lehmann 1990

The role of Statistics in Modelling A Reservoir of Models Model selection Theoretical And Empirical Models

"Model Specification: The Views of Fisherand Neyman, and Later Developments" by

E. L. Lehmann1

Luca Perdoni

April 21, 2015

1E. L. Lehmann was Emeritus Professor of Statistics at University ofCalifornia, Berkeley


Fisher’s view on the problem of statistics

1. Problems of specification, which are analysed in this paper2. Problems of estimation, which deal with point estimation

of the parameters embedded in the model chosen in 13. Problems of distribution, which refers to the distribution

of the estimator derived in 2

"As regards problems of specification, these are entirely amatter for the practical statistician"


Neyman’s view on the problem of statistics

• Models of complex phenomena are constructed bycombining simple building blocks

• These elementary bricks "partly through experience andpartly through imagination, appear to us familiar, and,therefore, simple."

• A distinction must be done between theoretical andempirical models


What contribution statistical theory has to make tomodel specification or construction?

1. A reservoir of models2. Model selection3. Classification of models


A reservoir of models

• Univariate distributions• Multivariate distributions• Stochastic processes• Linear models and general linear models

We must be careful in our characterization


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√

2πe−

(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., thedensity is the same at all points equidistant from the origin.


A reservoir of models

• It is better to define models via simple and practicalfeatures rather than formulas

• Assumption of independence must not be taken forgranted

• A good substitute for independence can be De Finetti’sExchangeability


Exchangeability

if we assume independence we can write the joint distributionof identical distributed random variables as

p(x1, ..., xn) =n∏

i=1

p(xi) (1)

while if we assume exchangeability ( which is equivalent toindependence conditional on θ)

p(x1, ..., xn) =

∫ n∏i=1

p(xi |θ) dP(θ) (2)


Model Selection

• Given data, a model is selected from a certain family ofmodels

• As example, the appropriate number of regressors, k, canbe chosen minimizing the Mean squared prediction error,(MSPE)

• Anyway, a preliminary step of choosing the right familyhas been omitted

• A characterization of different kinds of models can beuseful


Theoretical Vs. Empirical Models

They have a different purpose• Theoretical models explain the basic mechanismunderlying the process being studied; they constitute aneffort to achieve understanding.

• Empirical models are used as a guide to action, oftenbased on forecasts of what to expect from futureobservations.


Theoretical Vs. Empirical Models

The framework in which they are applied is often different• "The technologist is not concerned with truth at all, Themark of the technologist is that he must act; everythingthat he does has some sort of deadline. He has tomanage therefore, with as much truth as is available tohim, with the scientific theories current in his time."

• "If all we need to do is either to estimate the behaviour ofthe process under various experimental conditions or tofind optimum operating conditions, we do not necessarilyneed a mechanistic model. In some circumstances, anattempt to discover a mechanism merely to develop anoperable system would be needlessly time consuming"


Mendel’s Inheritance Model


Mendel’s Inheritance Model

Law of SegregationDuring gamete formation, the alleles for each gene segregatefrom each other so that each gamete carries only one allele for

each gene.

Law of Independent AssortmentGenes for different traits can segregate independently during

the formation of gametes.


Pearson distribution

A Pearson density is defined to be any valid solution to thefollowing differential equation

p′(x)

p(x)+

a + x − λb2(x − λ)2 + b1(x − λ) + b0

= 0. (3)

The great use of this distribution in empirical research is dueto the fact that it is uniquely determined by the first fourmoments

µn =

∫ ∞−∞

(x − c)n f (x) dx (4)


Pearson distribution1. First empirical moment is the empirical Expected Value2. Second empirical moment is related to Variance3. Third empirical moment is related to Skewness4. Fourth empirical moment is related to Kurtosis

Within the framework of Pearson distribution we can find thefollowing distributions:• beta distribution• chi squared distribution• uniform distribution• exponential distribution• gamma distribution• F-distribution• inverse chi squared distribution• inverse-gamma distribution• normal distribution• Student’s t-distribution


Skewness


Kurtosis


Galileo’s Trial

Perché il dire, che supposto che la Terra si muova e il Sole siafermo si salvano tutte le apparenze meglio che con porre glieccentrici et epicicli, è benissimo detto, e non ha pericolonessuno; e questo basta al mathematico: ma volere affermareche realmente il Sole stia nel centro del mondo e solo si rivoltiin sé stesso senza correre dall’oriente all’occidente, e che laTerra stia nel terzo cielo e giri con somma velocità intorno alSole, è cosa molto pericolosa non solo d’irritare i filosofi etheologici scolastici, ma anco di nuocere alla Santa Fede conrendere false le Scritture Sante

lehmann 1990

Education

model selection3

berkeleythe role of

reservoir of models2

empirical modelsexample

empirical modelsfishers

empirical modelsneymans

problems of specification

problems of distribution