All Questions
230 questions
82
votes
4
answers
60k
views
Linear model with log-transformed response vs. generalized linear model with log link
In this paper titled "CHOOSING AMONG GENERALIZED LINEAR MODELS APPLIED TO MEDICAL DATA" the authors write:
In a generalized linear model, the mean is transformed, by the link
function, instead of ...
- 3,814
63
votes
3
answers
25k
views
Which has the heavier tail, lognormal or gamma?
(This is based on a question that just came to me via email; I've added some context from a previous brief conversation with the same person.)
Last year I was told that the gamma distribution is ...
- 290k
32
votes
3
answers
17k
views
Extreme Value Theory - Show: Normal to Gumbel
The Maximum of $X_1,\dots,X_n. \sim$ i.i.d. Standardnormals converges to the Standard Gumbel Distribution according to Extreme Value Theory.
How can we show that?
We have
$$P(\max X_i \leq x) = P(...
- 1,271
73
votes
4
answers
191k
views
How do you calculate the probability density function of the maximum of a sample of IID uniform random variables?
Given the random variable
$$Y = \max(X_1, X_2, \ldots, X_n)$$
where $X_i$ are IID uniform variables, how do I calculate the PDF of $Y$?
- 871
10
votes
1
answer
3k
views
MLE/Likelihood of lognormally distributed interval
I have a variable set of responses that are expressed as an interval such as the sample below.
...
- 103
26
votes
1
answer
9k
views
Whether distributions with the same moments are identical
Following are similar to but different from previous posts here and here
Given two distributions which admit moments of all orders, if all the moments of two distributions are the same, then are they ...
- 19.8k
83
votes
3
answers
105k
views
How is the minimum of a set of IID random variables distributed?
If $X_1, ..., X_n$ are independent identically-distributed random variables, what can be said about the distribution of $\min(X_1, ..., X_n)$ in general?
- 941
3
votes
2
answers
1k
views
How to find normal and lognormal moments, given partial information?
$Y=\ln(X)$. $X$ is lognormal and $Y$ is normal. If all I know is the arithmetic mean of $Y$ and the standard deviation of $X$. What is the formula to calculate the arithmetic mean of $X$ and the ...
- 101
26
votes
4
answers
46k
views
The sum of independent lognormal random variables appears lognormal?
I'm trying to understand why the sum of two (or more) lognormal random variables approaches a lognormal distribution as you increase the number of observations. I've looked online and not found any ...
- 1,779
24
votes
2
answers
11k
views
Distribution of the maximum of two correlated normal variables
Say I have two standard normal random variables $X_1$ and $X_2$ that are jointly
normal with correlation coefficient $r$.
What is the distribution function of $\max(X_1, X_2)$?
- 2,295
18
votes
1
answer
22k
views
Correlation of log-normal random variables
Given $X_1$ and $X_2$ normal random variables with correlation coefficient $\rho$, how do I find the correlation between following lognormal random variables $Y_1$ and $Y_2$?
$Y_1 = a_1 \exp(\mu_1 T +...
- 2,799
27
votes
7
answers
26k
views
How do I calculate a confidence interval for the mean of a log-normal data set?
I've heard/seen in several places that you can transform the data set into something that is normal-distributed by taking the logarithm of each sample, calculate the confidence interval for the ...
- 677
11
votes
1
answer
4k
views
Using bootstrap to obtain sampling distribution of 1st-percentile
I have a sample (of size 250) from a population. I do not know the distribution of the population.
The main question: I want a point estimate of the 1st-percentile of the population, and then I want ...
- 69.5k
7
votes
2
answers
2k
views
What is the best point forecast for lognormally distributed data?
I believe that the values I am forecasting are lognormally distributed with log-mean $\mu$ and log-variance $\sigma^2$. I need a point forecast (i.e., a one-number summary) that minimizes the expected ...
- 131k
32
votes
3
answers
9k
views
Difference of two i.i.d. lognormal random variables
Let $X_1$ and $X_2$ be 2 i.i.d. r.v.'s where $\log(X_1),\log(X_2) \sim N(\mu,\sigma)$. I'd like to know the distribution for $X_1 - X_2$.
The best I can do is to take the Taylor series of both and ...
- 421
13
votes
1
answer
15k
views
Expectation, Variance and Correlation of a bivariate Lognormal distribution
If $Y \sim N(\mu,\sigma^2)$ is normally distributed, then $X=\mathrm{e}^Y$ is lognormally distributed. To get the log-$\mu$ and log-$\sigma$ of this lognormal distribution you calculate
$$\sigma^2 = \...
- 1,781
69
votes
9
answers
8k
views
Taleb and the Black Swan
Taleb's book "The Black Swan" was a New York Times best seller when it came out several years ago. The book is now in its second edition. After meeting with statisticians at a JSM (an annual ...
