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Mathematics and Informatics ICTAMI 2005 - Alba Iulia, Romania

ON THE INVARIANCE PROPERTY OF THE FISHER INFORMATION FOR A TRUNCATED LOGNORMAL

DISTRIBUTION (I)

Ion Mihoc, Cristina I. F˘atu

Abstract.The Fisher information is well known in estimation theory. The objective of this paper is to give some definitions and properties for the trun-cated lognormal distributions. Then, in the next paper ([2]), we shall deter-mine some invariance properties of Fisher’s information in the case of these distributions.

Keywords: Fisher information, lognormal distribution, truncated distribu-tion, invariance property

2000 Mathematics Subject Classification: 62B10, 62B05.

1.Normal and lognormal distributions

Let X be a normal distribution with density function

f(x;mx, σx2) =

1

√

2πσx

exp (

−1

2

_x−_m

x

σx

2)

, x_∈R, (1) where the parameters mx and σx2 have their usual significance, namely:

mx =E(X), σx2 =V ar(X), mx ∈R, σx >0.

Definition 1.1.If X is normally distributed with mean mx and variance

σ_x2, then the random variable

Y =eX, orX = lnY, Y >0, (2) is said to be lognormally distributed. The lognormal density function is given by

g(y;mx, σx2) =

1

√

2πσx

1 yexp

  

−1₂ lny_σ−mx

x

!2  

E(Y) =my = exp mx+

σ_x2 2

!

, V ar(Y) =σ_y2 = exp_{2mx+σx2}

eσx2−1_. (4)

Remark 1.1.From the relations (4), we obtain

mx = ln

 

m2_y q

m2

y +σy2



, σ2_x= ln "

m2_y+σ2_y m2

y

#

. (5)

2.The truncated lognormal distributions

Definition 2.1. We say that the random variable Y has a lognormal distribution truncated to the left at Y = a, a _∈ R, a > 0 and to the right at Y = b, b _∈ R, a < b, denoted by Ya↔b, if its probability density function ,

denoted by ga↔b(y;mx, σy2), is of the form

ga↔b(y;mx, σx2) =

        

k(a,b)

√

2πσ

1

y exp

−12

_lny

−mx

σx 2

if 0< a_≤y_≤b, 0 if 0< y < a

0 if y > b,

(6)

where the constant

k(a, b) = _Φ(lnb₋mx 1 σx )−Φ(

lna₋mx

σx ) (7)

results from the conditions

10 _g

a↔b(y;mx, σx2)≥0if y ∈(0,∞),

20 +R∞

0

ga↔b(y;mx, σ2x)dx= 1.

(8)

Remark 2.1.The function

Φ(z) = √1

2π

z

Z

−∞

exp ₋t

2

2 !

for which we have the following relations

Φ(_−∞) = 0,Φ(0) = 1

2,Φ(+∞) = 1,Φ(−z) = 1−Φ(z), (10) is the normal distribution function corresponding to the standard normal ran-dom variable

Z = X−mx σx

, M(Z) = 0, V ar(Z) = 1 (11) which has the probability density function

f(z; 0,1) =f(z) = √1

2π exp − t2

2 !

, z _∈(_−∞,+_∞). (12)

Remark 2.2. A such truncated probability distribution can be regarded as a conditional probability distribution in the sense that if X has an un-restricted distribution with the probability density function g(y;mx, σx2), then

ga↔b(y;mx, σx2),as defined above, is the probability density function which

gov-erns the behavior of subject to the condition that Ya↔b is known to lie in [a, b].

Also, in such case, we say that the random variable Ya↔b has a bilateral

trun-cated lognormal distribution with the limits a and b.

