Abstract
The weighted distributions are used when the sampling mechanism records observations according to a nonnegative weight function. Sometimes the form of the weighted distribution is the same as the original distribution except possibly for a change in the parameters that is called the form-invariant weighted distribution. In this paper, by identifying a general class of weight functions, we introduce an extended class of form-invariant weighted distributions belonging to the non-regular exponential family which included two common families of distribution: exponential family and non-regular family as special cases. Some properties of this class of distributions such as the sufficient and minimal sufficient statistics, maximum likelihood estimation and the Fisher information matrix are studied.
Highlights
The weighted distributions have been used when the sampling mechanism records observations according to a certain chance
We study the form-invariant property of the extended class of distributions, termed as the non-regular exponential family with pdf, given by p f (x; θ) = exp bi(θ) di(x) + a(x) + c(θ)
For further insights on the form-invariance, we have extended the study to the two-parameter distributions that belong to the non-regular exponential family and proved the necessary and sucient conditions for the form-invariance property
Summary
The weighted distributions have been used when the sampling mechanism records observations according to a certain chance. Patil & Ord (1976) proved that a necessary and sucient condition for X to be form-invariant under size-biased weight function of order β (i.e. w(x, β) = xβ) is that its distribution belongs to log-exponential family with pdf f (x; θ) = exp{θ log x + a(x) − c(θ)}. We study the form-invariant property of the extended class of distributions, termed as the non-regular exponential family with pdf, given by p f (x; θ) = exp bi(θ) di(x) + a(x) + c(θ). For further insights on the form-invariance, we have extended the study to the two-parameter distributions that belong to the non-regular exponential family and proved the necessary and sucient conditions for the form-invariance property. Some properties of the distributions belong to the nonregular-exponential family such as the sucient and minimal sucient statistics, maximum likelihood estimation and the Fisher information matrix are given
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