A family of goodness-of-fit tests for the Cauchy distribution

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A FAMILY OF GOODNESS-OF-FIT TESTS FOR THE CAUCHY DISTRIBUTION

RODZINA TESTÓW ZGODNOŚCI Z ROZKŁADEM CAUCHY’EGO

* Dr Jan Pudełko, Instytut Matematyki, Wydział Fizyki, Matematyki i Informatyki Stosowanej, Politechnika Krakowska.

A b s t r a c t

A new family of goodness-of-fit test for the Cauchy distribution is proposed in the paper. Every member of this family is affine invariant and consistent against any non Cauchy distribution.

Results of the Monte Carlo simulations performed to verify finite sample behaviour of the new tests are presented.

Keywords: Cauchy distribution, goodness-of-fit test, empirical characteristic function S t r e s z c z e n i e

W artykule zaproponowano nową rodzinę testów zgodności z rozkładem Cauchy’ego. Każdy test z tej rodziny jest afinicznie niezmienniczy i zgodny przeciwko każdej alternatywie nie będąacej rozkładem Cauchy’ego. Zaprezentowano także wyniki symulacji numerycznych przeprowadzonych w celu zbadania zachowania nowych testów

dla skończonych prób.

Słowa kluczowe: Rozkład Cauchy’ego, test zgodności, empiryczna funkcja charakterystyczna

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1. Introduction

Let X₁, X₂, ... be a sequence of i.i.d. random variables with distribution function F. We consider the problem of testing the hypothesis:

against

where F is the family of the Cauchy distributions, i.e.

with

The location parameter m is the median and the scale parameter σ represents half of the interquartile range in this case. In recent years there were several papers devoted to this problem.

Gürtler and Henze [5] and Matsui and Takemura [9] considered the test statistics of the form

where ϕ_n is the empirical characteristic function

of the standardized with suitable estimators data

φ₀(t) = exp(–|t|) is the theoretical characteristic function of the standard Cauchy distribution and w(t) is the weight function. The weight w(t) = exp(–λ|t|) considered in [5, 9] results with simply and closed form of the test statistics, namely

Gürtler and Henze [5] showed that with the sample median and half of the interquartile range as the estimators of location and scale, respectively, the test based on D_n,λ is consistent against each alternative distribution having a unique median and unique upper and lower quartiles.

In this paper we propose another test statistics of the form (1). Since the most important properties of a distribution are determined by the behaviour of the characteristic function in a neighbourhood of zero, especially in the case of heavily tailed distribution like the Cauchy one, one should use the weight putting more mass around zero. For this reason we will use unbounded in zero weight function. With the weight

H F0: ∈F

H F₁: ∉F

F =

{

F F x F x m:

( )

⁼ ₀

( (

⁻

)

/σ

)

,

(

m,σ

)

′∈ × ∞

( )

0,

}

F x0 x

1 2 1

( )

⁼ ^/ ⁺^π⁻ ^{arctan .}

( )1 D n_n=

∫

^ϕ_n

( )

t ⁻^ϕ₀

( ) ( )

t w t dt² ,

ϕ_n _j

j

t n

n itY

( )

⁼

( )

∑

=

1

exp

( )2 Y_j=

(

X m_j− ^ˆ_n

)

/^σ^ˆ_n,j⁼1,..., ,n

D n Y Y Y

n

j k

j k n

j j n ,

, λ .

λ λ

λ

λ λ

=²

^∑

⁼¹ ²+

(

−

)

²⁻⁴

^∑

⁼¹

⁽

¹⁺¹⁺

⁾

²⁺ ²⁺²²⁺

w t( ) exp(= −λ| |)/ | | ,t t^γ

(3)

where λ > 0 and γ ∈ (0; 1), the test statistic again has closed form, namely

Since the family of the Cauchy distributions is closed with respect to the affine transformations one is interested in affine invariant test. To obtain an affine invariant test statistic of the form (1) it is enough to standardize the sample with equivariant estimators in (2), i.e. estimators mˆ_n = mˆ_n( ₁,...,X_n) and σˆ_n = σˆ_n(X₁,...,X_n) such that for every a > 0 and b ∈  we have

and

The previous authors considered the sample median and half of the interquartile range [5], the maximum likelihood estimators (MLE) and the EISE estimators [9]. Since the use of EISE do not improve the power of the test complicating the calculations at the same time we do not consider these estimators in this paper.

