An admissible estimator for the r th power of a bounded scale-parameter in a subclass of the exponential family under entropy loss function
We consider an admissible estimator for the rth power of a scale parameter that is lower or upper bounded in a subclass of the scale-parameter exponential family under entropy loss function. An admissible estimator of a bounded parameter in the family of transformed chi-square distributions is also...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508583913848832 |
|---|---|
| author | Alikhani, S. Mahmoudi, E. Torabi, H. Аліхани, С. Махмуді, Є. Торабі, Х. |
| author_facet | Alikhani, S. Mahmoudi, E. Torabi, H. Аліхани, С. Махмуді, Є. Торабі, Х. |
| author_sort | Alikhani, S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:31:48Z |
| description | We consider an admissible estimator for the rth power of a scale parameter that is lower or upper bounded in a subclass
of the scale-parameter exponential family under entropy loss function. An admissible estimator of a bounded parameter in
the family of transformed chi-square distributions is also given. |
| first_indexed | 2026-03-24T02:27:31Z |
| format | Article |
| fulltext |
UDC 517.23
E. Mahmoudi, H. Torabi, S. Alikhani (Yazd Univ., Iran)
AN ADMISSIBLE ESTIMATOR OF THE rth POWER
OF A BOUNDED SCALE-PARAMETER IN A SUBCLASS
OF THE EXPONENTIAL FAMILY UNDER ENTROPY LOSS FUNCTION
ДОПУСТИМА ОЦIНКА ДЛЯ r-ГО СТЕПЕНЯ ОБМЕЖЕНОГО ПАРАМЕТРА
МАСШТАБУ У ПIДКЛАСI ЕКСПОНЕНЦIАЛЬНОЇ СIМ’Ї
З ЕНТРОПIЙНОЮ ФУНКЦIЄЮ ВТРАТ
We consider an admissible estimator for the rth power of a scale parameter that is lower or upper bounded in a subclass
of the scale-parameter exponential family under entropy loss function. An admissible estimator of a bounded parameter in
the family of transformed chi-square distributions is also given.
Розглянуто допустиму оцiнку для r-го степеня параметра масштабу, обмеженого зверху або знизу у пiдкласi експо-
ненцiальної сiм’ї параметрiв масштабу з ентропiйною функцiєю втрат. Наведено також допустиму оцiнку обмеже-
ного параметра у сiм’ї трансформованих розподiлiв хi-квадрат.
1. Introduction. The first result in the case of truncated parameter space for minimax estimation
of the scale parameter λ and the recioprocal of the scale parameter λ−1, in gamma distribution, was
obtained by Zubrzycki [19] who applied the well-known method of Lehmann (cf. Lehmann [9]).
Using Karlin method (cf. Karlin [7]), Ghosh and Singh [3] proved admissibility of the estimator
(s− 2)X−1 of the gamma parameter λ. They also gave the minimax estimator of λ, but this result
is contained in that of Zubrzycki [19]. Singh [16] showed that
Γ(s− r)
Γ(s− 2r)
X−r is an admissible
estimator of λr under squared error loss, where r is an integer, r <
s
2
. Ghosh and Meeden [4] and
Ralescu and Ralescu [14] have found admissible estimators of λ and λ−1 in the gamma distribution.
Also, Kaluszka [6] obtained an admissible minimax estimator of the parameter λr under the scale-
invariant squared error loss, where r 6= 0 is an integer and λ ∈ (λ0,∞) or λ ∈ (−∞, λ0) with given
constant λ0.
Minimaxity and admissibility results for lower-bounded parameters can be found in Katz [8],
Berry [1] and van Eeden [17, 18]. Using scale-invariant squared-error loss, van Eeden [17] gives an
admissible minimax estimator of the scale-parameter θ of the gamma distribution with known shape
parameter, where θ ∈ [a,∞).
Jafari Jozani et al. [5] extended the results of van Eeden [17]. They obtained an admissible
minimax estimator of a bounded scale parameter in a subclass of the exponential family under scale-
invariant squared error loss. Also, they studied the admissibility and minimaxity in the family of
transformed chi-square distributions due to Rahman and Gupta [13].
