5.4 The linear exponential family A large parametric family that includes many of the distributions of interest to actuaries has special use in Bayesian analysis (Section 15.3) and in credibility (Section 18.7). The definition is as follows.
What is the difference between linear and exponential models?
- It is its own derivative d/dx (e^x)= e^x
- It is also its own integral
- It exceeds the value of any finite polynomial in x as x->infinity
- It is continuous and differential from -infinity to +infinity
- It's series representation is: e^x= 1 +x +x^2/2! + x^3/3! ...
- e^ix=cosx + isinx
- It is the natural solution of the basic diff.eq. ...
What is exponential family?
Interpretation
- T(x) is a sufficient statistic of the distribution. ...
- η is called the natural parameter. ...
- A(η) is called the log- partition function because it is the logarithm of a normalization factor, without which f X ( x ; θ ) {\displaystyle f_ {X} (x;\theta )} ...
What is the difference between a linear and exponential graph?
- If the value decreased linearly, what was its annual rate of decrease?
- If the value decreased exponentially, what was its annual decay factor? What was its annual percent depreciation?
- Calculate the value of your car when it is 5 5 years old under each assumption, linear or exponential depreciation.
Is Weibull distribution an exponential family?
This person is not on ResearchGate, or hasn't claimed this research yet. A three parameter probability model, the so called Weibull-exponential distribution was proposed using the Weibull Generalized family of distributions. Some important models in the literature were found to be sub models of the new model.
What is a regular exponential family?
Exponential families have sufficient statistics that can summarize arbitrary amounts of independent identically distributed data using a fixed number of values. Exponential families have conjugate priors, an important property in Bayesian statistics.
How do you show a distribution is an exponential family?
0:017:53Normal distribution: in the exponential family - YouTubeYouTubeStart of suggested clipEnd of suggested clipCase here for simplicity one over the square root of 2 Pi Sigma squared times the exponential ofMoreCase here for simplicity one over the square root of 2 Pi Sigma squared times the exponential of minus X minus mu to be squared. Over 2 Sigma squared.
How do you show Poisson is an exponential family?
Page 4CHAPTER 8. THE EXPONENTIAL FAMILY: BASICS. Thus the Poisson distribution is an exponential family distribution, with:η = log λ (8.14)T(x) = x. (8.15)A(η) = λ = eη (8.16)h(x) = ... x! .(8.17) Moreover, we can obviously invert the relationship between η and λ:λ = eη. (8.18)More items...
Is gamma an exponential family?
The gamma distribution is a two-parameter exponential family with natural parameters k − 1 and −1/θ (equivalently, α − 1 and −β), and natural statistics X and ln(X). If the shape parameter k is held fixed, the resulting one-parameter family of distributions is a natural exponential family.
Is Rayleigh distribution an exponential family?
For selected values of the scale parameter, run the simulation 1000 times and compare the empirical density function to the true density function. Finally, the Rayleigh distribution is a member of the general exponential family.10-Aug-2020
Is Beta exponential family?
The family of beta(α,β) distributions is an exponential family.
Is multinomial exponential family?
One example of exponential family is multinomial distribution. For given data x = (x1,x2, ..., xk) , xi ∼ Multi(1, πi ) and Σ πi = 1, we can write the probability of the data as follows.
Is Laplace distribution exponential family?
The Laplace distribution is also a member of the general exponential family of distributions. Suppose that X has the Laplace distribution with known location parameter a∈R and unspecified scale parameter b∈(0,∞).02-Feb-2021
Is exponential family convex?
The cumulant function of an exponential family is a lower semicontinuous proper convex function.23-Jan-2013
What is curved exponential family?
Curved exponential families are a subset of the class of exponential families where the dimension of the parameter space does not match the dimension of the exponential family.04-Mar-2019
Is lognormal and exponential family?
The lognormal and Beta distribution are in the exponential family, but not the natural exponential family.
Is uniform distribution an exponential family?
The uniform(0,θ) family is not an exponential family since the support Yθ = (0,θ) depends on the unknown parameter θ.
EXAMPLE 5.11
Show that the normal distribution is a member of the linear exponential family.
EXAMPLE 5.12
Show that the gamma distribution is a member of the linear exponential family.