Remark 2.3.It is easy to see that from the relations (6), (7) and (10) follows

lim

a→0ga↔b(y;mx, σ 2

x) =

  

1 Φ(lnb₋mx

σx )

.g(y;mx, σx2) if 0< y≤b

0 if y > b, (13)

lim

b→+∞

ga↔b(y;mx, σx2) ==

  

1 1−Φ(lna−mx

σx )

.g(y;mx, σx2) if y ≥a

0 if 0< y _≤a, (14) and

lim

a→−0,b→+∞

ga↔b(y;mx, σx2) = √₂1_πσx

1

y exp

−12

lny−mx

σx 2

where g_→b(y;mx, σx2) is the probability density function then when Y→b has

a lognormal distribution truncated to the right at Y = b; ga←(y;mx, σx2) is

the probability density function then when Ya← has a lognormal distribution

truncated to the left atY =aandg(y;mx, σ2x)is the probability density function

then when Y has an ordinary lognormal distribution.

Theorem 2.1.If Ya↔b is a random variable which follows a bilateral

trun-cated lognormal distribution, then its expected value E(Ya↔b) and second

mo-ment E(Y2

a↔b) have the following expressions

E(Ya↔b) = exp

( mx+

σ_x2 2

)h_Φlnb−mx

σx −σx

−Φlna−mx

σx −σx i h

Φlnb−mx

σx

−Φlna−mx

σx

i , (16)

E(Y_a2_↔b) = expn2mx+ 2σx2

o h

Φlnb−mx

σx −2σx

−Φlna−mx

σx −2σx i h

Φlnb−mx

σx

−Φlna−mx

σx

i . (17) Proof. Making use of the definition of E(Ya↔b), we obtain

E(Ya↔b) =

k(a, b)

√

2πσx b

Z

a

exp   

−1₂ lny_σ−mx

x

!2  

dy. (18)

By making the change of variables

t = lny−mx σx

, (19)

we obtain

y= exp_{mx+σxt}, dy=σxexp{mx+σxt}dt, (20)

and (18) can be rewritten as follows

E(Ya↔b) =

k(a, b) exp_{mx}

√

2π

lnb₋mx σx Z

lna₋mx σx

exp

−1

2(t

2

−2σxt)

= k(a, b) exp n

mx+σ

2

x

2

o

√

2π

lnb₋mx σx Z

lna₋mx σx

exp

−1₂(t₋σx)2

dt= (22)

= k(a, b) exp n

mx+σ

2

x

2

o

√

2π

lnb₋mx σx −σx

Z

lna₋mx σx −σx

exp

−1₂w2

dw, (23)

if we used a new change of variables, namely w=t₋σx.

From (23) we obtain just the form (16) for the expected value of Ya↔b.

Using an analogous method we can obtain the expression for the second moment E(Y2

a↔b).

Thus, we have E(Y2

a↔b) = k(a,b)

√

2πσx

b

R

a

yexp

−1 2

_lny

−mx

σx 2

dy= = k(a,b) exp{2mx+2σ2x}

√

2π

lnb₋mx σx R

lna₋mx σx

expn₋1₂(t₋2σx)2

o dt=

= k(a,b) exp{2mx+2σ2x}

√

2π

lnb₋mx σx −2σx

R

lna₋mx σx −2σx

expn₋1₂v2o_dv=

= exp_{2mx+ 2σx2}

[Φ(lnb₋mx

σx −2σx)−Φ(

lna₋mx σx −2σx)] [Φ(lnb−mx

σx )−Φ(

lna₋mx σx )]

,

if we used the change of variables (19), respectively the change of variables v =t₋2σx.