Recently, Mahmoudi and Zakerzadeh [10] obtained an admissible estimator of a lower bounded
scale-parameter under squared-log error loss function. Also Mahmoudi [11] studied an admissible
minimax estimator of θr in a subclass of the exponential family with truncated parameter space
under squared-log error loss function.
Assuming the entropy error loss, of the form
c© E. MAHMOUDI, H. TORABI, S. ALIKHANI, 2012
1138 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 8
AN ADMISSIBLE ESTIMATOR OF THE rTH POWER OF A BOUNDED SCALE-PARAMETER . . . 1139
L (δ, θr) =
δ
θr
− ln
δ
θr
− 1. (1.1)
Sanjari Farsipour [15] obtained an admissible estimator of the parameter λr, where λ ∈ (0, λ0)
or λ ∈ (λ0,∞) with given constant λ0 and r 6= 0 in the gamma distribution under entropy loss
function (1.1).
In this paper we consider a subclass of the scale-parameter exponential family. We obtain an
admissible estimator of the rth power of a lower or upper bounded scale-parameter (say θr), using
Karlin’s method, under entropy loss function (1.1). We show that the admissible estimator obtained
by Sanjari Farsipour [15] is a special case of our estimator. In fact, our paper generalizes the results
of Sanjari Farsipour [15]. The rest of the paper is as follow:
A subclass of the scale-parameter exponential family is introduced in Section 2. We give the
admissibility results in Section 3. An admissible estimator of the bounded parameter, for the family of
transformed chi-square distributions, introduced by Rahman and Gupta [13], is presented in Section 4.
2. A subclass of the exponential family. Let X1, X2, . . . , Xn be a random sample of size n from
a distribution with density (1/η) g (x/η), where g is known and η is an unknown scale parameter.
The joint density of X1, X2, . . . , Xn is denoted by f (x; η) =
1
ηn
f
(
x
η
)
. In some cases the above
model reduces to
f(x; θ) = c(x, n)θνe−θT (x), (2.1)
where c(x, n) is a function of x = (x1, . . . , xn)′ and n, θ = η−r (r > 0), ν is a function of n and
T (X) is a complete sufficient statistic for θ with a Γ(ν, θ) distribution.
Jafari Jozani et al. [5] have listed some distributions belonging to this subclass of the scale-
parameter exponential family such as gamma, inverse Gaussian with zero drift, normal, Weibull and
Rayleigh distribution. For example Γ(α, β) with known α belongs to this subclass of distributions
with:
θ = β−1(η = β, r = 1), ν = nα, T (X) =
n∑
i=1
Xi, c(x, n) =
n∏
i=1
xα−1i
Γ (α)
, (2.2)
where the joint density is given by
f (x, β) =
(
n∏
i=1
xα−1i
Γ (α)
)
β−nαe−
∑n
i=1 xi/β, xi > 0, i = 1, 2, . . . , n.
Some properties of the family of distributions (2.1) along an admissible linear estimator of θ = ηr,
under the entropy loss function, can be found in Parsian and Nematollahi [12].
In Section 3 an admissible estimator of θr, where r 6= 0 is constant, is given where θ is restricted
to (0, θ0) or (θ0,∞).
3. Admissible estimator of θr with truncated parameter space. Let us denote by γ(. , .),
Γ (., .), the incomplete gamma functions, i.e.,
γ (x, y) =
y∫
0
tx−1 exp (−t) dt, Γ (x, y) =
∞∫
y
tx−1 exp (−t) dt, x, y > 0.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 8
1140 E. MAHMOUDI, H. TORABI, S. ALIKHANI
We now give an admissible estimator of θr in the scale-parameter exponential family (2.1), where
θ is unknown parameter to satisfy the restrictions θ ∈ (0, θ0) or θ ∈ (θ0,∞) , for some known θ0 > 0
and integer r 6= 0 with r <
ν
2
. Consider the following two lemmas.
Lemma 3.1. We have
lim
b→0
θ0∫
b
θν−ρ−1e−(T (x)+k)θdθ = (T (x) + k)ρ−ν γ (ν − ρ, θ0 (T (x) + k)).
