A distribution describing the time between events in a Poisson process; a continuous analogue of the geometric distribution.

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Is there a general expression for ancillary statistics in exponential families?

It is known that an i.i.d sample $X_1,\cdots,X_n$ from a scale family with c.d.f. $F(\frac{x}{\sigma})$ has $S(X)$ as an ancillary statistics if $S(X)$ depends on the sample only through ...
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32 views

correlation coefficient for exponential model - the perils of Excel

Editing a non-peer-reviewed paper I came across a Excel scatter-plot with an "exponential trend-line" drawn, AND the value of the correlation coefficient presented alongside (calculated for the ...
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42 views

Self-Starting Nonlinear Regression Model

In the following two nonlinear regression models, is it possible for the dependent variable (y) or the independent variable (x) to be negative? \begin{align} y &= φ_1 \exp[-\exp(φ_2)x] + φ_3 ...
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46 views

Exponential regression and focus on small values

I have a set of data with 3 numeric variables: X, Y and Z. I have access to data with 15 ...
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68 views

Exponential distribution question

Let X and Y be independent exponential random variables with respective rate $\lambda$ and $\mu$ let c be a positive constant. I want to find out a) $E[min(X,Y)|X>c]$ b)$E[min(X,Y)|X>Y+c]$ by ...
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54 views

What is the ratio distribution of a spacing and the sample mean?

Let $X_1,\dots,X_n$ be a sample of iid exponential random variables with mean $\beta$, and let $X_{(1)},\dots,X_{(n)}$ be the order statistics from this sample. Let $\bar X = \frac{1}{n}\sum_{i=1}^n ...
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1answer
48 views

Relationship between λ (or μ) in Poisson and Exponential distribution?

Is there any relationship between λ (or μ) in Poisson and Exponential distribution? In other words, if I know λ (or μ) for one of the distributions, is it possible to calculate the corresponding value ...
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24 views

Sufficient statistic for expectation of exponential family

True or false: Let X $\sim$ f, where f is element of an exponential family. Then, $\frac{\sum_{i=1}^n x_i}n$ is a sufficient statistic for $E(X)$. For either case, please provide the ...
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29 views

Survival Analysis - Improbable Exp(B)?

I'm currently conducting a survival analysis on my dataset. I have data on adoption of a new service and am using a Cox Proportional Hazard Model in order to account for right-censored observations. ...
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26 views

How can I evaluate the upper or lower bound for a function of random variables?

Let $\Gamma_1$,$\Gamma_2$ and $\Gamma_3$ be random variables defined as: \begin{align} \Gamma_1 &= \frac{XY}{X + Y + 1} \\[10pt] \Gamma_2 &= \frac{XY}{aX + bY + c} \\[10pt] \Gamma_3 &= ...
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60 views

Time to second failure of independent exponential lifetimes

This maybe an easy question, but as I am a beginner, I need help. Suppose that an electronic system contains $n$ similar components that function independently of each other and that are connected ...
5
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98 views

Related to exponential distribution

I am a beginner in statistics.(and probability theory too).I studied Exponential distribution and just started doing problems when I got stuck in the following one: Suppose that $X_1, . . . , X_n$ ...
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1answer
26 views

exponential likelihood with normal prior

If I have a likelihood function based on the exponential distribution with parameter $\lambda$ , why would a normal prior with a very large variance be inappropriate for $\lambda$?
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1answer
19 views

Exponential Regression with x outside of exponential

I am trying to do exponential regression by matrix notation, and I am trying to figure out to create my $\mathbf{X}$ matrix to fit my model. I know that I need to use a model function of the form $c_1 ...
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0answers
47 views

Parametric Competing Risk Survival Analysis - Exponential Distribution

I am familiar with the literature on nonparametric estimation of the cumulative incidence function under a competing risk model. I'm looking for the more basic solution under exponential ...
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0answers
17 views

Compute the probability of k events within time period during overvation period

The Erlang distribution is the distribution of the sum of k independent and identically distributed random variables each having an exponential distribution. Thus, it can be used to compute the ...
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1answer
86 views