The paper is organized as follows. Section 2 contains a review of properties of estimators proposed in [10] by Pudełko. In Sections 3 and 4 there are main results of the paper, e.i. theorems concerning the weak convergence of D_n,λ,γ when the sample comes from the Cauchy distribution, its limit distribution and the consistency of corresponding test against each non-Cauchy alternative distribution. Section 5 presents the results of the numerical simulations performed to verify the finite sample behaviour of the new test.

2. Estimators of the parameters of the Cauchy distribution

The choice of the parameters used to standardize the data in (2) is very important to the performance of the test. In this paper besides MLE and order estimators (sample median and half of the quartile range) we will use estimators proposed by Pudełko in [10]. These estimators are defined as argument θˆ_n,α = (mˆ_n, σˆ_n) minimizing¹ the distance

1 Estimators defined as argument minimizing

ϕ_n

( )

t ⁻ϕ_θ

( ) ( )

t w t dt

∫

 ²

were proposed independently by [6] and [11] but these authors considered bounded weight func- tion w.

D_n n Y_j

j n , ,

( )/

( ) ( ) ( ) cos ( )

λ γ γ γ

γ λ λ γ

= −

(

+ ⁻⁻

(

⁺ ⁺

)

⁻ ⁻

∑

=

2 1 2 ¹ 2 1 ² ² ^{1 2} 1

1

Γ aarctan

( ) ⁽ ^)/ co

,

Y

n Y Y

j

j k

j k n

1

1 2 2 1 2

1

+



 





 



+

(

+ −

)

⁻

∑

=

λ

λ ^γ ss (γ )arctan .

−  λ−

 





 



1 Y Y_j _k

m aX bˆ_n( ₁+ …, ,aX b am X_n+ =) ˆ_n( ₁, ,…X_n)+b

σˆ_n(aX b₁+ …, ,aX b a_n+ =) σˆ_n(X₁, ,…X_n).

δ θ ϕ ϕ

α θ

( ) | ( ) α ( )|

| | ,

= −

∫

 ⁿ ^t_t¹+ ^t ²d^t

(4)

where φ_n is the empirical characteristic function, φ_θ(t) = e^itm–σ|t| is the characteristic function of the Cauchy distribution with the median m and the interquartile range 2σ. In [10] was showed that θˆ_n,α may be equivalently defined by

where Γ is the Gamma function and Z_j = arctan((X_j – m)/σ): As it was showed in [10] the family of the above estimators can be continuously closed by taking for α = 0 the ML estimators

Estimators θˆ_n,α are affine equivariant, strongly consistent, asymptotically normally distributed with the covariance matrix

where B is the Beta function and I₂ is the 2 × 2 identity matrix and have the following Bahadur representation

with

3. Asymptotic behaviour of the proposed test statistic

The following useful representation of D_n,λ,γ can be obtained by straightforward algebra

where

We will consider Zˆ_n(t) as a random element in the Frechet space C() of continuous functions on the real line endowed with the metric

θ _α α σ α

θ

α α α

n j

n

j j

n Z Z

, : argmin= (− )  − cos cos( ) ,

 



∈

− −

∑

=

Θ2Γ 2 1 1

1

ˆ

θ σ σ

n θ

j n

n X mj

,₀ argmax log log ( ) .

1

2 2

= ^ −1

(

+ −

)

 



∈_Θ

∑

=

ˆ

Σ( )θ ( σ) ( ) ( , ) ,

α α α α

0

0 2

2 2

2 1

1

3 2 2 2 1

= −  − − − −

 



B I

nm n l X o

n n l X o

n j

n

j P

n j

n

j P

,

( ) ( ),

( ) ( ) ( ),

α

σ α σ

= +

− = +

=

∑

1 1

0 2

1

ˆ

ˆ (3)

(5)

ˆ

l x z z

l x z

1 2

1

2 2

1 1

2

1 1

2

1 2

( ) cos sin(( ) ),

( ) cos c

= − −

− −

− − −

α α

α α α

α α

α

(

oos((1−^α) ) .^z

)

( ) ( )

| | ,

, ,

| |

4 ²

1

D Z t e

t t

n n

t n

n λ γ

λσ γ

γ

=

∫

ˆ ⁻ ^ˆ σˆ ⁻ d

Z t_n n tX_j tX_j e ^t tm_n tm_n

j n

( )=

(

cos( ) sin(+ )− ⁻ n^{| |}(cos( ) sin(+ ))

)

=

1

∑

σ

∑

^.

