The area of statistics that focuses on taking information from samples of a population, in order to derive information on the entire population.

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What is a easy way to draw from distribution $f(x|u) \propto x^{\alpha-1}I(x<u)$?

We know $u$ and want to draw $x$ from the conditional density $f(x|u) \propto x^{\alpha-1}I(x<u), \alpha>0$. One way is that first draw $r$ from uniform(0,1), and then set $x=u ...
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653 views

How to quantify the differencen between 2/4 and 20/40?

Assume I have two methods to do prediction. The first method makes 4 predictions and 2 out of 4 are correct. The second method makes 40 predictions and 20 out of 40 are correct. The prediction ...
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37 views

Is there a method to check if two curves (non-linear) are identical

I have two data sets of pollutant concentration on simultaneous days. I have to check whether these two curves follow similar pattern or not ( there might be some time lag between both) on daily ...
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70 views

Density functions and estimators

A random variable is said to have probability density function $$f_X(x)=\frac{\alpha k^\alpha}{x^{\alpha +1}},\quad \alpha , k>0 \; \text{ and }\; x>k.$$ 1. Compute the MLE estimators ...
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79 views

Fisher information matrix of MLE's

I know what it means to compute the fisher information matrix of a vector of parameters. However, how does one compute the fisher information matrix of a vector of MLE's? Specifically, I am working ...
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9 views

Finding the Optimal Function Composition to Maximize Goodness of Fit [on hold]

Assume I have one function $f(x)$, which optimally models empirical data in the range $[x_0, x_n]$ and a second function $g(t)$, which is optimal for the range $[x_{n+1}, x_{max}]$. $^1$ The function ...
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9 views

Topics under Model Based Cluster Analysis

Can anyone recommended topic(s) I could use for my thesis under "Model Based Cluster Analysis"? I initially used "Inference in Model Based Cluster Analysis" as my working topic but appears to be ...
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51 views

Infinite population mean?

When reading about the central limit theorem, the concept of infinite population mean arises.How can a population mean be infinite?
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28 views

Conditional expectation and rao-blacwell

I am studying on UMVUE, and I'm struggling to find that conditional expectation Let $X_1,\ldots,X_n$ random sample of $X\sim U[0,\theta]$. i) Show that $2X_1$ is a unbiased estimator for $\theta$ and ...
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7 views

Ellipsoid confedence intervales?

Are Bonforonni, Scheffe, Multivariate t, and Tukey for simultaneous Confidence intervals are ellipsoid? How can I tell from the form of the interval that it is ellipsoid or rectangular?
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Rao-Blackwell theorem and uniform distribution [on hold]

Let $X_1,...,X_n$ random sample of $X$~$U[0,\theta]$.Use the fact that $X_{(n)}=max(X_1,..,X_n)$ is a sufficient and complete statistic and Rao-Blackwell theorem for show that ...
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Rao-Blackwell theorem and conditional distribution

Let $X_1,..,X_n$ random sample of $X\sim\text{Exp}(\lambda)$ with $f(x;\lambda)=\frac{1}{\lambda}e^{-\frac{1}{\lambda}x}I_{[0,\infty]}(x)$ i) Find a unbiased estimator of $\lambda$ based ...
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11 views

Question regarding the density function of first n prediction

This is an example from Bertsekas' Introduction to Probability 2nd edition example 8.2 Consider now a variation involving the first $n$ dates. Assume that Juliet is late by random amounts $$X_1, ...
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10 views

Given a sample determine using Chi-squared test whether these values fit in an EXPONENTIAL distribution

Here I've got such a problem. I was given $n = 20$ values for time of good functioning of a robot between two consecutive defects. 1200, 1432, 1502, 1100, 3286, 4235, 1149, 5236, 2234, ...
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2answers
28 views

In Bayesian Statistic how do you usually find out what is the distribution of the unknown?

To estimate the posterior we have $$p(\theta|x) = \frac{p(\theta)*p(x|\theta)}{\sum p(\theta ')*p(x|\theta ')}$$ $x$ is usually the experimentally sampled data, and $\theta$ is the model, but both ...
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2answers
31 views

Calculate the confidence interval of parameter of exponential distribution?