Proof. Using the integration by parts we have
lim
b→0
θ0∫
b
θν−ρ−1e−(T (x)+k)θdθ = lim
b→0
1
(T (x) + k)ν−ρ
θ0(T (x)+k)∫
b(T (x)+k)
tν−ρ−1e−tdt =
=
1
(T (x) + k)ν−ρ
θ0(T (x)+k)∫
0
tν−ρ−1e−tdt =
= (T (x) + k)ρ−ν γ (ν − ρ, θ0 (T (x) + k)).
Thus, the proof is completed.
Lemma 3.2. We have
lim
b→∞
b∫
θ0
θν−ρ−1e−(T (x)+k)θdθ = (T (x) + k)ρ−ν Γ (ν − ρ, θ0 (T (x) + k)).
Proof. Proof of this lemma is similar to the proof of Lemma 3.1.
Remark 3.1. Setting ρ = r, 2r in Lemma 3.1 and Lemma 3.2 gives the following results:
(i) lim
b→0
∫ θ0
b
θν−r−1e−(T (x)+k)θdθ = (T (x) + k)r−ν γ (ν − r, θ0 (T (x) + k)) ,
(ii) lim
b→0
∫ θ0
b
θν−2r−1e−(T (x)+k)θdθ = (T (x) + k)2r−ν γ (ν − 2r, θ0 (T (x) + k)) ,
(iii) lim
b→∞
∫ b
θ0
θν−r−1e−(T (x)+k)θdθ = (T (x) + k)r−ν Γ (ν − r, θ0 (T (x) + k)) ,
(iv) lim
b→∞
∫ b
θ0
θν−2r−1e−(T (x)+k)θdθ = (T (x) + k)2r−ν Γ (ν − 2r, θ0 (T (x) + k)) .
Theorem 3.1. The estimator
δ̂ (X) =
γ (ν − r, θ0 (k + T (X)))
γ (ν − 2r, θ0 (k + T (X)))
(T (X) + k)−r , 0 < θ < θ0,
Γ (ν − r, θ0 (k + T (X)))
Γ (ν − 2r, θ0 (k + T (X)))
(T (X) + k)−r , θ0 < θ <∞,
(3.1)
in which k ≥ 0 is an arbitrary constant, is admissible for θr under the entropy loss function (1.1),
where θ has the improper prior density function
π (θ) = θ−r−1 exp (−kθ) , θ ∈ (0, θ0) or θ ∈ (θ0,∞) . (3.2)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 8
AN ADMISSIBLE ESTIMATOR OF THE rTH POWER OF A BOUNDED SCALE-PARAMETER . . . 1141
Proof. In the proof of Theorem 2.1 of Sanjari Farsipour [15], if we replace the values s, λ, x and
xs−1
Γ(s)
with ν, θ, T (x) and c(x, n) respectively, then using Remark 3.1, the proof is still established.
Thus, the proof of Theorem 3.1 can be derived parallel to the proof of Theorem 2.1 of Sanjari
Farsipour [15]. Here we only give the short proof of the first case of this theorem, where 0 < θ < θ0.