ˆ ^ˆ ˆ ˆ

(5)

where ρ_j(f; g) = sup_|t|≤j |f(t) – g(t)|.

Now we formulate the following theorem on the convergence of the process Zˆ_n.

Theorem 1. Let X₁ X₂, ... be a sequence of independent, identically, Cauchy distributed random variables. Then there exists a centered Gaussian process Z in C() such that

where “→^d ” denotes weak convergence. If mˆ_n and σˆ_n are the sample median and half of the interquartile range, respectively, than the covariance kernel of Z is

for all t, s ∈ , where

For the estimators θˆ_n,α we have

In particular, for the maximum likelihood estimators we have

Proof. In the case of the MLE, the sample median and half of the interquartile range this theorem was proved in [5, 9] respectively. Here we prove the case of estimators θˆ_n,α. Let S ⊂ . By C(S) we denote the space of real-valued continuous function on S with the supremum norm. Using the Theorem of Csörgő and the notation therein (Section 3. of [3]) we will show that Z_n(t) is weakly convergent in C(S) to the zero mean Gaussian process with the covariance kernel K_α(∙, ∙). Assumptions (i)*, (ii)* and (vi) do not depend on the choice of the estimators and were verified in [5].

Assumption (iv) is a consequence of the Bahadur representation of the estimators θˆ_n,α presented in the previous section. In order to verify the Assumption (v) we estimate

( ) ( , ) ( , )

( , ),

6 1

12 1

ρ ρ

f g ρ f g

f g

j j j

j

= ₌ +

∑

∞

Zˆ_n→ ( ),^d Z in C

K s t e e st st s t

e

t s t s

t

( , ) ^| ^| ^{| | | |} | | | | | |

|

= +  + + + −

 



−

− − − −

−

π π π

2 2 2 1

|| | |

(tJ s1( )+2| |t J s2( ))−e sJ t⁻^s( 1( )+2| |s J t2( )),

J s sx

x dx J s sx

x dx

1 0 2 2 0 2

1

1 1

( ) sin( )

, ( ) cos( )

= .

+ =

+

∫

∞

∫

( ) ( , )

( ) ( ) ( , )

| | | | | |

7

1 4

1

3 2 2 2 1

2

K s t

e e st

B

t s t s

α

α α α α

=

− −

− − − − −

− − − − 

 





 + −

−

− + −

2 1

2 2

2

2 2

α

α α

| | ( , | |) α

( ) | | ( , |

| | | |

t es Γ s s et t

Γ

Γ ||)

( ) | | | | Γ 2 .

0

− − − 0



 















⋅ >

⋅ <

α s t t s

t s if if

( )8 K s t e₀( , )= ^{− −}^|^{t s}^|− +

(

1 2(st st e+| |)

)

^{− −}^{| | | |}^{t s}.

(6)

Hence, Zˆ_n converges weakly in C(S) to the zero mean Gaussian process with the covariance kernel of the form

where

By the direct calculation we have

Let us now calculate next components of K(s, t).

thus,

sup , ,

sup max(|cos sin

| |

x u x

x u

l x D l x

z



θ θ

α

α α

0 0

2

2 1

1

( )

⁺

( )

≤ −⁻ ⁻ (( ( ))|,| cos cos( ( ))|

max(|cos si

z z z

z

1 2 1

2

1 1

2 2

− − −



 +

− −

α ^α ^α α

α α

n

n( ( ))|,|cos cos( ( ))|)

( ) .