How can I calculate the confidence interval for parameter $\alpha$ of exponential distribution ? I think I can use test-t. Knowing that: $$mean = {1\over\alpha}$$ I found that : $${1\over ...
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33 views

Confidence Interval for Nonlinear Regression using F-Test - lmfit

I am trying to understand the implementation for the lmfit confidence interval calculation - in the docs it is stated: "The F-test is used to compare our null model, which is the best fit we have ...
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1answer
18 views

Test of confidence intervals?

In one of my assignments I have to "test" if the confidence intervals for a set of parameters in a mixed effect model is accurate. I'm asked to simulate from fittet parameters and there after refit ...
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1answer
21 views

Defining bias function for n trial

Let a point estimate for the sample variance be given as $\hat{\sigma}^2 = \frac{1}{n}\sum\limits_{i=1}^n(X_i- \bar{X})^2$ where $n$ is the number of samples. What is the bias in this estimate as a ...
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1answer
8 views

mean square error comparison

Do you have any idea about how i can solve the question below? $X_1$ and $X_2$ are random variables that satisfy $E[X_1]=E[X_2]=\mu$ and $Var[X_1]=Var[X_2]=1$. Show that when $|\mu - 10| \leq ...
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30 views

MLE of Integer Valued Normal Distribution

If Z is a normal random variable on $\mathbb{R}^d$ with parameters $(\mu,\Sigma)$ and we know that $\mu\in \mathbb{Z}^d$ and $\Sigma \in \mathbb{Z}^{d+}$; then how can we solve this MLE problem for ...
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12 views

Correlated random effects?

Say I have a linear model where two random effects are possibly correlated, do I still have a random effect matrix for each of them? for instance if $Z_1X=U_1,Z_2W=U_2$ are two possibly correlated ...
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1answer
20 views

Converting Univariate Time Series to a Multivariate Time Series

$X_{t} =0.9X_{t-1}-0.7X_{t-2} + \epsilon_t$ Its clear that the above process does not exhibit the markov property, i.e the future depending on the present. How would I rewrite the above Time Series ...
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40 views

Series of sums of normal variables, likelihood principle

Suppose I have a series of normal variables $Y_i \sim \mathcal N(\theta, 1)$ for $1 \leq i \leq N$. Define: $$S_k = \sum\limits_{i=1}^kY_i$$ Since they're sums of normally distributed variables, ...
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inferring parameters from limting relative frequencies

I refer to my previous question concerning what i call the converse strong law of large numbers (instead o the normal SLLN given the probability=p that with prob1, the limiting relative frequency=p; ...
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1answer
22 views

Linear Regression quadratic terms

I have a hard time understanding the term 'linear regression'. For what I know, linear means polynomial of degree 1. But then, I found that in one of my lectures, the lecturers are saying that this ...
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1answer
17 views

What does the notation $\{ \pm 1 \}^X$ in relation to functions and hypothesis classes means in the context of PAC learning over half spaces?

I was reading the following paper (on PAC learning over half-spaces) and encountered the following notation for a hypothesis class (on page 4): $$\mathcal{H} \subset \{ \pm 1 \}^X$$ However, it was ...
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18 views

iid sequence of random vectors

If $W_1,...,W_N$ is an iid sequence of random vectors, with $W_i=(X_i,Y_i)^T$, does $W_1,...,W_N$ being an iid sequence imply that $X_i$ will be independent of $Y_i$? Does it imply that $(X_i,Y_i)$ ...
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10 views

Statistical Modeling with the combination of two models

I'm having a modeling problem now. Assume we have discrete random variable Y and continuous random variables X and Z. First, we assume a logistic regression between Y and Z.(Assumption One) Also, we ...
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X correlates with Y and Y correlates with Z when p-values are known

"How can I calculate the range of correlation of two variables X and Z given I have the correlations of X and Y, and Y and Z?" These are useful resources: ...
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1answer
21 views

Does scatterplot matrix “work” with quadratic variables?

basically I want to plot a scatterplot matrix using a few variables. For simplicity lets say my model is: $$z=\alpha_0 + \alpha_1w+\alpha_2x+\alpha_3y+\alpha_4y^2 + \epsilon$$ When I plot the matrix, ...
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22 views

Uniform most powerful Test for one-sided hypothesis

I am trying to understand this proof above. What I am confusing is (1) The whole theorem correspond to the hypothesis $H_0:θ\leθ_0 \, vs \, H_1:θ\gtθ_0$. But at the beginning of the ...
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1answer
17 views

Question about law of iterated expectations?