Suppose that there exist an estimator δ̃ which is better than δ̂. This implies that the inequality
∞∫
0
(
δ̃
θr
− ln
δ̃
θr
− 1
)
f (x, θ) dx ≤
∞∫
0
(
δ̂
θr
− ln
δ̂
θr
− 1
)
f (x, θ) dx
holds for all θ ∈ (0, θ0) or θ ∈ (θ0,∞) with strict inequality for some θ. Using above inequality, we
get
∞∫
0
(
δ̃
δ̂
− ln
δ̃
δ̂
− 1
)
f (x, θ) dx ≤
∞∫
0
(
δ̃
δ̂
− δ̃
θr
+
δ̂
θr
− 1
)
f (x, θ) dx. (3.3)
Integration both sides of (3.3) with respect to the improper prior density function (3.2) gives
θ0∫
b
∞∫
0
(
δ̃
δ̂
− ln
δ̃
δ̂
− 1
)
f (x, θ)π (θ) dxdθ≤
≤
θ0∫
b
∞∫
0
(
δ̃
δ̂
− δ̃
θr
+
δ̂
θr
− 1
)
f (x, θ)π (θ) dxdθ. (3.4)
By interchanging the order of integration in the right-hand side of (3.4) and substituting δ̂ from (3.1)
into (3.4) we have
θ0∫
b
∞∫
0
(
δ̃
δ̂
− δ̃
θr
+
δ̂
θr
− 1
)
f (x, θ)π (θ) dxdθ =
=
∞∫
0
δ̃γ (ν − 2r, θ0 (T (x) + k))
γ (ν − r, θ0 (T (x) + k))
(T (x) + k)r c (x, n)
θ0∫
b
θν−r−1e−(T (x)+k)θdθdx−
−
∞∫
0
δ̃c (x, n)
θ0∫
b
θν−2r−1e−(T (x)+k)θdθdx+
+
∞∫
0
γ (ν − r, θ0 (T (x) + k))
γ (ν − 2r, θ0 (T (x) + k))
(T (x) + k)−r c (x, n)
θ0∫
b
θν−2r−1e−(T (x)+k)θdθdx−
−
∞∫
0
c (x, n)
θ0∫
b
θν−r−1e−(T (x)+k)θdθdx. (3.5)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 8
1142 E. MAHMOUDI, H. TORABI, S. ALIKHANI
Using Lemma 3.1 with ρ = r for θ ∈ (0, θ0) , (3.5) tends to zero and we have
∞∫
0
(
δ̃
δ̂
− ln
δ̃
δ̂
− 1
)
f (x, θ) dx = 0,
i.e., δ̃ = δ̂ a.e., and the admissibility of δ̂ (X), in the first case is completed.
The proof of the second case, where θ ∈ (θ0,∞), is quite similar. So according to the first case,∫ ∞
0
(
δ̃
δ̂
− ln
δ̃
δ̂
− 1
)
f (x, θ) dx = 0, i.e., δ̃ = δ̂ a.e., and the admissibility of δ̂ (X), in this case, is
completed.
Remark 3.2. For the untruncated case θ > 0, it can be easily show that
Γ (ν − r)
Γ (ν − 2r)
(T (X)+k)−r
is an admissible estimator of θr under the entropy loss function (1.1) and the improper prior π (θ) =
= θ−r−1 exp (−kθ) , θ > 0.
The following remark shows that the result of Sanjari Farsipour [15] is a special case of our
result.
Remark 3.3. In the special case where the random variable X has Γ(s, λ) distribution, choosing
θ = λ, ν = s, T (X) = X, c(x, n) =
xs−1
Γ (s)
,
gives the admissible estimator
δ̂ (X) =
γ (s− r, λ0 (k +X))
γ (s− 2r, λ0 (k +X))
(X + k)−r , 0 < λ < λ0,
Γ (s− r, λ0 (k +X))
Γ (s− 2r, λ0 (k +X))
(X + k)−r , λ0 < λ <∞,
which is an admissible estimator of θr, r <
s
2
, obtained by Sanjari Farsipour [15].
According to Theorem 3.1 and Remark 3.2 we have the following examples.
Example 3.1. Consider a sample of size n from gamma distribution with pdf
f (x, λ) =
λα
Γ (α)
xα−1e−λx, x > 0,
where α is known and λ ∈ (λ0,∞) is unknown.
(i) An admissible estimator of the scale parameter λr, r <
nα
2
, under the loss function (1.1) and
improper prior (3.2), is given by
δ̂ (X) =
Γ
(
nα− r, λ0
(
k + nX̄n
))
Γ
(
nα− 2r, λ0
(
k + nX̄n
)) (nX̄n + k
)−r
.
(ii) An admissible estimator of the scale parameter λr, r <
nα
2
, for the untruncated case λ > 0,
is of the form
δ̂ (X) =
Γ (nα− r)
Γ (nα− 2r)
(
nX̄n + k
)−r
.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 8
AN ADMISSIBLE ESTIMATOR OF THE rTH POWER OF A BOUNDED SCALE-PARAMETER . . . 1143
Example 3.2. Suppose that X1, X2, . . . , Xn is a sample of size n from inverse Gaussian with
zero drift, having the pdf
f (x, λ) =
(
2πx3
)−1/2
λ1/2e−λ/2x, x > 0.