- 43.2k
28
votes
1
answer
49k
views
Expected value and variance of log(a)
I have a random variable $X(a) = \log(a)$ where a is normal distributed $\mathcal N(\mu,\sigma^2)$. What can I say about $E(X)$ and $Var(X)$? An approximation would be helpful too.
- 435
15
votes
2
answers
21k
views
What is the distribution for the maximum (minimum) of two independent normal random variables?
Specifically, suppose $X$ and $Y$ are normal random variables (independent but not necessarily identically distributed). Given any particular $a$, is there a nice formula for $P(\max(X,Y)\leq x)$ or ...
- 355
12
votes
3
answers
32k
views
Calculating distribution from min, mean, and max
Suppose I have the minimum, mean, and maximum of some data set, say, 10, 20, and 25. Is there a way to:
create a distribution from these data, and
know what percentage of the population likely lies ...
- 123
52
votes
2
answers
36k
views
Gamma vs. lognormal distributions
I have an experimentally observed distribution that looks very similar to a gamma or lognormal distribution. I've read that the lognormal distribution is the maximum entropy probability distribution ...
- 1,257
26
votes
1
answer
47k
views
Is a log transformation a valid technique for t-testing non-normal data?
In reviewing a paper, the authors state, "Continuous outcome variables exhibiting a skewed distribution were transformed, using the natural logarithms, before t tests were conducted to satisfy the ...
- 361
26
votes
6
answers
53k
views
Can mean plus one standard deviation exceed maximum value?
I have mean 74.10 and standard deviation 33.44 for a sample that has minimum 0 and maximum 94.33.
My professor asks me how can mean plus one standard deviation exceed the maximum.
I showed her ...
- 261
25
votes
5
answers
6k
views
What exactly are moments? How are they derived?
We are typically introduced to method of moments estimators by "equating population moments to their sample counterpart" until we have estimated all of the population's parameters; so that, in the ...
- 1,427
6
votes
1
answer
435
views
Confidence interval for GLM or the maximum of a function?
Imagine I have a set of (xi,yi) measures.
I can show it on a scatter plot
I want to choose the value of x that maximizes y,
or I could fit a function and find the values of the parameters that ...
- 1,094
6
votes
2
answers
11k
views
Moment Generating Function for Lognormal Random Variable
I'm working through the proof of a lognormal random variable and am having some difficulty in moving through it. I understand the following:
Our CDF is $\Phi(\frac{logx - \mu}{\sigma})$, and thus our ...
- 61
4
votes
1
answer
2k
views
How to find the $(a_n,b_n)$ for extreme value theory
In the solution to this question (Extreme Value Theory - Show: Normal to Gumbel), the OP asked for the sequence $(a_n, b_n)$ such that $\Phi(a_nx+b_n)$ converges to the Gumbel CDF. Not only did I not ...
- 569
26
votes
2
answers
2k
views
Bias of moment estimator of lognormal distribution
I am doing some numerical experiment that consists in sampling a lognormal distribution $X\sim\mathcal{LN}(\mu, \sigma)$, and trying to estimate the moments $\mathbb{E}[X^n]$ by two methods:
Looking ...
- 363
14
votes
4
answers
1k
views
Unbiased estimator for the smaller of two random variables
Suppose $X \sim \mathcal{N}(\mu_x, \sigma^2_x)$ and $Y \sim \mathcal{N}(\mu_y, \sigma^2_y)$
I am interested in $z = \min(\mu_x, \mu_y)$. Is there an unbiased estimator for $z$?
The simple estimator ...
- 141
13
votes
1
answer
15k
views
Multivariate log-normal probabiltiy density function (PDF)
The Multivariate Gaussian pdf is given by
$$(2\pi)^{-\frac{K}{2}} \det(\Sigma)^{-\frac{1}{2}} \exp({-\frac{1}{2}}(X-\mu)' \Sigma^{-1} (X-\mu)) $$
The wikipedia for multivariate Gaussians is here
...
- 1,235
12
votes
1
answer
23k
views
How to calculate a mean and standard deviation for a lognormal distribution using 2 percentiles
I am trying to calculate a mean and standard deviation from 2 percentiles for a lognormal distribution.
I was successful in performing the calculation for a normal distribution using ...
- 123
11
votes
1
answer
22k
views
The product of two lognormal random variables
Let $X_1$ and $X_2$ be two normal random variables. Write $X_1\sim N(\mu_1, \sigma^2_1)$ and $X_2\sim N(\mu_2, \sigma^2_2)$, to fix ideas.
Consider the corresponding log-normal random variables: $...
- 221
11
votes
2
answers
4k
views
Asymptotic distribution of maximum order statistic of IID random normals
Is there a nice limiting distribution of $\max( X_1,X_2,...,X_n) $ as $n$ goes to $\infty$, assuming that they are iid normal distributions with variance $\sigma^2$.