I have this question: Let Y = a + bX + U, where X and U are random variables and a and b are constants.Assume that E(U|X) = 0, and that Var(X) > 0. I need to find E[UX] The answer is zero, found ...
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Is this an instance of the base-rate fallacy?

The following line of probability reasoning is supposedly fallacious, and is an instance of the base-rate fallacy. The argument is that $(1)-(3)$ don't give us enough reason to conclude that $(C)$. ...
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25 views

Distribution of Normal distribution

suppose $X \sim Normal(\mu, \sigma^{2})$. What is the distribution of $Y := N(X)$? where $N$ denotes Standard Normal Cumulative Distribution Function? e.g. in a special case when $\mu = 0$ and $\sigma ...
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36 views

A woman has n keys

A woman has n keys, of which one will open her door. After trying one she discards it and tries again if it does not work. What is the expected number of attempts needed? Its straight forward to see ...
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1answer
11 views

Nonparametric Skew of Data

Recently in my studies of statistics, I have come across the second skewness coefficient to determine the skewness of the set of data. The formula is given by: $$ \frac{3(\mu - \nu)}{\sigma}$$ ...
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13 views

function of independent random variables

I have following question: If $X$ and $Y$ are independent, then are $g(X)$ and $g(Y)$ independent as well, for any real function $g$?
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Finding probability of Type I & II error

Past studies have shown that 35% of drivers received parking citations while on vacation. Paula believes the actual percentage is higher. She decides to carry out a test with sample size n. Let Y be ...
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20 views

Hypothesis testing: does it have the highest frequency?

Let's suppose, for example, that a fisherman catches $N$ fishes, with a total of $r$ different species: $E_1$, $E_2$, ..., $E_r$. The fisherman caught $O_1$ fishes of species $E_1$, $O_2$ fishes of ...
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What measurement should be used to determine per unit capacity of a given volume ish question

I am reselling data packages and would like to add a sense of scale to the packages. Without disputing actual numbers, an email message is between (3-20)Kbytes, a photo 270kb, a web page ...
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52 views

Asymptotically normal but biased estimator

This is the problem 2.11 from Lehman book "Theory of point estimation" 2-nd edition. Construct a sequence $\{\delta_{n}\}$ of estimators of $g(\theta)$, satisfying $$ \sqrt{n}[\delta_{n} - ...
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26 views

Statistical Multiple Linear Regression Log Transformation

If for example we have a multiple linear regression as follows: $$hydrcarb=x_1+x_2tanktemp+x_3disptemp+x_4tankpres+x_5disppres+x_6tankpres^2+x_7dispres^2$$ And I am trying to do a backward ...
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Minimizing Type I error population proportion estimate

Past studies have shown that 35% of drivers received parking citations while on vacation. Paula believes the actual percentage is higher. She decides to carry out a test with sample size $ n $. Let $ ...
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1answer
30 views

Hypothesis testing for difference of two means

The weight of 11 schoolchildren was measured before and after six months on the proposed lunch plan. The weights before were: 132, 146, 135, 141, 139, 162, 128, 137, 145, 151, 131 The weights ...
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34 views

Maximum likelihood estimator in terms of $\sum \frac{(x_i-\mu)^2}{x_i}$

I'm trying to solve this problem Let X be a random absolutely continuous variable with probability density function $$f_{\lambda\mu}(x) = \sqrt{\frac{\lambda}{2\pi ...
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1answer
49 views

Chi-square test of independence: show that sum of squared standard normals has chi-square distribution

I'm studying the chi-square test of independence. According to my understanding, we first hypothesize independence between variables and consider them as being normally distributed. Then we go on to ...
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22 views

Correction factor for the variance

In an exercise they asked me: "Why could we use the following correction factor? $\text{varianceX} = \frac{n-1}{n}*\text{varianceY}$ What I said was basically, because the unbiased sample variance ...
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1answer
22 views

Statistics - Unbiased estimators

Let $X_1, X_2, \ldots, X_N$ be an i.i.d. random sample of Bernoulli random variables, with $\mathbb{P}(X_i =1) = p$ and $\mathbb{P}(X_i = 0) = 1 − p$. I'm confused as to why $1 − X^{-}$ is an ...
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30 views

Minimizing Unintegrable Exponential Function

I am trying to develop an algorithm which minimizes an unintegrable function. I don't have a strong mathematics background and am unaware of such strategies. My integral is of the following form: ...