Questions tagged [truncated-distributions]
A truncated distribution is one that is cut off at some value, either at the low or high end of the distribution, or both.
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Truncated distribution and hazard rates in a microeconomic model
I'm trying to prove something related to a labor microeconomic model:
t∈{1,2,3}, A∼N(0,1) (A is fixed across periods),
εt∼N(0,1) (εt are independent for every εt),
c>0 is the cutoff for promotion ...
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1
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Rejection sampling to obtain a random sample from a truncated version of a multivariate probability density
Suppose I have a multivariate probability density $f(\mathbf{y}|\boldsymbol{\theta})$ with support $\mathbb{R}^d$ that is analytically tractable, and I know how to randomly sample from $f(\mathbf{y}|\...
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Distribution of a random variable conditional on its being a maximum or not
Consider the random variables $\epsilon_1,\dots, \epsilon_D$ defined on the probability space $(\Omega, \mathcal{F}, P)$. Assume they are continuous. Let
$$
Y=\sum_{d=1}^D d\times \mathbb{1}\{\...
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Difference between truncated and unseen data
I have 2 related questions.
Assume that we want to build a model to study of some random discrete variable $x$ that follows some known distribution with PMF $P(x)$, yet with unknown parameters that we ...
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Distribution supported on $(0,\infty)$ for which moments of its truncated distribution are elementary functions of the truncation point and power
I am looking for a distribution with a differentiable PDF $f:(0,\infty)\rightarrow (0,\infty)$ for which for any $\delta>1,z>0$, the two following integrals are finite elementary functions of $\...
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What is the PMF of the Zero-Truncated Skellam distribution?
Vaguely related to the notion of a Lindley equation,
I am considering a recurrence
$$L_{t+1} := L_t + \max \left( 0, A_t - S_t \right)$$
where
$$L_t$$
is the number of customers in a queue system at ...
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glmmTMB truncated models with zero inflation
everyone. I am fitting a glmm model using the R library glmmTMB for predicting a count response variable with excess-zeros and overdispersion (...
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A question related to the convergence of mathematical expectations restricted to an Interval centered on zero
Let $(X_j)_{j= \mathbb 0}^\infty$ a fixed realization of strictly stationary AR(1) process:
$$X_j = 0.9 \,X_{j-1}+ \eta_{j}, \quad (\eta_j) \overset{iid}{\sim} N(0,1)$$
For each $n$, consider $B_n\sim ...
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Develop a model for theoretical best performances at different ultra running distances/times
Goal
Develop a model for theoretical best performance for running distances from marathon to around 1000 km. Partly to compare the strength of ultrarunning world records, but more importantly, to get ...
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Generalized likelihood ratio test for a left-truncated exponential distribution [duplicate]
I am doing self study in statistical inference and am rather confused about how to approach generalized likelihood ratio test (GLRT) problems. I am trying the traditional approach by definition and ...
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Mixed model with censored data in R?
My objective is to see if there is a significant difference in BHB concentration between age categories in farm animals. Farm should be a random effect in the model. The issue is that BHB ...
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Truncated lognormal distribution calibration with MME
To estimate the parameters of a truncated distribution (lognormal for example), we can use the Maximum Likelihood Estimation or Method of Moments.
For the Method of Moments Estimation, one needs to ...
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Expected value of Truncated Normal Distribution [duplicate]
For the truncated normal distribution below:
$$
{f_X(x; σ, a, b)} = \frac{1}{\sigma}\frac{φ(\frac{x-µ}{σ})}{Φ(\frac{b-µ}{σ})-Φ(\frac{a-µ}{σ})}
$$
$$
a = 1; b = ∞; σ = 2
$$
I need to calculate the ...
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How to use Truncated Normal for observation distribution in GLM model?
I'm trying to setup a Bayesian GLM with Truncated Normal, $\mathcal{N}_+(\mu, \sigma, 0)$ truncated at $0$.
I want to specify $\mathbf{E} (y\mid x) = ax + b$ but it looks like $\mathbf{E} (y\mid x)$ ...
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How to combine two truncated distributions
We want to combine two truncated distributions to better model one phenomenon. For example, we have a Gaussian distribution, but we want to modify the right hand side tail to make it heavier. So we ...
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Central limit theorem : relaxing assumption of all finite moments
Consider $S_n = \sum_{i = 1}^n b_{i,n} X_{i,n}$ where
$X_{i,n}$ are random variable neither independent neither identically distribution and $b_{i,n}$ are weights satisfying the Lindeberg condition. I ...