)

z2 z z 2

2

1 2 1 2

2

1 2

− −

− + + <∞

−

− − −

α α

α

α α α



K s t e e H s E l X l X H t H

s t s t T T

( , ) ( , ) ( ( ) ( ) ) ( , )

(

| | | | | |

= − +

−〈

− − − − θ₀ ₁ ₁ θ₀

tt,θ₀), k x s l x dF x( , ) ( ) ₀( ) H s( ,θ₀), k x t l x dF x( , ) ( ) ₀( ) ,

 

∫

^{〉 −〈}

∫

^〉

H t( , )θ =

∫

k x t d F x( , ) ∇_θ ( , ).θ

H t( ,θ₀) (= te⁻^{| |}^t, | |− t e⁻^{| |}^{t T}) .

E l X l X( ( ) ( ) )^T E( (X , )( (X , )) )^T C ( )

1 1

1

1 0

1

1 0

1 1

0

= =

− − − −

Λ Λ Λ Λ

Σ

ψ θ ψ θ

θ 22

1

3 2 2 2 1

2 2

( ) ( ) ( , ) ,

α−  − α −α −α −

 



B I

H s E l X l X H t

B

T T

( , ) ( ( ) ( ) ) ( , )

( ) ( ) ( , )

θ θ

α α α α

0 1 1 0

2

2 1

1

3 2 2 2 1

= −  − − − −

 

 +

= −

− −

∫

−

e st st

k x s l x dF x

t s

| | | |

( | |).

( , ) ( ) ( )

( ) (cos

 1 0

22

1

α

π α (( ) sin( ))cos sin( ( ))

( ) cos

∫

 ⁺ ⁻ ₊

= −

−

− −

sx sx z z dx

x

1

2

3

1 0

1 1

2 1

α

α α

α

π α

ππ α

π α

/ sin( ( ))sin( tan ) ,

( , ) ( ) ( )

( ) (

2

2 0

1 2 1

∫

−

= −

z z s z dz

k x s l x dF x

 ccos( ) sin( ))

( ) (cos( ) sin( ))cos



∫

+ +

− −⁻ +

sx sx dx

x

sx sx

1 2

1

2

2α

π α

1 1

1 2

1

− −

+

αz z α dx

sin( ( )) x

(7)

(comp. [4] formula 3.723.2).

Hence,

In the above integral there is minus when s ∙ t > 0 and plus in another case.

In the case of s ∙ t > 0 using the formulas 3.718.6 and 9.224 of [4] we have

where W is the Whittaker function, and Γ(∙, ∙) denotes the incomplete Gamma function. In the second case (s ∙ t < 0) by formula 3.718.5 of Gradshteyn, Ryzhik [4] we obtain

Thus K_α(∙, ∙) is of the form (7).

Since the convergence of Zˆ_n in C(S) was showed for any compact set S ⊂ ; Zˆ_n converges to Z also in the Frechet space C() with the metric ρ (comp. [8], p. 62).

Now we present the theorem on the convergence of the test statistic D_n,λ,γ. □ Theorem 2. Under the assumptions of Theorem 1 we have

= − −

− −

∫

⁻

∫

⁻

4 1

2

1 1

0

2 3

1 0

2

π α ^π π ^αα ^π ^α α

( ) cos( tan )

( ) cos cos( ( )

/ /

s z dz z z ))cos( tan )

( ) ^{| |} ( ) ^/ cos cos( (

s z dz

e ^s z z

= − −

− −

∫

−

2 1

2

1 1

3

1 0

2

α π ^αα ^π ^α α)))cos( tan )s z dz,

〈 〉

= −

∫

− −

−

H t k x s l x dF x

te ^t z

( , ), ( , ) ( ) ( )

( ) cos s

| | /

θ

π α

α π α

0 0

3

1 0

2 2

1

iin( ( ))sin( tan ) | |

( )

| | (

| | | |

| |

z s z dz t e e

t e

t s

t

1 2

1 2³

− −

− +

− −

α α

π

α

1

1 1

2 1

1 0

2

3

− −

= −

−

− −

α

∫

α

π α

π α α

) cos cos( ( ))cos( tan )

| | (

/

| |

z z s z dz

t e ^t

)) cos cos ( ) tan | |

( ) .

