\theta}$ be the MLE of $\theta$. I am attempting to identify the asymptotic distribution of the odds ratio. I believe that I ...
0
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1answer
12 views

Unbiased estimator and variance

A random sample of n people are asked whether they are against smoking or not. Suppose x are against smoking. What is the distribution of the random variable X (number of those against smoking). State ...
2
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2answers
26 views

Principal Component Analysis Question

In a previous year exam about data analysis, there was a question that I'm confused about. Here's the question: Explain why it is natural to expect that in real-world high dimensional data the ...
0
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1answer
26 views

Which two values on this curve will give the largest uncertainty?

Hello I'm sorry if this question is too basic for this site, I was asked in a question which two points on this curve give the largest uncertainty value from reading off the graph. and to give 2 ...
2
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0answers
22 views

Box-Jenkins Forecasting With ARIMA(p,d,q) models

I want to check that I understand the general theme of forecasting with ARIMA models using box-jenkins, so I am going to take an example and then proceed from there. We will use $B$ notation for the ...
2
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0answers
28 views

Showing a Time Series is Regular

Note: this is not an assessed question, it is a practice question from a past exam. If we are given the following time series: $X_t=\alpha^2X_{t-1}+Z_t-\alpha Z_{t-1}+2\alpha^2$ I am asked to find ...
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1answer
21 views

Auto-covariance in a stationary time series

I am trying to calculate the auto-covariance of time series $Z_t$. Given a weakly stationary process $Y_t$; $t \in \mathbb{Z}$: $$Z_t=a+Y_t$$ Now, my goal is to show that $Z_t$ is also a stationary ...
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1answer
48 views

Can I say that residuals are white noise?

I want to check whether residuals are white noise or not. When I look at the plot, all lags do not pass(exceed) the significance band except for fourth lag. However, fourth lag's p-value of 0.228 is ...
1
vote
1answer
30 views

Rewriting RegressionSS

I am having troubles in deriving the RHS of this formula: $$RSS= \hat{\beta}^{T}Z^{T}y-n\bar{y}^{2}$$ where $Z$ is the design matrix I am given ${y}^{T}y$ (what exactly does ${y}^{T}y$ stand for and ...
0
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0answers
20 views

Information gain of flipping coins

I'm studying for an exam on Bishop's Pattern Recognition and Machine Learning (ISBN: 978- 0387310732). One of the questions in the mock exam paper is: Three fair coins are flipped sequentially; you ...
2
votes
1answer
27 views

Efficiency becomes 0 when sample size becomes big?

I am trying to solve Robert Hogg's mathematical anaysis 6th exercise 6.2.11. The problem says. Let $\bar{X}$ be the mean of a random sample of size n from a $N(\theta,\sigma^2)$ distribution, ...
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0answers
8 views

Equally Weighted Portfolio

My question is how to create an equally weighted portfolio with three stock indices. Would it be that i first calculate their daily return separately by the formula: P2-P1/P1. And call it R1 R2 R3 ...
0
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0answers
10 views

long run relationship cointegration [on hold]

I have been given a question for my assignment: "Test for the presence of a long run relationship (or cointegration) between the three financial assets. Estimate the joint relationship using the ...
2
votes
1answer
33 views

Prove the loglikelihood is strictly concave for ABO allele frequency blood type data

I am working through the problems in Kenn Lange's book Numerical Analysis for Statisticians. I am going to try and do all of the problems in the book, though none of them are specifically assigned for ...
3
votes
1answer
40 views

Probability that one sum of squared standard normals is greater than a constant times another such sum

Let $Y_1,Y_2,..., Y_n$ $i.i.d$ $\mathcal{N}(0,\sigma^2)$ random variables and $n>4$. Find the probability $\mathbb{P}\{Y_3^2 + Y_4^2+ ... + Y_n^2 \geq \alpha ( Y_1^2 + Y_2^2)\}$ for $\alpha ...
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0answers
24 views

Likelihood Ratio Test for two Poisson variables

I am trying to put together a Likelihood Ratio Test for two sets of Poisson variables: $X_i$ and $Y_i$ (both with n points of data), doing a hypothesis test on their parameters $H_o: ...
2
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1answer
22 views

Solving the probability of independent events without the complement

Suppose that virus transmission in 500 acts of intercourse are mutually independent events and that the probability of transmission in any one act is $\frac{1}{500}$. What is the probability of ...
0
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0answers
19 views

How to interpret a discrete variable in regression model

I have a regression model where the dependent variable is logarithm of miles per gallon and several IVs. One of the IVs is ...
0
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1answer
19 views

Transformation of any normal distribution into a standardized t-distribution

What will be the transformed Mean and transformed standard deviation if any normal distribution is transformed into a standardized t-distribution? Does t force ...
2
votes
1answer
22 views

Dependent or Independent samples? I cant decide! Help

I'm confused i have assignment where i have to determine if the samples are dependent or independent. The sample is of 100 people, where their reaction times from their dominate and non-dominate hands ...
2
votes
2answers
49 views

What will be the t-value if my sample size increases to an infinity compare to z value?

