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The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models
"Model Specification: The Views of Fisher
and Neyman, and Later Developments" by
E. L. Lehmann1
Luca Perdoni
April 21, 2015
1
E. L. Lehmann was Emeritus Professor of Statistics at University of
California, Berkeley
The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models
Fisher’s view on the problem of statistics
1. Problems of specification, which are analysed in this paper
2. Problems of estimation, which deal with point estimation
of the parameters embedded in the model chosen in 1
3. Problems of distribution, which refers to the distribution
of the estimator derived in 2
"As regards problems of specification, these are entirely a
matter for the practical statistician"
The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models
Neyman’s view on the problem of statistics
• Models of complex phenomena are constructed by
combining simple building blocks
• These elementary bricks "partly through experience and
partly through imagination, appear to us familiar, and,
therefore, simple."
• A distinction must be done between theoretical and
empirical models
The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models
What contribution statistical theory has to make to
model specification or construction?
1. A reservoir of models
2. Model selection
3. Classification of models
The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models
A reservoir of models
• Univariate distributions
• Multivariate distributions
• Stochastic processes
• Linear models and general linear models
We must be careful in our characterization
The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models
Example 1
Let’s compare a model defined with the following statement
X1, . . . , Xn are i.i.d with normal distribution N(0, σ2
) so that
f (xi ) = 1
σ
√
2π
e−
(xi )2
2σ2
with another one characterized by the following two properties
the X’s are independent
and
the joint density of the X’s is spherically symmetric, i.e., the
density is the same at all points equidistant from the origin.
The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models
A reservoir of models
• It is better to define models via simple and practical
features rather than formulas
• Assumption of independence must not be taken for
granted
• A good substitute for independence can be De Finetti’s
Exchangeability
The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models
Exchangeability
if we assume independence we can write the joint distribution
of identical distributed random variables as
p(x1, ..., xn) =
n
i=1
p(xi ) (1)
while if we assume exchangeability ( which is equivalent to
independence conditional on θ)
p(x1, ..., xn) =
n
i=1
p(xi |θ) dP(θ) (2)
The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models
Model Selection
• Given data, a model is selected from a certain family of
models
• As example, the appropriate number of regressors, k, can
be chosen minimizing the Mean squared prediction error,
(MSPE)
• Anyway, a preliminary step of choosing the right family
has been omitted
• A characterization of different kinds of models can be
useful
The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models
Theoretical Vs. Empirical Models
They have a different purpose
• Theoretical models explain the basic mechanism
underlying the process being studied; they constitute an
effort to achieve understanding.
• Empirical models are used as a guide to action, often
based on forecasts of what to expect from future
observations.
The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models
Theoretical Vs. Empirical Models
The framework in which they are applied is often different
• "The technologist is not concerned with truth at all, The
mark of the technologist is that he must act; everything
that he does has some sort of deadline. He has to
manage therefore, with as much truth as is available to
him, with the scientific theories current in his time."
• "If all we need to do is either to estimate the behaviour of
the process under various experimental conditions or to
find optimum operating conditions, we do not necessarily
need a mechanistic model. In some circumstances, an
attempt to discover a mechanism merely to develop an
operable system would be needlessly time consuming"
The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models
Mendel’s Inheritance Model
The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models
Mendel’s Inheritance Model
Law of Segregation
During gamete formation, the alleles for each gene segregate
from each other so that each gamete carries only one allele for
each gene.
Law of Independent Assortment
Genes for different traits can segregate independently during
the formation of gametes.
The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models
Pearson distribution
A Pearson density is defined to be any valid solution to the
following differential equation
p (x)
p(x)
+
a + x − λ
b2(x − λ)2 + b1(x − λ) + b0
= 0. (3)
The great use of this distribution in empirical research is due
to the fact that it is uniquely determined by the first four
moments
µn =
∞
−∞
(x − c)n
f (x) dx (4)
The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models
Pearson distribution
1. First empirical moment is the empirical Expected Value
2. Second empirical moment is related to Variance
3. Third empirical moment is related to Skewness
4. Fourth empirical moment is related to Kurtosis
Within the framework of Pearson distribution we can find the
following distributions:
• beta distribution
• chi squared distribution
• uniform distribution
• exponential distribution
• gamma distribution
• F-distribution
• inverse chi squared distribution
• inverse-gamma distribution
• normal distribution
The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models
Skewness
The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models
Kurtosis
The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models
Galileo’s Trial
Perché il dire, che supposto che la Terra si muova e il Sole sia
fermo si salvano tutte le apparenze meglio che con porre gli
eccentrici et epicicli, è benissimo detto, e non ha pericolo
nessuno; e questo basta al mathematico: ma volere affermare
che realmente il Sole stia nel centro del mondo e solo si rivolti
in sé stesso senza correre dall’oriente all’occidente, e che la
Terra stia nel terzo cielo e giri con somma velocità intorno al
Sole, è cosa molto pericolosa non solo d’irritare i filosofi e
theologici scolastici, ma anco di nuocere alla Santa Fede con
rendere false le Scritture Sante

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Lehmann 1990