/ | | | |

1 0

2 1 2

1

− − −

∫

^π ^α^z

⁽

^z ⁻^α ^∓^s ^{z dz}

⁾

⁻ ^{t e}₋_α^{s t}

〈 〉

= −

∫

− −

−

H t k x s l x dF x t e s^t W

( , ), ( , ) ( ) ( )

| | ^{| |}| |⁽ ^)/ ( )/

θ

α

α α

0 0

1 2

2 1

1

 2

2 2 2 2

2

2 1 2

1

,( )/ | | | |

| | | |

( | |)

( )

| |

( )

| |

− − −

− +

− −

−

= −

α

α α

α

s t e

t e

s t

t s

Γ ΓΓ

Γ

( , | |)

( )

| |

( )

| |

| | | |

|

2 2

2

2 1 2

1

2

−

− −

−

= −

− −

α

α α

α

s t e

t e e

s t

t s s|| ( , | |)

( ) ,

Γ Γ

2 2

2 1

−

− −



 

 α

α s

〈^{H t}( ,θ₀),

∫

k x s l x dF x( , ) ( ) ₀( )〉 =0.



( ) ( )

| | : ( )

, , |

| |

,

| |

9

2 1 2

D Z t e

t t D Z t e

n n n n t d t t

λ γ

γ σ λ

γ λ γ

σ λ

=

∫

⁻ ⁻ d→ = ⁻

||_γ .

∫

^dt

ˆ ˆ ^ˆ

(8)

Proof. Since

and

by the Tonelli Theorem we have

Thus, D_λ,γ is finite with probability 1. By the following Taylor expansion

where θ θ^*_n− ₀ ≤θ θˆ_n− ₀ →0 with probability 1, Zˆ_n has the form

where Z_n^* is the following process

By straightforward calculations it is easy to show that the process Z_n^* has zero mean function and the same covariance kernel as the process Z and that Z_n^* converges weakly to Z in C(S).

Since this convergence takes place for any compact set S ⊂ , Z_n^* converge weakly do Z in the Frechet space C().

Further in this proof the following convergences will be needed K t t e

t ^t dt

∫

 ^{( , )}_{| |}⁻^λ^γ^{| |} ^{< ∞}

K t t e t ^t dt

α λ

 γ

∫

^{( , )}_{| |}⁻^{| |} ^{< ∞}^,

ED Z t e

t ^t dt

λ γ

λ γ ,

| |

( )| | .

=

∫

 ² ⁻ < ∞

F x( ,θˆ_n)−F x( ,θ₀)= 〈 −θ θˆ_n ₀,∇_θF x( ,θ^*_n) ,〉

Z t k x t d n F x F x k x t d n F x F x

n n n

n

( ) ( , ) ( ( ) ( , ))

( , ) ( ( ) ( , ))

= −

= − +

∫

θ

θ₀ kk x t d n F x F x k x t d n F x F x n

n n

∫

−

= − −〈

( , ) ( ( , ) ( , ))

( , ) ( ( ) ( , )) (

θ θ

θ

0

0 θθ θ θ

θ

n θ n

n

k x t d F x k x t d n F x F x

n

− ∇ 〉

= − −〈

∫

0

1

), ( , ) ( , ) ( , ) ( ( ) ( , ))

*

ll X H t

n l X H t n H t

j n

j

j n

=

∑

〉

+〈 〉 −〈 −

1

0

1

0 0

1

( ), ( , )

( ), ( , ) ( ), ( ,

θ

θ θ θ θθ

θ θ θ θ

n

n n n

j n

j

Z t n H t H t

n l X n

*

* *

)

( ) ( ), ( , ) ( , )

( ) (

〉

= +〈 − − 〉

+〈 −

∑

=

0 0

1

1 θθ θ θ

θ θ θ θ

n

n n n P

H t

Z t n H t H t o

− 〉

= +〈 − − 〉 +〈

0 0

0 0 1

), ( , )

( ) ( ), ( , ) ( , ) ( )

* *

,,H t θ( , 0) ,〉

ˆ ˆ

Z t k x t d n F x F x

n l X H t

n

n n

j n

j

*( ): ( , ) ( ( ) ( )) ( ), ( , )

cos

= − −

=

∫ ∑

 0 =

1

0

1

θ

((tX_j) sin(tX_j) e^{| |}^t te l X^{| |}^t ( _j) | |t e l X^{| |}^t ( _j) .