I do not know where to check this. Any reference/help is much appreciated. If the sample size, n increases to an infinity then will the t-value be larger/smaller ...
0
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0answers
15 views

Log-Log transformation of a regression model

I am creating a regression model project to my econometrics lecture. Because of the heteroskedasticity I have to transform the model somehow. The way I used is log-log transformation. My question ...
1
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1answer
22 views

Clarification on Notation

I'm using Andrew Gelman's 3rd edition of Bayesian Data Analysis and am going through the exercises. For one of the exercises, he supposes that if $\theta = 1$, then $y$ has a normal distribution with ...
0
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1answer
21 views

test on a proportion with the hypotheses

Suppose you are doing a test on a proportion with the hypotheses $H_0$: p = 0.4, $H_a$: p $\neq$ 0.4. In addition, you plan to use a sample of 50 values, and a significance level of $\alpha$ = ...
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0answers
34 views

Independent RVs theorem: rigorous?

I am reproducing here theorem (#3.30) from "All of Statistics" by Larry Wasserman: Let X and Y have joint pdf $f_{X,Y}$ . Then $X\perp Y$ if and only if $f_{X,Y}(x,y)=f_{X}(x)f_{Y}(y)$ for all ...
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1answer
37 views

What is the value of π in this experiment(Bernoulli)?

Experiment: You roll a fair 6-sided die 5 times. Define the random variable x = number of times you rolled an even number. The probability of exactly X successes in n trials for a Bernoulli process ...
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0answers
39 views

How to calculate mean and standard deviation of a range variable (grades) when the raw data is based on frequency categories?

Results of an exam indicate that: 13% of the class earned an A 20% earned a B 48% earned a C 10% earned a D 9% earned F's The grade C ranges from 70-79%. Assuming that the data is normally ...
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0answers
15 views

likelihood ratio testing

Let $Y_{1}, Y_{2}, \ldots, Y_{n}$ denote a random sample from a $N(\theta,\ \sigma^{2})$ population. Consider testing $H_{o}: \theta\geq\theta_{o}$ versus $H_{a}: \theta<\theta_{o}.$ If ...
3
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1answer
27 views

Karlin-Rubin with Beta distribution [duplicate]

Suppose $Y$ is one observation from a population with a Beta $(\theta,\ 1)$ pdf. Use the Karlin-Rubin theorem to find a UMP level $\alpha$-test (based on $\mathrm{Y}$) of $H_{0}$ : $\theta\leq 1$ ...
7
votes
4answers
136 views

Suppose $X_1, X_2, \dotsc, X_n$ are i.i.d. random variables. When is the sequence expected to decrease for the first time?

As suggested in the title. Suppose $X_1, X_2, \dotsc, X_n$ are continuous i.i.d. random variables with pdf $f$. Consider the event that $X_1 \leq X_2 \dotsc \leq X_{N-1} > X_N$, $N \geq 2$, thus ...
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0answers
18 views

Limiting distribution of a Markov chain?

I have the problem below. There are n identical machines. They are all operational at time 0. The lifetime of each one is an exponential random variable with rate L. There are r repairmen (1 ≤ r ≤ ...
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0answers
17 views

2-dimensional density of Brownian bridge?

I know that a $1$-dimensional Brownian bridge $B(t)$ just follows a normal distribution with mean $0$ and variance $t(1-t)$. But how do I compute the 2-dimensional density? I mean, $\{B(s), B(t)\}$ ...
0
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1answer
29 views

Dummy Variable problem

I am doing a regression project based on this dataset. I wonder whether wouldn't it be better to transform the IV origin from 1,2,3 to three dummy variables like this: When the car would be from ...
0
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1answer
16 views

MPG Dataset UCI Repository

I want to use this dataset for my regression project. Does anybody know in what units is the variable weight? Pounds or kilos? Thanks a lot!
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0answers
17 views

Spatially inseparable data - what to do? [closed]

I'm new to machine learning and trying to solve this problem. In all tutorials, samples and so on the data is usually plotted in 2D and you can see some kind of structure which then the algorithm ...
4
votes
3answers
129 views

In which case $\mathbb E[X]=\sum _ix_i P[x_i]$ can be $0$ when all $x$'s are not zero ($0$)?