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I want the function that defines truncated lognormal distribution
Problem, I have a process(water level in chamber), it perfectly fits with lognormal.
But the chamber has a maximum water level, after which no effect of water must be there.
I guess I can use the ...
PSA
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random number generation of truncated multivariate normal distribution
I want to generate random numbers from truncated multivariate normal distribution specified as follows:
$ \begin{bmatrix}
Y \\
X
\end{bmatrix} \sim N
\begin{pmatrix}
\begin{bmatrix}
\mu_Y \\
\mu_X
\...
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Maximum likelihood fit of left truncated Weibull distribution
I want to fit some samples to the right tail of a Weibull distribution. To fix the notation:
the samples are $\{X_i\}_{i=1,\ldots,n}$,
all samples are greater than a fixed threshold $L$,
the $X_i$'...
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How to choose priors for bounds on circular truncated distributions?
I am considering choices of priors for truncated distributions on a circle. Let's take the truncated normal distribution on the unit circle as an example. It has parameters $\mu \in [-\pi, \pi]$ and $\...
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Left censored regression bounded at zero
I've been handed some data that is obviously left-censored. Many zeroes, probably due to insensitivity of the assay. However, since it is a protein level assay, it is theoretically impossible for any ...
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Sufficiency and completeness of truncated distribution
[From Theory of Point Estimation (Lehmann and Casella, 1999, Exercise 6.37)]
Let $P=\{P_\theta:\theta \in \Theta\}$ be a family of probability
distributions and assume that $P_\theta$ has pdf $p_\...
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Is the truncated squared expected value less than the variance?
Let the continuous random variable $X$ be distributed with mean $\mu_x$ and variance $\sigma^2_x$ with support $[0, \infty]$.
Let the random variable $Y$ be the right truncation of $X$, with ...
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Fitting truncated and censored data
I have data that is truncated on the left and censored on the right. The reason is that this is claims data, which for a claim gives the amount of the claim. The claim appears in the data:
Only if it ...
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Is the mean of the left-truncated binomial distribution convex in p?
The expectation of the binomial distribution of successes in $G$ trials, left-truncated at $R$, with success probability $p$, is
$$
E[X|p] = \frac{\sum_{l=R}^Gl\phi(l)}{\sum_{l=R}^G\phi(l)}
$$
where
$$...
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Back Transformed Truncated Negative Binomial Model Results Less Than One
I'm using a truncated negative binomial model to describe my count data where all values are >=1. I have attempted to back-transform my model results using emmeans. However, all of my back ...
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Generate a truncated lognormal distribution given mean, variance, lower bound and upper bound?
Basically, I would like to generate a sample of truncated lognormal distribution given mean, variance, lower bound and upper bound. Note that the mean and variance here are the mean and variance of ...
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Understanding truncated distributions & simulations [duplicate]
I have a data set which follows lognormal distribution (parameters $μ$, $σ$ known - estimated by maximum likelihood estimation). I have to generate random numbers within range $[a:b]$ from this known ...
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Fitting truncated normal mixtures in R
I have a vector x, lower_bound < x < upper_bound. I would like to fit a truncated normal mixture distribution to ...
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Knowing the sum, the n(), and the bound parameters of a truncated-Pareto distributed variable, how I identify the alpha (shape) parameter?
I know that there would be a fancy command on R to do the estimation of $\alpha$ given the inputs, but I am also curious about the relationship between $\alpha$ to $...
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Conditional expectation of $X_t$ in a time series, given that other draws were below $c$
I'm interested in the moments of a given draw, $X_t$, of a time series conditional on the knowledge that all other draws within some window before and after $t$ were below a fixed threshold, $c$. For ...
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Difference of means with a truncated distribution
Let's say I'm measuring viral load post-infection. The two groups I have are vaccinated and unvaccinated.
We expect the distribution of the unvaccinated cohort's viral load to resemble a normal ...
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Why are Truncated Probability Distributions important in Statistics?
Why are Truncated Probability Distributions important in statistics?
Recently, I was reading about "Truncated" Probability Distributions. As the name suggests, a Truncated Probability ...
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Why I cannot generate random numbers having a truncated lognormal distribution?
My deduction is:
When the distribution is truncated, a normalization factor should be introduced:
\begin{equation}
g(x) = \frac{C}{x\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{\ln{x}-\mu}{\sigma}\...