Corollary 2.1.IfYa↔b is a random variable with a lognormal distribution

truncated to the left at Y =a and to the right at Y =b, then Var(Ya↔b) = E(Ya2↔b)−[E(Ya↔b)]2 =

=exp_{2mx+ 2σx2}

[Φ(lnb−mx

σx −2σx)−Φ(

lna₋mx σx −2σx)] [Φ(lnb−mx

σx )−Φ(

lna₋mx

σx )] −

−exp_{2mx+σx2}

[Φ(lnb₋mx

σx −σx)−Φ(

lna₋mx σx −σx)]

2

[Φ(lnb₋mx σx )−Φ(

lna₋mx σx )]

Corollary 2.2.For the unilateral truncated lognormal random variables Ya← and Y→b, respectively for the lognormal random variable Y we have the

following expected values and variances

E(Ya←) = lim b→+∞

E(Ya↔b) = exp

( mx+

σ2

2 )h₁

−Φlna−mx

σx −σx i h

1₋Φlna−mx

σx

i , (24)

VarYa←= lim_b

→+∞V ar(Ya↔b) =

= exp_{2mx+ 2σx2}

[1−Φ(lna−mx

σx −2σx)] [1−Φ(lna−mx

σx )] −

exp_{2mx+σx2}

[1−Φ(lna−mx

σx −σx)]

2

[1−Φ(lna−mx

σx )]

2 ,

E(Y_→b) = lim

a→0E(Ya↔b) = exp

n

mx+σ

2 2

oΦ(lnb−mx σx −σx)

Φ(lnb₋mx σx )

,

VarY_→b = lim

a→0V ar(Ya↔b) =

= exp_{2mx+ 2σx2}

Φ(lnb−mx

σx −2σx)

Φ(lnb−mxσx) −exp{2mx+σ

2

x}

Φ2₍lnb₋mx σx −σx)

Φ2₍lnb₋mx σx )

, respectively

E(Y)=my = lim a→−∞,b→+∞

E(Ya↔b) = exp

mx+σ

2

x

2

,

Var(Y)=σ2

y =_a lim

→0,b→∞V ar(Ya↔b) = exp{2mx+

σ_x2_}eσ2

x−1

.

Starting with the second page the header should contain the name of the author and a short title of the paper

3.Fisher’s information measures for a truncated lognormal distribution.

Case: mx-an unknown parameter,σ2x-a known parameter

Theorem 3.1.If the random variable Ya↔b has a bilateral truncated

log-normal distribution,that is its probability density function is of the form

ga↔b(y;mx, σx2) =

        

k(a,b)

√

2πσ

1

yexp

−12

_lny

−mx

σx 2

if 0< a_≤y _≤b, 0 if 0< y < a

0 if y > b,

where

k(a, b) = 1

Φlnb−mx

σx

−Φlna−mx

σx

, (26)

then the Fisher’s information measure, about the unknown parameter mx,

has the following form

IYa_↔b(mx) = 1 σ2

x −

[f(lnb;mx, σ2x)−f(lna;mx, σ2x)]2

h

Φlnb−mx

σx

−Φlna−mx

σx

i2 −

−[(lnb−mx)f(lnb;mx, σ

2

x)−(lna−mx)f(lna;mx, σx2)]

σ2

x

h

Φlnb−mx

σx

−Φlna−mx

σx

i , (27)

where

f(lna;mx, σx2) =

1

√

2πσx

exp   

−1₂ lna_σ−mx

x

!2  

∈R+

f(lnb;mx, σx2) =

1

√

2πσx

exp   

−1₂ lnb_σ−mx

x

!2  

∈R+.

Proof. For a such continuous random variable Ya↔b the Fisher information

measure, about the unknown parameter mx,has the form

IXa_↔b(mx) = E  

∂lnga_↔b(y;mx, σ

2

x)

∂mx

!2 =

=

b

Z

a

∂lnga_↔b(y;mx, σ

2

x)

∂mx

!2

ga_↔b(y;mx, σx2)dy=

=₋

b

Z

a

∂2_lng

a_↔b(y;mx, σ

2

x)

∂m2

x

ga_↔b(y;mx, σx2)dy=−E

" ∂2_lng

a_↔b(y;mx, σ

2

x)

∂m2

x

Using (25), we obtain lnga

↔b(y;mx, σ

2

x) =−ln(

√

2πσx) + lnk(a, b)−lny−1₂

_lny

−mx

σx 2

, respectively

∂lnga

↔b(y;mx, σ

2

x)