(i) An admissible estimator of λr, r <
n
4
and λ ∈ (λ0,∞), under the loss (1.1) is
δ̂ (X) =
Γ
(
n
2
− r, λ0
(
k +
1
2
∑n
i=1
1
Xi
))
Γ
(
n
2
− 2r, λ0
(
k +
1
2
∑n
i=1
1
Xi
)) (1
2
n∑
i=1
1
Xi
+ k
)−r
.
(ii) For the untruncated case λ > 0, this estimator is replaced by
δ̂ (X) =
Γ (n/2− r)
Γ (n/2− 2r)
(
1
2
n∑
i=1
1
Xi
+ k
)−r
.
4. Admissibility results in the family of transformed chi-square distributions. In this section,
we use a subfamily of the one-parameter exponential family of distributions called the transformed
chi-square family of distributions, introduced by Rahman and Gupta [13], to derive an admissible
estimator of the rth power of the unknown bounded parameter.
Let X = (X1, . . . , Xn)′ be a random vector whose joint density function belongs to the one-
parameter exponential family, i.e.,
f(x, η) = ea(x)b(η)+c(η)+h(x). (4.1)
Rahman and Gupta [13] proved the following theorem for this family of distributions.
Theorem 4.1. In a one-parameter exponential family (4.1), the function −2a(X)b(η) has a
Γ(j/2, 2)-distribution if and only if
2c′(η)b(η)
b′(η)
= j, (4.2)
where j is positive and free from η. In the case that j is an integer, −2a(X)b(η) follows a central
chi-square distribution with j degrees of freedom.
The one-parameter exponential family (4.1) satisfying the condition (4.2) is called the family of
transformed chi-square distributions, provided j is a positive integer. Note that if a(x) > 0 then b(η)
must be a negative. From condition (4.2) we get
c(η) =
j
2
ln |b(η)|+ k1. (4.3)
Let θ = −b (η) > 0, then (4.3) reduces to ec(η) = [−b (η)]j/2 ek1 = θj/2ek1 . So the family of
distributions (4.1) can be written in the form
f(x, η) = e−a(x)[−b(η)]+c(η)+h(x) = c (x,m) θj/2e−θa(x),
where c (x,m) = eh(x)+k1 . Also note that 2θa(X) ∼ Γ (j/2, 2) or a(X) ∼ Γ (j/2, θ). Therefore,
if condition (4.2) holds then the one-parameter exponential family (4.1) is in the form of the scale-
parameter exponential family (2.1) with ν =
j
2
, T (X) = a(X) and θ = −b (η).
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 8
1144 E. MAHMOUDI, H. TORABI, S. ALIKHANI
Jafari Jozani et al. [5] listed some distributions belong to the family of transformed chi-square dis-
tributions in Table 1. This table contains normal, lognormal, exponential, gamma, Rayleigh, Weibull,
Maxwell and inverse Gaussian distributions.
Pareto, Burr X, Burr XII, Laplace, generalized Laplace, generalized gamma and etc. are other
distributions belonging to this family of distributions which did not list by Jafari Jozani et al. [5].
Some of these distributions with associated a(x), θ = −b(η) and j are:
(i) Pareto distribution: Pareto(α, β) with α known,
a(x) =
n∑
i=1
ln
xi
α
, b(β) = −β, c(β) = n lnβ, θ = β, j = 2n,
and the joint density is given by
f (x, α, β) =
βn∏n
i=1
xi
exp
(
−β
n∑
i=1
ln(xi/α)
)
, xi > α, i = 1, 2, . . . , n,
where −2b(β)a(X) = 2β
∑n
i=1
ln
Xi
α
∼ Γ(n, 2).
(ii) Generalized Laplace distribution: GL(α, β) with α known,
a(x) =
n∑
i=1
|xi|α, b(β) = −β−α, c(β) = −n lnβ, θ = β−α, j = 2n/α,
and the joint density is given by
f (x, α, β) =
αn
(2β)nΓn(1/α)
exp
(
− 1
βα
n∑
i=1
|xi|α
)
, xi ∈ R, i = 1, 2, . . . , n,
where −2b(β)a(X) = 2
∑n
i=1
|Xi/β|α ∼ Γ(n/α, 2).