  • 1. The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models "Model Specification: The Views of Fisher and Neyman, and Later Developments" by E. L. Lehmann1 Luca Perdoni April 21, 2015 1 E. L. Lehmann was Emeritus Professor of Statistics at University of California, Berkeley
  • 2. The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models Fisher’s view on the problem of statistics 1. Problems of specification, which are analysed in this paper 2. Problems of estimation, which deal with point estimation of the parameters embedded in the model chosen in 1 3. Problems of distribution, which refers to the distribution of the estimator derived in 2 "As regards problems of specification, these are entirely a matter for the practical statistician"
  • 3. The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models Neyman’s view on the problem of statistics • Models of complex phenomena are constructed by combining simple building blocks • These elementary bricks "partly through experience and partly through imagination, appear to us familiar, and, therefore, simple." • A distinction must be done between theoretical and empirical models
  • 4. The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models What contribution statistical theory has to make to model specification or construction? 1. A reservoir of models 2. Model selection 3. Classification of models
  • 5. The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models A reservoir of models • Univariate distributions • Multivariate distributions • Stochastic processes • Linear models and general linear models We must be careful in our characterization
  • 6. The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models Example 1 Let’s compare a model defined with the following statement X1, . . . , Xn are i.i.d with normal distribution N(0, σ2 ) so that f (xi ) = 1 σ √ 2π e− (xi )2 2σ2 with another one characterized by the following two properties the X’s are independent and the joint density of the X’s is spherically symmetric, i.e., the density is the same at all points equidistant from the origin.
  • 7. The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models A reservoir of models • It is better to define models via simple and practical features rather than formulas • Assumption of independence must not be taken for granted • A good substitute for independence can be De Finetti’s Exchangeability
  • 8. The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models Exchangeability if we assume independence we can write the joint distribution of identical distributed random variables as p(x1, ..., xn) = n i=1 p(xi ) (1) while if we assume exchangeability ( which is equivalent to independence conditional on θ) p(x1, ..., xn) = n i=1 p(xi |θ) dP(θ) (2)
  • 9. The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models Model Selection • Given data, a model is selected from a certain family of models • As example, the appropriate number of regressors, k, can be chosen minimizing the Mean squared prediction error, (MSPE) • Anyway, a preliminary step of choosing the right family has been omitted • A characterization of different kinds of models can be useful
  • 10. The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models Theoretical Vs. Empirical Models They have a different purpose • Theoretical models explain the basic mechanism underlying the process being studied; they constitute an effort to achieve understanding. • Empirical models are used as a guide to action, often based on forecasts of what to expect from future observations.
  • 11. The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models Theoretical Vs. Empirical Models The framework in which they are applied is often different • "The technologist is not concerned with truth at all, The mark of the technologist is that he must act; everything that he does has some sort of deadline. He has to manage therefore, with as much truth as is available to him, with the scientific theories current in his time." • "If all we need to do is either to estimate the behaviour of the process under various experimental conditions or to find optimum operating conditions, we do not necessarily need a mechanistic model. In some circumstances, an attempt to discover a mechanism merely to develop an operable system would be needlessly time consuming"
  • 12. The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models Mendel’s Inheritance Model
  • 13. The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models Mendel’s Inheritance Model Law of Segregation During gamete formation, the alleles for each gene segregate from each other so that each gamete carries only one allele for each gene. Law of Independent Assortment Genes for different traits can segregate independently during the formation of gametes.
  • 14. The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models Pearson distribution A Pearson density is defined to be any valid solution to the following differential equation p (x) p(x) + a + x − λ b2(x − λ)2 + b1(x − λ) + b0 = 0. (3) The great use of this distribution in empirical research is due to the fact that it is uniquely determined by the first four moments µn = ∞ −∞ (x − c)n f (x) dx (4)
  • 15. The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models Pearson distribution 1. First empirical moment is the empirical Expected Value 2. Second empirical moment is related to Variance 3. Third empirical moment is related to Skewness 4. Fourth empirical moment is related to Kurtosis Within the framework of Pearson distribution we can find the following distributions: • beta distribution • chi squared distribution • uniform distribution • exponential distribution • gamma distribution • F-distribution • inverse chi squared distribution • inverse-gamma distribution • normal distribution
  • 16. The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models Skewness
  • 17. The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models Kurtosis
  • 18. The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models Galileo’s Trial Perché il dire, che supposto che la Terra si muova e il Sole sia fermo si salvano tutte le apparenze meglio che con porre gli eccentrici et epicicli, è benissimo detto, e non ha pericolo nessuno; e questo basta al mathematico: ma volere affermare che realmente il Sole stia nel centro del mondo e solo si rivolti in sé stesso senza correre dall’oriente all’occidente, e che la Terra stia nel terzo cielo e giri con somma velocità intorno al Sole, è cosa molto pericolosa non solo d’irritare i filosofi e theologici scolastici, ma anco di nuocere alla Santa Fede con rendere false le Scritture Sante