j

n

(

+ − ⁻ − ⁻ + ⁻

)

∑

= ¹ ²

1

(9)

In order to obtain (10) we calculate

where τ_n1= nmˆ_n and τ_n2= n

(

σˆ_n−1

)

. H_i are bounded and continuous on the set S × Θ₀, where S ⊂  is any compact set and Θ₀ is closure of certain neighborhood of θ₀, the sequences τ_n1 and τ_n2 are tight and H_i(t, θ₀) – H_i(t; θ θ^*_n) converge to 0 with probability 1 for i = 1, 2; thus, in − ₀ ≤θ θˆ_n− ₀ →0 the consequence, we obtain (10).

Convergence (11) can be obtained analogously.

Using the Taylor expansion

where |δ_n – 1| ≤ | σˆ_n – 1| and the Schwarz inequality we estimate

As it was showed in Gürtler i Henze [5] the sequence

( ) (Z ( ) ( ))

| | ,

( ) ( ( ) ( ))

*

| |

*

10 0

11

2

n n

t P

n n

t Z t e t dt Z t Z t

∫

−

−λ →

γ

ee t

e t dt

Z t e

n

t t P

n

t

− −

−

 −

 

 →

∫

λσ γ

λ γ

λσ

| | | |

*

|

| | | |

( ) ( )

0

12 ²

and

||

*

| |

| | ( )

| | .

t dt Z t e t dt

n

t P

γ

λ

−

∫

⁻γ →

2 0

ˆ

( ( ) ( ))

| |

( ( ), ( , ) ( ,

*

| |

*

Z t Z t e t dt

n H t H t

n n

t

n n

∫

−

= 〈 − −

− 2

0 0

λ γ

θ θ θ θ )) ( ), ( , ) )

| | ( ( , )

| |

,

〉 +〈 〉

=

−

∑

=

∫

o H t e

t dt H t

P

t

i j ni nj i

1 ₀ ²

1 2

0

θ

τ τ θ

λ γ

−− −

+

−

∑

=

H t H t H t e

t dt o

i n j j n

t

i j ni P

( , )( ( , ) ( , )

| | (

* *

| |

,

θ θ θ

τ

λ γ 0

1 2

2 1)) ( ( , ) ( , ) ( , )

| | ( )

*

| |

,

H t H t H t e

t dt

o H

i i n j

t

i j P i

∫

∑

−

+

−

=

θ₀ θ θ₀ ^λ_γ

1 2

1

∫

^{( ,}^t^θ⁰⁾^{H t}^j^{( ,}^θ⁰⁾^e_{| |}_t⁻^λ^γ^{| |}^t ^dt^,

ˆ ˆ

e⁻^λσ^ˆⁿ^{| |}^t =e⁻^λ^{| |}^t−λ| |t e⁻^{λ δ}^{| |}^t ⁿ(σˆ_n−1),

Z t e

t dt Z t e t dt Z

n

t

n t

n n

*

| |

*

| |

( ) | | ( )

| |

( ) | |

∫

⁻ ⁻

∫

⁻

−

2 2

13 1

λσ γ

λ γ

λ σ nn

t

n n t

t et dt

Z t e dt e

n

*

| |

* | | /

( ) | |

| | ( )

∫

−

− −

−

( )

2 1

4 1 2

1

λ δ γ

λ λ

λ σ ^{|| |(} ⁾

/

| | .

t n

t₂² ₂¹dt

δ 1 2 γ

−

∫

−



 

 ˆ

ˆ

Z t e dt_n^*( ) ^{| |}^t ^/

∫

 ⁻

(

⁴ ^λ

)

^{1 2}

(10)

is tight. Since σˆ_n → 1 with probability 1, the last integral in (13) converge with probability 1 to 2Γ(3 – 2γ)/λ^3–2γ and in the consequence we obtain (12).