Note that in a special case, GL(2,
√
2σ) ∼ N(0, σ2) and GL(1, σ) has Laplace distribution.
(iii) Generalized gamma distribution: GG(p, α, λ), in which p and α are known and pα > 0,
a(x) =
n∑
i=1
xαi , b(λ) = −λ, c(λ) =
p
α
lnλ, θ = λ, j = 2np/α,
and the joint density is given by
f (x, p, α, λ) =
|α|n
∏n
i=1
xp−1i
Γn(p/α)
λnp/α exp
(
−λ
n∑
i=1
xαi
)
, xi > 0, i = 1, 2, . . . , n,
where −2b(λ)a(X) = 2λ
∑n
i=1
Xα
i ∼ Γ(np/α, 2).
Note that this distribution contains Maxwell, Weibull, Rayleigh and others in a special case.
For the above distributions which belong to the family of transformed chi-square distributions
and have been applied by Sanjari Farsipour [15] in Section 3, the following theorem, which is the
extended version of Theorem 3.1, gives an admissible estimator of parameter θr, in which r 6= 0 is
an integer and θ ∈ (θ0,∞) or (0, θ0) , under entropy loss function (1.1).
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 8
AN ADMISSIBLE ESTIMATOR OF THE rTH POWER OF A BOUNDED SCALE-PARAMETER . . . 1145
Theorem 4.2. LetX = (X1, . . . , Xn)′ with joint density function (4.1), satisfies condition (4.2).
An admissible estimator of θr, r <
j
4
, n the truncated parameter space θ ∈ (θ0,∞) or θ ∈ (0, θ0),
under entropy loss function (1.1) and the improper prior (3.2), is given by
δ̂ (X) =
γ (j/2− r, θ0 (k + a (X)))
γ (j/2− 2r, θ0 (k + a (X)))
(a (X) + k)−r , 0 < θ < θ0,
Γ (j/2− r, θ0 (k + a (X)))
Γ (j/2− 2r, θ0 (k + a (X)))
(a (X) + k)−r , θ0 < θ <∞.
(4.4)
Proof. The proof is quite similar to the proof of Theorem 3.1.
Example 4.1. With a random sample of size n from N(0, σ2) distribution, the joint density is
of the form (4.1), with
a(x) =
1
2
n∑
i=1
x2i , b(σ2) = − 1
σ2
, c(σ2) = −n
2
lnσ2, θ = σ−2, j = n.
and
−2b(σ2)a(X) =
1
σ2
m∑
i=1
X2
i ∼ Γ
(n
2
, 2
)
.
So, an admissible estimator of σ−2r, r <
n
4
, is given by
δ̂ (X) =
γ
(n
2
− r, σ20
(
k +
n
2
X̄2
))
γ
(n
2
− 2r, σ20
(
k +
m
2
X̄2
)) (n
2
X̄2 + k
)−r
, σ20 < σ2 <∞,
Γ
(n
2
− r, σ20
(
k +
n
2
X̄2
))
Γ
(n
2
− 2r, σ20
(
k +
n
2
X̄2
)) (n
2
X̄2 + k
)−r
, 0 < σ2 < σ20.
By choosing r = −1 an admissible estimator of σ2 for the untrancated case 0 < σ2 < ∞, under
entropy loss function (1.1) and the improper prior (3.2), has the form
δ̂ (X) =
Γ(n/2 + 1)
Γ(n/2 + 2)
(
1
2
n∑
i=1
X2
i + k
)
.
Putting k = 0 gives an admissible estimator
δ̂ (X) =
Γ(n/2 + 1)
Γ(n/2 + 2)
(
1
2
n∑
i=1
X2
i
)
,
for σ2 under the noninformative improper prior π(σ2) = 1, σ2 > 0.
Example 4.2. Suppose thatX1, X2, . . . , Xn is a random sample of size n from BurrXII(c, λ)-
distribution with known c and unknown λ. The joint density function is
f(x, c, λ) =
cnλn
(∏n
i=1
xi
)c−1
∏n
i=1
(1 + xci )
n∏
i=1
(1 + xci )
−λ,
which is of form (4.1) with
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 8
1146 E. MAHMOUDI, H. TORABI, S. ALIKHANI
a(x) =
n∑
i=1
ln(1 + xci ), b(λ) = −λ, c(λ) = n lnλ, θ = λ, j = 2n.
and
−2b(λ)a(X) = 2λ
n∑
i=1
ln(1 +Xc
i ) ∼ Γ (n, 2).