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How to compute the PDF of a conditional distribution [duplicate]
Let $T \sim Unif(0, 1)$. Then, $f_T(t) = 1 \text{ for } t \text{ in [0, 1] (0 elsewhere)}$.
How do we formally compute $f_{T \mid T > 0.5}$?
Intuitively, $f_{T \mid T > 0.5}(t) = 2 \text{ for } ...
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True mean of a truncated distribution?
I use C++ GSL library to generate random numbers now. The numbers obey a distribution, (e.g. normal or lognormal distribution). This library requires the input of expected value ${\mu}$ (i.e. mean) ...
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Output Distribution of ReLU given a Laplace Distribution as its Input
If input to a ReLU function (Max(X, 0)) is a Laplace Distribution, what would be the output distribution? will it have a density function? how would it look like? assuming that mean of the Laplace is ...
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Understanding the pdf of a truncated normal distribution
Suppose $\boldsymbol{x} = (x_1, \ldots, x_m)^T$ follows a multivariate normal distribution with 2-sided truncation $a_i \leq x_i \leq b_i$. This is a truncated multivariate normal defined by $TN(\mu, \...
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Data count regression with a truncated distribution
Imagine that we are conducting an experiment to test the effectiveness of a treatment, where the «level of illness» is measured by a count that is distributed as a negative binomial (NB). The plan is ...
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Enforcing conditions on truncated exponential distribution
The CDF for an exponential distribution of rate $\lambda$ truncated at T is
$F(t) = \frac{1-e^{-\lambda t}}{1-e^{-\lambda T}}$. (for $t<T$, else 0).
I would like to determine $\lambda$ and $T$ such ...
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how to compute marginal effects of predictors in NBSTRAT (truncated & endogenously stratified negative binomial) model? (Stata)
I'm using STATA 16.0 to develop recreational demand function via using NBSTRAT model. I have several factor and continuous variables that force me to use "xi:" prefix in the model syntax ...
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Finding Confidence Interval for Lower Bounded Truncated Normal Distribution
I am working on finding a confidence interval for data that follows a lower bounded truncated normal distribution (lbtnd) bounded from 0 to $\infty$. I am having difficulty completely understanding ...
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Mean preserving spread and truncated distributions
Take two distributions $F_B(x)$, $F_A(x)$ with the same support. Assume that B is a mean-preserving spread of A.
What I want to understand is whether $E_{A}[x | x \leq t] \geq E_{B}[x | x \leq t]$, ...
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Truncated expectation of sum of independent random variables
Take three random variables $X$, $Y$, $Z$ s.t. $E[X]>0$, $E[Y|X]=0$, $Z = X+Y$.
What can I say about $E[x| x> k]$ vs. $E[z| z>k]$ where $k>0$? Intuitively, the latter should be bigger ...
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Generating random samples obeying the exponential distribution with a given min and max
Random samples obeying the exponential distribution can be generated by the inverse sampling technique by using the quantile function of the exponential distribution:
$$
x = F^{-1}(u) = - \frac{1}{\...
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Dominance of truncated means
If for two random variables, the truncated mean of one is always larger than the other, i.e. $E(Y|Y<x)>E(X|X<x)$ for all $x$, can we infer that $Y$ first-order stochastically dominate $X$?
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Completeness of a statistic in a truncated distribution
Suppose a random sample $x_1,\dots, x_n$ (i.i.d.) from a random variable $X$ defined over $(\Omega,\mathcal{F},P)$ whose probability density function is $f(x_1,\dots, x_n;\theta)$ and $T(x_1,\dots, ...
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Truncated normal distribution without scaling
My understanding of a truncated normal distribution $\mathcal{N}(\mu,\sigma;a,b)$ is that it results from scaling the values of a normal distribution within the bounds $[a; b]$ such that the area ...
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Is it Sufficient to Truncate a Left Censored Distribution?
A colleague explained their approach to dealing with left censored data in an analysis, and while I don't think it is the best approach, I am not sure if it is insufficient or not.
My colleague has ...
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Confidence interval for the maximum likelihood estimate of the minimum of a left truncated exponential distribution
I am currently working on a problem in which I have observations $y_{i}$ that are distributed,
$y_{i} \sim \textrm{Exponential}(\beta = ax_{i})\cdot T[b, \infty)$
where, $\beta$ is the rate parameter ...
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