Convergence

can be proved analogously to Henze and Wagner ([7], proof of 2.17, pp. 10-12) By (10) and (11) we have

thus

and in consequence we have

Finally, applying (12), (14), (15) and the Slutsky Lemma we obtain

hence

The covariance kernel of the process Z determine an integral operator on the space L²()

Theorem 1(iii) of Buescu ([1]) guarantees that the kernel of this operator has the representation as absolutely and uniformly convergent series

( ) ( )

| | ( )

| |

*

| | | |

14 Z t e² ²

t dt Z t e t dt

n

t d t

 

∫

⁻^λ^γ ^→

∫

⁻^λ^γ

Z t e

t dt Z t e

t dt

n

t

n

n nt

∫

⁻

∫

⁻



 

 −



( ) | | ( )

| |

| | /

*

| | 2

1 2

2 λσ

γ

λσ γ





 −

 

 →

∫

⁻

1 2

2

1 2

0

/

*

| | /

( ( ) ( ))

| | ,

Z t Z t e

t dt

n n

t P

n

λσ γ

ˆ

ˆ ˆ

ˆ

Z t e

t dt Z t e

t dt

n

t

n

n nt

∫

⁻

∫

⁻



 

 =



( ) | | ( )

| |

| | /

*

| | 2

1 2

2 λσ

γ

λσ γ



 +

1 2

1

/

o_P( ), ˆ

ˆ ˆ

( ) ( )

| | ( )

| | ( ).

| |

*

| |

15 Z t e² ² 1

t dt Z t e

t dt o

n

t

n

t P

n n

∫

^ˆ ⁻^λσ^ˆ^γ ⁼

∫

⁻^λσ^ˆ^γ ⁺

Z t e

t dt Z t e

n

t

n

t

n

n n

∫

^{( )}² ⁻_{| |}^λσ^γ^{| |} ⁼

∫

^{( )}² _{| |}⁻^λσ^γ^{| |} ⁻

∫

^*^{( )}² ⁻^λσσ^γ

λσ γ

λ

n

t

n

t

n t

t dt Z t e

| |

*

| |

*

| |

( ) | | ( )



 



+

∫

⁻ −

∫

⁻

2 2

|| |

( ) | | ( )

| | ,

*

| | | |

t dt Z t e

t dt Z t e t dt

n

t d t

γ

λ γ



 



+

∫

⁻ →

∫

⁻

2 2

ˆ

ˆ ˆ

ˆ

D t e

t dt Z t e

t dt D

n n Zn

t d t

n , ,

| | | |

( ) | | ( )

λ γ γ λσ | |

γ

λ

γ λ

σ

=ˆ¹⁻

∫

ˆ ² ⁻ →

∫

² ⁻ = ,,_γ.

ˆ

K: ( ) ( , ) ( )

( ) ( ).

(| | | |)/

L f K s t f t e /

st^{s t} dt L

2

  →

∫

_ ^α ⁻^λ ⁺^γ ∈ 2 

( ) ( , )

( ) ( ) ( ),

(| | | |)/

16 /

2

2 1

K s t e

st^{s t} _j s t

j j j

α λ

γ η φ φ

− +

=

∑

∞

□

(11)

where η_j are eigenvalues of the operator K ordered nonincreasingly (1 ≥ 2 ≥ ... ≥ 0), and ϕ_j are the corresponding eigenfunctions. Let us define the following stochastic process

where N₁, N₂, is a sequence of independent random variables distributed according to the standard normal low. Since the series (17) is convergent in mean, Y is centered Gaussian process with the covariance function (16), there is the covariance function of the process

Taking into account orthonormality of the eigenfunctions we obtain

where =^L denote equality of probability laws. Hence the limit distribution of statistics D_n,λ,γ is the same as the distribution of η_j _j

j₌₁ N²

∑

∞ ^.

4. Consistency

In order to obtain consistent goodness-of-fit test for the Cauchy distribution the following procedure can be applied: first we estimate the parameters and then we compute the test statistics and compare its value with critical value for fixed signicance level. In [10] it was showed that estimators θˆ_n,α cannot be computed if, and only if, #{k : X_k = m}/n ≥ 2^α–1 for some m. For Cauchy distributed samples the probability of such event is equal to 0. Thus, in that case the hypothesis H₀ should be rejected.

The following theorem guarantees consistency of the test based on the statistic D_n,λ,γ against any non Cauchy alternative. Let us stress that this theorem does not impose any restrictions on the alternative distribution. Theorem 2.3. of Gürtler and Henze [5] can be proved analogously. In this way one can obtain consistency of test considered by Gürtler and Henze [5] against any non Cauchy alternative without assumptions on uniqueness of the median and interquartile range.