Distribution of $\sum_{i=1}^d | \mathbf{u}^H \mathbf{F} \mathbf{v}_i |^2$ if $| \mathbf{u}^H \mathbf{F} \mathbf{v}_i |^2$ is exponentially distributed

Let $\mathbf{u}$ and $\mathbf{v}_i$ be an $M \times 1$ and $N \times 1$ vectors of unit norm, respectively. $\mathbf{u}$ is a column of a unitary matrix $\mathbf{U}$ and the $\mathbf{v}_i$ are ...
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52 views

Fit linear model with a spatially structured random effect in R

There are n sampling units (with explicit spatial coordinates (pos_x[i], pos_y[i]), i = 1..n). Data is generated as: ...
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19 views

Distribution of Spacings

If I'm understanding my notes correctly, the distribution for any finite collection of spacings is approximately Exponential with mean 1/n(f(F^-1(k/n)). Can anyone help me understand the proof of ...
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59 views

Quantile regression on non linear data

R and statistics beginner here, trying to do a quantile regression on a non-linear dataset. I want to identify datapoints that have a higher y axis value that expected given their value on the x ...
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2answers
52 views

Work order completion time is exponentially distributed

I have a work order system that serves the operational needs of a big organization. It has tickets and it gets completed in x days (0 < x < 1000) . I pulled out the data for completion time in ...
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1answer
57 views

Understanding a passage written about an exponential function

I read the following in the book Forecasting, Time Series, and Regression - Fourth Edition by Bowerman - O'Connell - Koehler (P. 296), and it said the following: It can be shown that as the power ...
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1answer
88 views

expectation of $e^{-x}$ when x is log-normal

I'm trying to find the expected value of $ e^{-x} $ when $ x$ is log-normal. I know that if $ x \sim N(\mu, \sigma) $ then $ E[e^x] = e^{\mu + \frac{1}{2}\sigma^2} $, the expectation of a log-normal. ...
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62 views

How to perform goodness of fit test for ex-Gaussian distribution?

I used "timefit" from the R package "retimes" to fit an exponentially modified Gaussian distribution to reaction time data. This function estimates ex-Gaussian parameters based on maximum likelihood ...
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1answer
17 views

Average duration of a double outage in a system with exponentially-distributed failure and repair times

Assume that we have a system with two units. For each unit, its failure frequency follows an exponential distribution with mean $\lambda_1$ and its repair time follows an exponential distribution with ...
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1answer
57 views

Exponential Distribution with possible Binomial Probability

You have a system with 6 components. In order for the system to work the following must be met: Component 1 must work. At least one of components 2, 3, 4 must work. At least one of components 5, 6 ...
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36 views

Mean for grouped data when midpoints are unknown [duplicate]

I have a dataset with disease frequency in age groups. <1 <3 <5 <7 >=7 14 19 15 3 10 And now, when actual data lost, I've been ...
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1answer
60 views

Max n for which sum of exponential distribution is bigger then gamma variable

I am currently preparing to the actuarial exam and it is one of the exercises from previous years I encountered and have no idea how to deal with: Let us assume that $X_1, X_2, ..., X_n$ are ...
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1answer
76 views

Fit a non-linear exponential model to using R?

I have interest rate data $r_k$ and the maturity times $k$ given and the problem I want to solve is that: $\min_a\sum(r(a,k)-r_k)^2$, where $a$ is a vector and $r(a,k)=a_0+a_1 e^{-k/a_3}+a_2 ...
7
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1answer
92 views

Distribution of the exponential of an exponential random variable

Let $X$ be a real valued random variable with exponential distribution. Let $a$ be a complex number. What is the distribution of $Y = e^{aX}$? Can Y be written in the form of another known ...
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45 views

Relationship between Poisson arrival rate and Exponential inter-arrival rate

A poisson distribution has a lambda 1.5 per minute. I would like to know as to how would i use this for the lambda in exponential distribution? Would both the lambda values be the same?
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1answer
14 views

Hierarchical process of exponentials

I'd like to work with a what I believe is a called a "hierarchical process" -- given by the multiplication of a pair of exponential distributions such that the random variable from one process is the ...
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2answers
69 views

Process with parameters that are themselves statistical

I'd like to work with a pair of statistical processes such that the random variable from one process is the parameter of the second process. The simplest case I can imagine (and which is still ...
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1answer
55 views

Finding the best fitted distribution for an experimental data with R

I have read most of the similar questions and answers, but still can not solve my problem... I have an experimental medical time-depending data, which I want to analyse and classify. I'm trying to ...
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2answers
2k views

Why is nls() giving me “singular gradient matrix at initial parameter estimates” errors?