∂mx

= ∂lnk(a, b) ∂mx

+lny−mx σ2

x

= 1

k(a, b)

∂k(a, b) ∂mx

+lny−mx σ2 x . (29) Because ∂k(a, b) ∂mx = ∂ ∂mx   1 Φlnb−mx

σx

−Φlna−mx

σx  = = − ∂ ∂mx h

Φlnb−mx

σx i

+ ∂ ∂mx

h

Φlna−mx

σx i h

Φlnb−mx

σx i

+hΦlna−mx

σx

i2 , and

∂ ∂mx

"

Φ lna−mx σx !# = ∂ ∂mx     1 √ 2π

lna₋mx σx Z

−∞

exp ₋t

2 2 ! dt    = = ∂ ∂mx     1 2 + 1 √ 2π

lna₋mx σx Z

0

exp ₋t

2 2 ! dt    =

=₋1√2πσxexp

  

−1

2

lna₋mx

σx

!2  

=₋f(lna;mx, σ2x), (30)

∂ ∂mx

"

Φ lnb−mx σx

!#

=₋f(lnb;mx, σ2x), (31)

we obtain that

∂k(a, b) ∂mx

= f(lnb;mx, σ

2

x)−f(lna;mx, σx2)

h

Φlnb−mx

σx i

+hΦlna−mx

σx

∂lnga_↔b(y;mx, σx2)

∂mx

= f(lnb;mx, σ

2

x)−f(lna;mx, σ2x)

h

Φlnb−mx

σx i

+hΦlna−mx

σx

i +

lny₋mx

σ2

x

.

Then, from this last relation, we obtain ∂2lnga_↔b(y;mx, σx2)

∂m2

x

=

−_σ1₂

x

+k(a, b) σ2

x

h

(lnb₋mx)f(lnb;mx, σ2x

−lna₋mx)f(lna;mx, σx2)

i +

+[f(lnb;mx, σ

2

x)−f(lna;mx, σ2x)]

2

h

Φlnb−mx

σx i

+hΦlna−mx

σx

i2 ,

then when we have in view the relations (30), (31) as well as the following relations

∂ ∂mx

h

f(lna;mx, σx2)

i

= lna−mx σ2

x

f(lna;mx, σ2x)

∂ ∂mx

h

f(lnb;mx, σ2x)

i

= lnb−mx σ2

x

f(lnb;mx, σ2x).

Because ∂2lnga_↔b(y;mx,σ2x)

∂m2_x is independent of the variable y, from (3.3b) we

obtain just the Fisher’s information (27), if we have in view thatga_↔b(y;mx, σ

2

x)

is a probability density function. This completes the proof of the theorem.

References

[1] Mihoc, I., F˘atu C. I., Fisher′ _{s Information Measures and Truncated}

Normal Distributions (II), Seminarul de teoria celei mai bune aproxim¯ari, con-vexitate ¸si optimizare, 1960-2000, ,,Omagiu memoriei academicianului Tiberiu Popoviciu la 25 de ani de la moarte, 26-29 octombrie,2000, pp.153-155.

[3] Mihoc, I., F˘atu, C.I., Calculul probabilit˘at¸ilor ¸si statistic˘a matematic˘a, Casa de Editur˘a Transilvania Pres, Cluj-Napoca, 2003.

[4] Schervish,M.J.,Theory of Statistics, Springer-Verlag New York,Inc. 1995. Ion Mihoc

Faculty of Economics

Christian University Dimitrie Cantemir , 3400 Cluj-Napoca, Romania Teodor Mihali 56, 3400 Cluj-Napoca, Romania

email:imihoc@cantemir.cluj.astral.ro Cristina F˘atu

Faculty of Economics

Christian University Dimitrie Cantemir , Cluj-Napoca, Romania Teodor Mihali 56, 3400 Cluj-Napoca, Romania