Therefore, an admissible estimator of λr, r <
n
2
, is given by
δ̂ (X) =
γ
(
n− r, λ0
(
k +
∑n
i=1
ln(1 +Xc
i )
))
γ
(
n− 2r, λ0
(
k +
∑n
i=1
ln(1 +Xc
i )
)) (∑n
i=1
ln(1 +Xc
i ) + k
)−r
,
λ0 < λ <∞,
Γ
(
n− r, λ0
(
k +
∑n
i=1
ln(1 +Xc
i )
))
Γ
(
n− 2r, λ0
(
k +
∑n
i=1
ln(1 +Xc
i )
)) (∑n
i=1
ln(1 +Xc
i ) + k
)−r
, 0 < λ < λ0.
For the untruncated case 0 < λ < ∞, an admissible estimator of λr under the entropy loss func-
tion (1.1) and the noninformative improper prior π(λ) = 1, λ > 0, is given by
δ̂ (X) =
Γ (n− r)
Γ (n− 2r)
(
n∑
i=1
ln(1 +Xc
i )
)−r
.
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Received 01.02.11,
after revision — 29.06.12
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 8
|
| id | umjimathkievua-article-2647 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:27:31Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/38/3afe2072f96b9ae88f4035bbef77d838.pdf |
| spelling | umjimathkievua-article-26472020-03-18T19:31:48Z An admissible estimator for the r th power of a bounded scale-parameter in a subclass of the exponential family under entropy loss function Допустима оцiнка для r-го степеня обмеженого параметра масштабу у пiдкласi експоненцiальної сiм’ї з ентропiйною функцiєю втрат Alikhani, S. Mahmoudi, E. Torabi, H. Аліхани, С. Махмуді, Є. Торабі, Х. We consider an admissible estimator for the rth power of a scale parameter that is lower or upper bounded in a subclass of the scale-parameter exponential family under entropy loss function. An admissible estimator of a bounded parameter in the family of transformed chi-square distributions is also given. Розглянуто допустиму оцiнку для r-го степеня параметра масштабу, обмеженого зверху або знизу у пiдкласi експоненцiальної сiм’ї параметрiв масштабу з ентропiйною функцiєю втрат. Наведено також допустиму оцiнку обмеженого параметра у сiм’ї трансформованих розподiлiв хi-квадрат. Institute of Mathematics, NAS of Ukraine 2012-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2647 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 8 (2012); 1138-1147 Український математичний журнал; Том 64 № 8 (2012); 1138-1147 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2647/2053 https://umj.imath.kiev.ua/index.php/umj/article/view/2647/2054 Copyright (c) 2012 Alikhani S.; Mahmoudi E.; Torabi H. |
| spellingShingle | Alikhani, S. Mahmoudi, E. Torabi, H. Аліхани, С. Махмуді, Є. Торабі, Х. An admissible estimator for the r th power of a bounded scale-parameter in a subclass of the exponential family under entropy loss function |
| title | An admissible estimator for the r th power of a bounded scale-parameter in a subclass of the exponential family under entropy loss function |
| title_alt | Допустима оцiнка для r-го степеня обмеженого параметра масштабу у пiдкласi експоненцiальної сiм’ї з ентропiйною функцiєю втрат |
| title_full | An admissible estimator for the r th power of a bounded scale-parameter in a subclass of the exponential family under entropy loss function |
| title_fullStr | An admissible estimator for the r th power of a bounded scale-parameter in a subclass of the exponential family under entropy loss function |
| title_full_unstemmed | An admissible estimator for the r th power of a bounded scale-parameter in a subclass of the exponential family under entropy loss function |
| title_short | An admissible estimator for the r th power of a bounded scale-parameter in a subclass of the exponential family under entropy loss function |
| title_sort | admissible estimator for the r th power of a bounded scale-parameter in a subclass of the exponential family under entropy loss function |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2647 |
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