I have some basic data on emission reductions and cost per car: ...
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45 views

Expected time in exponential distribution?

I'm using the exponential distribution to calculate a probability of an event to occur. $\lambda = 0.007$ failures/year. I am to calculate an expected time between 2 failures. My approach was to ...
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1answer
193 views

Central Limit Theorem for Exponential Distribution

I've spent so long, and I think I'm missing something super obvious. The question: Let $X_1,\ldots,X_5$ be five independent variables from the exponential distribution with mean $2$. Write the pdf ...
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1answer
50 views

Why is the exponential distribution chosen to model service time in Queuing theory?

Why is the exponential distribution chosen to model service time in Queuing theory ?
3
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1answer
71 views

Confidence interval for exponential distribution

Let $X_1,...,X_n$ random sample of $X$~$exp(\theta)$. i) Find a exact confidence interval for $\theta$ with coefficient of confidence equal to $\gamma$ ii)Find a asymptotic confidence ...
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482 views

What is my lambda here

The life of automobile voltage regulators has an exponential distribution with a mean life of six years. You purchase a six-year-old automobile, with a working voltage regulator and plan to own it for ...
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0answers
47 views

Adapt Kolmogorov-Smirnov Test to Lilliefors Test

I'm using EasyFit to fit distributions to a sample of data and test with the Kolmogorov-Smirnov test whether the null hypothesis may be rejected or not. But as this test estimates the parameters from ...
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1answer
57 views

Translate exponential distribution into normal distribution [closed]

I have a bunch of inventory management formulas that are supposed to be used with normal distributions, however my demand data fits an exponential distribution. Is there any way to translate the ...
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1answer
52 views

Heteroskedasticity- is everything over for my model?

So, I've got this exponential model: Which, when tested via Pagan- Breusch, got heteroskedasticity detected. Breusch-Pagan / Cook-Weisberg test for heteroskedasticity Ho: Constant ...
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2answers
101 views

Exponential Regression to Forecast Future Growth

I need to use Exponential Regression to forecast the future earnings of a company. I have the past 10 years of quarterly data. I can do linear regression but the data is not in a linear fashion. ...
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3answers
245 views

Difference between rexp and qexp in R

I need to write a simulation that requires the use of exponential distribution. I was wondering what is the difference between the following two approaches in drawing random numbers from an ...
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1answer
56 views

Lower bound on the mean for an exponential distribution at a confidence level of 95%

for the life of me I simply cannot wrap my head around confidence intervals and upper/lower bounds on parameters. E.g $$ f(t;\tau) = \frac{1}{\tau} e^{-t/\tau}, \qquad t \geq0 $$ If $t=1$ (a single ...
3
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65 views

Show that MLE of $\lambda = \frac{n-T_n}{S_n+cT_n}$

$X_i$ are i.i.d exponential, mean $\lambda^{-1}$ for $1 \leq i \leq n$ and, the values are measured such that $X_i = c$ if $X_i \geq c$ and $X_i$ otherwise. Show that MLE of $\lambda = ...
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1answer
167 views

Understanding the mean and rate of an exponential distribution

Background: In the field of Civil Traffic Engineering the arrival of the vehicles in a road is random process modelled by exponential distribution [1]. Illustrative example: Along this line, let's ...
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70 views

Superposition of two renewal processes

Usually, superposition of two independent renewal processes may not lead to a renewal process. However, in my problem, the interval time between two events of the renewal process is more specific ...
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2answers
130 views

Linear graph turning exponential at a particular point

For a line graph, it behaves linearly upto a particular point and varies exponentially after it. Please suggest me a statistical approach/test to know this threshold point. When I plot a scatter ...