This is almost certainly a well ...
- 1,511
8
votes
3
answers
49k
views
How can I convert a lognormal distribution into a normal distribution?
I have a sample of data that follows a lognormal distribution. I would like to represent the distribution as a "Gaussian" histogram and overlayed fit (along a logarithmic x-axis) instead of a ...
user146123
6
votes
1
answer
2k
views
Covariance between a normally distributed variable and its exponent
Given $X \sim \mathcal{N}(\mu,\,\sigma^{2})$, what is the covariance between $X$ and $e^X$?
- 327
6
votes
1
answer
2k
views
What's the story behind the log-normal distribution?
I have been playing around with datasets for the past while for practice. I've noticed that a distribution that looks something like the following appears:
This shape appears frequently! I can guess ...
user46925
4
votes
1
answer
2k
views
Lognormal distribution using binned or grouped data
I understand the Max likelihood estimators for mu and sigma for the lognormal distribution when data are actual values. However I need to understand how these formulas are modified when data are ...
- 41
2
votes
2
answers
25k
views
Determine density of $\min(X,Y)$ and $\max(X,Y)$ for independently uniform distributed variables
Two independent random variables, $X$ and $Y$, are uniformly distributed on the unit interval $(-1,1)$.
Determine the density for $U=\min(X,Y)$ and for $W=\max(X,Y)$
- 23
27
votes
5
answers
59k
views
How to specify a lognormal distribution in the glm family argument in R?
Simple question: How to specify a lognormal distribution in the GLM family argument in R?
I could not find how this can be achieved. Why is lognormal (or exponential) not an option in the family ...
- 1,635
23
votes
3
answers
3k
views
Distribution of the largest fragment of a broken stick (spacings)
Let a stick of length 1 be broken in $k+1$ fragments uniformly at random. What is the distribution of the length of the longest fragment?
More formally, let $(U_1, \ldots U_k)$ be IID $U(0,1)$, and ...
- 14.9k
19
votes
2
answers
12k
views
What is the variance of the maximum of a sample?
I'm looking for bounds on the variance of the maximum of a set of random variables. In other words, I'm looking for closed-form formulas for $B$, such that
$$
\mbox{Var}(\max_i X_i) \leq B \enspace,
$$...
- 273
14
votes
3
answers
13k
views
Priors for log-normal models
I am trying to determine what the most appropriate non-informative priors are for the two parameters of a log-normal distribution (for an accelerated failure time model). I had been working with a ...
- 729
12
votes
1
answer
11k
views
Is it possible to analytically integrate $x$ multiplied by the lognormal probability density function?
Firstly, by analytically integrate, I mean, is there an integration rule to solve this as opposed to numerical analyses (such as trapezoidal, Gauss-Legendre or Simpson's rules)?
I have a function $\...
- 123
12
votes
3
answers
35k
views
Exponential of a standard normal random variable
We know that $Z\sim N(0, 1)$. How do I prove that $e^Z$ has a mean of $e^{0.5}$? I have tried integrating $e^z$ times the pdf of $Z$ but I don't know why it isn't working out.
Also what is the pdf ...
- 121
11
votes
2
answers
13k
views
Variance of $X$ and Variance of $\log(X)$. How to relate them?
I have the variance of a random variable $X$ and I want to obtain the variance of $\log(X)$. Is it possible if I dont know its PDF?
If I assume that $X$ has a lognormal PDF, how variances should be ...
- 111
9
votes
1
answer
442
views
Intuition about the coupon collector problem approaching a Gumbel distribution
The coupon collector's problem
Let there be $n$ different types of coupons and we try to collect all of the types.
We do this by independent random draws of coupons in which each type of coupon has an ...
- 86.5k
9
votes
1
answer
6k
views
Can I get the parameters of a lognormal distribution from the sample mean & median?
I have the mean and median values for a sample drawn from a lognormal distribution. Note that this is not the mean and median of the logs of the variable, though I can of course calculate the logs of ...
- 3,247
7
votes
2
answers
52k
views
Plot log-normal distribution in R [closed]
I need to plot lognormal distribution with mean 1 and variance 0.6 in R.
I tried to do this using rlnorm function in ...
- 71
7
votes
2
answers
12k
views
How do I estimate the parameters of a log-normal distribution from the sample mean and sample variance
Given the sample mean $\bar{x}$ and the sample variance $s^2$ of a random variable $X$, is it possible to estimate the shape $\sigma^2$ and log-scale $\mu$ of the log-normal distribution, with ...
- 671
6
votes
1
answer
2k
views
How to transform one PDF into another graphically?
To understand what I mean, let's use two well-known distributions: the normal and lognormal ones.
From the dataset point of view, if you take normally-distributed data and take their exponential, you ...
- 830