Parametric Distributions: Role, Scale, Families, Mixtures, and Data-Dependent, Study notes of Applied Statistics

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The Role of Parameters

1 Parametric and scale distributions

2 Parametric distribution families

3 Finite mixture distributions

4 Data-dependent distributions

The Role of Parameters

1 Parametric and scale distributions

2 Parametric distribution families

3 Finite mixture distributions

4 Data-dependent distributions

The scale distribution: Definition

• (^) Definition: A parametric distribution is a scale distribution if, when

a random variable from that set of distributions is multiplied by a positive constant, the resulting random variable is also in that set of distributions.

• (^) For instance, the Weibull distribution is a scale distribution
• (^) Other examples are:
  • exponential (see textbook),
  • gamma (see textbook),
  • normal (why?)

The scale distribution: Definition

• (^) Definition: Let X be a random variable with nonnegative support which has a scale distribution. If a parameter of that scale distribution satisfies the following two conditions:

1. When a member of that scale distribution is multiplied by a positive constant, the scale parameter is multiplied by the same constant, while
2. all the other parameters remain the same, that parameter is called the scale parameter.

• For the exponential (we do have a scale parameter θ)
• For gamma (we again have a scale parameter θ)
• Weibull - see the problem from class
• For the lognormal we do not have a scale parameter (according to the parametrization used in the textbook) - although this is a scale distribution (why??)

Parametric distribution families

• (^) “Definition:” A parametric distribution family is a set of parametric distributions that are related in some meaningful way.
• (^) Most importantly, we can do the following:
• (^) the set of parameters is finite - but we can decrease the exact number of parameters by setting some of them to be
• (^) constant, e.g., exponential is a special type of gamma (how?)
• (^) equal to each other, e.g., paralogistic is a special type of distribution

from the transformed beta distribution family with τ = 1 and α = γ

The Role of Parameters

1 Parametric and scale distributions

2 Parametric distribution families

3 Finite mixture distributions

4 Data-dependent distributions

Variable-component mixture distribution

• (^) Definition: A variable-component mixture distribution has a distribution function F that can be written as

F (x) =

∑^ K

j=

aj Fj (x)

for some K ∈ { 1 , 2 ,... } and where

∑^ K

j=

aj = 1, aj > 0 , j = 1, 2 ,... , K.

• (^) This is a type of a semiparametric model

The Role of Parameters

1 Parametric and scale distributions

2 Parametric distribution families

3 Finite mixture distributions

4 Data-dependent distributions