Say $X$ is a random variable and $x$'s are realizations of $X$ . Say , $\mathbb E[X]=\sum _ix_i P[x_i]=0$ . But I do not understand in which case $\mathbb E[X]=\sum _ix_i P[x_i]$ can be $0$ when all ...
0
votes
4answers
151 views

Why is $\mathbb E(X)=\sum_{i=1}^{n}x_i P(x_i)$?

If $X$ is a random variable and $x$'s are the realizations form $X$ and $N$ is the population size $n$ is the sample size Which one is correct $\mathbb E(X)=\sum_{i=1}^{N}x_i P(x_i)$ or ...
3
votes
2answers
36 views

Errors and Residuals

In Wikipedia , it is written that : the sum of the residuals within a random sample is necessarily zero, and thus the residuals are necessarily not independent. The statistical errors on the other ...
0
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1answer
32 views

For the model $y_i=\beta_0+\beta_1x_{1i}+e_i,\quad i=1,\ldots,n$ , does $e_1=e_2$ imply $y_1=y_2$?

Which one notation is correct and why ? $y_1=\beta_0+\beta_1x_{11}+\epsilon_1$ or, $y_1=\beta_0+\beta_1x_{11}+e_1$ or, $Y_1=\beta_0+\beta_1x_{11}+\epsilon_1$ or, ...
0
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0answers
46 views

How to indicate a 95% confidence interval for the total amount to be paid

Can you please help me to solve this question! Peter who is soon to be married, has for a long time been thinking about the perfect morning gift for his bride, Kate. As the couple is going to run a ...
0
votes
1answer
23 views

Computing Issues with Kriging

I am having some issues with Kriging in R, and I was looking for some idea where I am going wrong. From what I can tell, I done a decent job removing the trend, and I believe my transformed data is ...
4
votes
1answer
131 views

Minimization of the Sum of Absolute Deviations

My particular task is to show $|Y_i-B_0-B_1X_{i}-B_2X_{i}^2 |$ has more than one min value. We are given $x_1=1$, $x_2=2$, $y_1=3$ and $y_2=4$. I am truly lost, I need to show there are more than ...
2
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2answers
68 views

Constructing alternate hypothesis: How to determine if Ha > H0 or Ha < H0

In Chapter 8, Test of Hypotheses based on a Single Sample, in Devore's Probability & Statistics for Engineering and Sciences, he states that the null hypothesis, $H_{0}$, is the a priori claim ...
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0answers
16 views

Finding conditional distribution in graphical model (undirected graph)

Given that I have a graph $G=(V,E)$ and a set of random variables $X:=(X_v: v\in V)$. I also have the joint distribution of $X\sim p(x)$. What are the ways to find out the conditional distribution of ...
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0answers
30 views

neural network multiple layers feed forward

NN on figure below has two nodes (N0,0 and N0,1) in input layer, two nodes in hidden layer (N1,0 and N1,1) and one node in output layer (N2,0). Input layer nodes are connected to hidden layer nodes ...
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0answers
22 views

Conditional Expectation of Order Statistics

Given $X_1,...,X_n \sim f(x)$ How do I find $E(X_{(1)} | X_{(2)})$? Would I have to find the conditional pdf and integrate wrt x? I get the conditional distribution to be $f_{X|Y}(x|y) ...
1
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1answer
25 views

positive price coefficient after instrumentation in demand estimation

I need to complete an assignment for Industrial Organization course where one of the tasks is to estimate a discrete choice demand model. This means I basically need to estimate a linear model: ...
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2answers
46 views

$X \sim N(0, \sigma_1^2)$, $Y \sim N(0, \sigma_2^2)$, $U = X+Y$. What are $E[X|U], E[Y|U]$?

$X \sim N(0, \sigma_1^2)$, $Y \sim N(0, \sigma_2^2)$, $U = X+Y$. What are the values of $E[X|U], E[Y|U]$? I understand $E[X|U] + E[Y|U] = U$, but I'm not sure how to move forward...
1
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0answers
151 views

Likelihood and sufficient statistics

a)Find the maximum likelihood estimador for $a$ in the density $f(x;a)=\frac{2}{a^2}(a-x)I_{(0,a)}(x)$. b)Is it a sufficient statistics? I did $$\prod f(x;a)=\prod_{i=1}^2 ...