New answers tagged self-study
0
votes
How should I analyze my Likert scale data?
welcome to CV! A common and relatively simple way to analyze pre-post data with small samples is a paired samples t-test. That you can do in almost any software, including Excel. That will test ...
- 1,086
2
votes
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Is it possible to adjust a logistic regression model when the false positive rate is too high for observations belonging to a specific category?
If your classification model is based on a certain output variable surpassing a specific boundary, then you can adjust that output variable with a certain amount such that it changes the amount of ...
- 71.5k
2
votes
Joint distribution of a random variable and the sample maximum
A fundamental result in order statistics is Theorem 2.4.2 of Balakrishnan & Cohen:
Let $X_1, X_2, \ldots, X_n$ be i.i.id. random variables from a population with cdf $F$ and pdf $f,$ and let $X_{...
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0
votes
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Is the random variable normal?
No. Imagine a truncated normal distribution, it would still fulfill these criteria and would not be normal.
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2
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Finding P-value and power of the Most Powerful Test
The indicator function sets the limits of integration, but does not have to be carried through since the alternate hypothesis fits within the range of null hypothesis and the maximum sample fits ...
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4
votes
How to prove $s^2$ is a consistent estimator of $\sigma^2$?
Vanishing variance (and resulting convergence) occurs if the underlying distribution has finite kurtosis
The other answer here considers the case of a sample variance of IID normally distributed ...
- 118k
4
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How to prove $s^2$ is a consistent estimator of $\sigma^2$?
I have found a much simpler proof using the weak law of large numbers (This requires finite second moment):
$\begin{aligned} \frac{1}{n-1}\sum\left(x_i-\bar{x}_n\right)^2 & =\frac{1}{n-1}\left(\...
- 41
3
votes
Regarding Least Angle Regression
$\newcommand{\aset}{\mathcal{A}_k}$
What you need to show is that $|\operatorname{Corr}(x_j, r_k(\alpha) - \alpha u_k)| = \frac{|\langle x_j, r_k - \alpha u_k\rangle|}{\|r_k - \alpha u_k\|}$ are ...
- 16.5k
7
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expectation value, distribution function and the central limit theorem
You don't use CLT to get this result. What is needed is a direct evaluation of the term $E[S_n^3]$.
To begin with, note that for $n \geq 3$:
\begin{align*}
S_n^3 = (X_1 + X_2 + \cdots + X_n)^3 = \...
- 16.5k
5
votes
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Intuition and reasoning why LASSO can only select $n$ features when $n \ll p$
I think you can understand why LASSO selects at most $n$-features intuitively if you think about the contours of $rss = |X\beta - y|_2^2$ and $l_1 = |\beta|_1$ on the parameter space $\mathbb R^p$. ...
- 2,197
0
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Joint distribution of $Y$ and $S^2-Y^2$
we know, $$\frac{(n-1)S^2}{\sigma^2} \sim \chi^2_{n-1}$$
$$Y = \sum_{i=1}^{n} b_iX_i$$
notice, this is a linear combination of normal random variables so, this should follow normal distribution with
$$...
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Intuition and reasoning why LASSO can only select $n$ features when $n \ll p$
This is an excellent question, and approach to building intuition. Unfortunately, the answer is silly: your grid search doesn't include cases where $b_1$ or $b_2 = 0$:
...
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5
votes
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Finding the limiting distribution of $\sqrt{n} (\hat{\tau} - \tau)$ as $n \rightarrow \infty$ for $N(\mu, \mu^2 \tau)$
Whuber's comment is to the point, since it matters whether you estimate two parameters or one.
Algebraically, you are correct up to and including the computation of the expected value of the negative ...
- 57.4k
0
votes
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Data taken from survey where survey-takers self report a continous variable
You say:
I want to somehow correct this bias, or at least artificially modify the data so it is distributed rationally.
and also:
I'm going to do a multiple regression with sleep as the DV
In that ...
- 56.5k
1
vote
Calculate the variance of $\sum\limits_{i=1}^{n-1} \sum\limits_{j=i+1}^n S(X_i - X_j)$ for $X_1,\ldots,X_n$ i.i.d. random variables
Since the function $S$ is symmetric about $0$, if we denote $S(x - y)$ by $\phi(x, y)$, then the $U$ in your question can be viewed as a (scaled) U-statistic with the kernel $\phi$:
\begin{align*}
U = ...
- 16.5k
1
vote
How can I prove mathematically that the mean of a distribution is the measure that minimizes the variance?
$$
E[(X-M)^2]=E[X^2]-2ME[X]+M^2\Rightarrow M=E[X]
$$
because minimum of quadradic polynomial $aM^2+bM+c$ is at $M=-b/(2a)$
- 11
2
votes
Accepted
How can I deduce the probability of auto death from a death rate per billions of miles?
I think you are missing some zeros here. First of all, 20,000 miles travelled in 60 years is 1.2 million, not 1.2 billion. Second, I think the assumption which is hinted at in part b) is about how ...
- 36
2
votes
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What are the degrees of freedom to consider for a G-test when some cells have expected values of 0?
The contribution of cells with zero counts is zero in the G-test since each such term is evaluated as 0 in this situation. This is based on the limit of x, which is a standard calculus result.
The ...
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3
votes
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Showing incompleteness of density
First, let's calculate $E[X^2]$ and $E[X^4]$. By evaluating the integral with the given density directly, we have
\begin{align}
& E[X^2] = 2C\sqrt{\lambda}\int_0^\infty x^2\exp(-\lambda x^2- \...
- 16.5k
11
votes
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Show that the sample Mean is not complete
$n\bar{X} \sim \mathcal{N}(-n\sigma^2/2, n\sigma^2)$, so its mgf is
$$
M_{n\bar{X}}(t)
= \mathbb{E}[e^{tn\bar{X}}]
= \exp\left[-nt\sigma^2/2 + n\sigma^2 t^2/2\right].
$$
At $1$ we have $\mathbb{E}[\...
- 20.2k
5
votes
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Given $r\gt 0$, how to get $\mu_r = E[|U|^r]$ where $U\sim N(0,1)$?
It is just a matter of integration:
\begin{align*}
& E[|U|^r] = \int_{-\infty}^\infty |u|^r\frac{1}{\sqrt{2\pi}}e^{-u^2/2}du \\
=& 2\int_0^\infty u^r\frac{1}{\sqrt{2\pi}}e^{-u^2/2}du \\
=& ...
- 16.5k
2
votes
Accepted
Convergence in product of sequence of random variables
The following general result would come handy in deducing what the authors did above.
Result $1.$ Let $(\Omega,\boldsymbol{\mathfrak A},\mu)$ be a measure space. Let $\langle f_n\rangle_{n\in\mathbb N}...
- 6,897
1
vote
Convergence in product of sequence of random variables
If the $\limsup$ is less than $\epsilon$ for every positive $\epsilon$, then it's zero. It can't be, for example, $1/3, $ because you could take $\epsilon=1/4$. It can't be $0.00042,$ because you ...
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