Theorem 3. Let X₁, X₂, ... be a sequence of independent, identically distributed random variables with common characteristic function φ, mˆ_n and σˆ_n be any earlier considered estimators. Than

with probability 1.

(17) ( ) ( ) ,

1

Y t _j t N

j j j

=

∑

∞ ^{η φ}

Z t e t

t

( ) .

| |/

/

−λ γ 2

2

D Z t e

t dt

Y t dt t N

t

j j j j

λ γ

η φ

,

| |

( ) | |

( ) ( )

=

= = 

 



∫

∫ ∑

−

=

∞



 

2

1

 2

∫∫

⁼

∑

=

dt ∞ _jN

j η j

1 2,

( ) liminf _{, ,} inf ( / )

( , )

/ | |

|

18 1

0

2

n n m

itm t

nD e t e e

→ ∈

− − −

∫

−

λ γ σ

σ λ

 _Θ _ ϕ σ t ^tt^|dt

| |^γ

(12)

Let us notice that right-handside of (18) is equal to 0 if, and only if, φ is characteristic function of the Cauchy distribution.

Proof. For positive constants T and K we will denote

Using substitution s = t/σˆ_n and applying the Minkowski inequality we estimate as follows

Thus, we obtain

The first integral in the last inequality can be estimated in the following way

1 1

1

2

nD

n e e e

t dt e

n it X m

j

n t

T

T t

ism

j n n

, ,

( )/ | |

| |

λ γ | |

σ λ

⁻ γ

=

−

∫ ∑

⁻

= ⁿⁿ ^j ⁿ

n

n n

n e e e

s ds

e

isX j

n s

T

T n s

ism

1

2 1

=

−

− −

−

∫

σ

∑

⁻ ^σ

σ γ λσ

γ

| | σ

/

/ | |

| |

ⁿⁿ ^j ⁿ

n K

n

n e e e

s ds

e n e

isX j

n s

R

n s

ism

1

2 1

=

− − −

−

∫

_σ_,

∑

⁻ ^σ^{| |} ^σ ^γ_{| |}^γ^λσ^{| |}

^iisX

j

n m s

R

n s

j n

n K

e s e n

s ds

=

− − −

∫ ∑

⁻













1

2 1 1

| |

| | /

( ) | |

,

ϕ σ

σ

γ λσ

γ

2 2

2 1







 −

 



− − − −

∫

R ^e ^{m s} ^{s e} ^s ^e_s ^ds

n s

n n

n K

n

| | | |

| |

( ) | |

,

ϕ _σ σ ^γ ^λσ

σ γ

1 1 2/ 2

.







ˆ

ˆ ˆ

ˆ ˆ ˆ ^ˆ

ˆ

ˆ ˆ

ˆ

ˆ ˆ

ˆ

ˆ ˆ

1 2 1 1 2

nD s s e

s ds

n R n

n s

n K

n , ,

| | /

( ) ( )

| |

λ γ ,

γ λσ

ϕ ϕ σ γ

 σ −

 









∫

⁻ ⁻

ee s e e

s ds

m s s

R

n s

n n

n K

− − − − n

 −

 

 



∫

^{| |} ^{| |} ^{| |} 

/

( ) | |

,

ϕ _σ σ ^γ ^λσ

σ γ

2 1 1 2



2

. ˆ

ˆ

ˆ ˆ ˆ

ˆ

(19)

ϕ ϕ σ

ϕ ϕ

σ

γ λσ

n γ R

n s

s R n

s s e

s ds

s s

n K

n

n K

( ) ( )

| | sup ( ) ( )

,

| |

−

∫

⁻ ⁻

∈

2 1

2 σσ

γ ϕ ϕ

γ

γ σ

n

T K

s R n

s ds

T s s

n

n K 1

0

1

2 1

−

∈

∫

− −

| | sup ( ) ( ) .

min( / , )

,

ˆ

ˆ ˆ

ˆ

R_σ_,_K: [= −T/ , / ] [σT σ∩ −K K, ].