Moment-type estimators for a weighted exponential family
Roberto Vila Helton Saulo

	Checkpoint and Restart: An Energy Consumption Characterization in Clusters
    Marina Morán1 Javier Balladini1 Dolores Rexachs2 Emilio Luque2
    September 9, 2024
==========================================================================

§ ABSTRACT 
In this paper, we propose and study closed-form moment type estimators for a weighted exponential family. We also develop a bias-reduced version of these proposed closed-form estimators using bootstrap techniques. The estimators are evaluated using Monte Carlo simulation. This shows favourable results for the proposed bootstrap bias-reduced estimators.
	
	Keywords. Weighted exponential family · Moment method · Monte Carlo simulation ·  software.
	

	Mathematics Subject Classification (2010). MSC 60E05 · MSC 62Exx · MSC 62Fxx.
	

	
	

§ INTRODUCTION

	
	






In this work we provide closed-form estimators for the parameters of probability distributions that belongs to the following weighted exponential family <cit.>:








    f(x;ψ)
    =
    (μσ)^μ+1 (σ+δ_ab)Γ(μ+1) 
    [1+δ_abT(x)] | T'(x)| T(x) exp{-μσ T(x)+μlog(T(x))},
       
    x∈ (0,∞),

where ψ=(μ,σ), μ,σ>0, T:(0,∞)→ (0,∞) is a real strictly monotone twice differentiable function, δ_ab is the Kronecker delta function and T'(x) denotes the derivative of T(x) with respect to x.


The probability function f(x;ψ) in (<ref>) can be interpreted as a mixture of two distributions that belongs to the exponential family, that is,

    f(x;ψ)   =
    	σσ+δ_ab 
    	f_1(x)
    	+
    	δ_abσ+δ_ab 
    	f_2(x),
    	   x∈ (0,∞),  μ, σ>0,

where

    f_j(x)
    	=
    	(μσ)^μ+j-1Γ(μ+j-1) | T'(x)| T(x) exp{-μσ T(x)+(μ+j-1)log(T(x))},
    	   
    	x∈ (0,∞),    j=1,2.

Densities of form f_j(x), j=1,2, have appeared in <cit.> and <cit.>.

If X has density in (<ref>), from (<ref>) and (<ref>) it is simple to show that the random variable X defined as

    X≡(1-B)T^-1(Z_1)+BT^-1(Z_2),

has density in (<ref>), where  T^-1 denotes the inverse function of T, B∈{0,1} is a Bernoulli random variable with success parameter δ_ab/(σ+δ_ab), independent of Z_1 and Z_2, such that Z_j∼ Gamma(μ+j-1,1/(μσ)), j=1,2, that is, the density function of Z_j is given by

    f_Z_j(z)
    	=
    	1Γ(μ+j-1)[1/(μσ)]^μ+j-1  z^(μ+j-1)-1exp{-z[1/(μσ)]},    z>0, j=1,2.



Table <ref> <cit.> presents some examples of generators T(x) for use in (<ref>) with a=b.



For  a≠ b,
Table <ref> <cit.> provides some examples of generators T(x) for use in (<ref>).






There are a number of different methodological proposals in the literature for obtaining close-form estimators; see, for example, the moment-based type <cit.> and the score-adjusted approaches <cit.>. Closed-form estimators based on the likelihood function have also been suggested. A type of likelihood-based estimator is obtained by considering the likelihood equations of the generalized distribution obtained by a power transformation and taking the baseline distribution as a special case; see for example, <cit.>, <cit.>, <cit.>, <cit.> and <cit.>. For other proposals for likelihood-based estimators, the reader is referred to <cit.> and <cit.>.


This paper develops moment-based closed-form estimators for the parameters of probability distributions of the weighted exponential family (<ref>) in the special case where T(x)=x^-s, x>0 and s≠ 0. The main motivation for choosing this type of generator is that it provides closed-form expressions for moments (of functions) of X in (<ref>), which allows finding moment-based estimators for the corresponding parameters. Note that this type of generator includes the T(x) generators of Nakagami, Maxwell-Boltzmann, Rayleigh, gamma, inverse gamma, δ-gamma, Weibull, inverse Weibull, generalized gamma, generalized inverse gamma, chi-squared, scaled inverse chi-squared, weighted Lindley, weighted inverse Lindley, weighted Nakagami and weighted inverse Nakagami; see Tables <ref> and <ref>.


The rest of the paper is structured as follows. In section <ref> we briefly present some preliminary results. In Sections <ref> and <ref>, we describe the newly proposed moment-based estimators and some asymptotic results respectively. In Section <ref>, we perform a Monte Carlo simulation study to assess the
performance of a bootstrap bias-reduced version of the proposed estimators.












































§ PRELIMINARY RESULTS

In this section, closed-form expressions for moments (of functions) of X in (<ref>) are provided.




 §.§ Moments




If X has density in (<ref>), then, for any measurable function g:(0,∞)→ℝ, we have

    𝔼[g(X)]
    	=
    		σσ+δ_ab 𝔼
    	[
    	g(T^-1(Z_1))
    	]
    	+
    	δ_abσ+δ_ab 𝔼
    	[
    	g(T^-1(Z_2))
    	].



	Taking into account the characterization (<ref>) of X  and by using the independence of B∼ Bernoulli(δ_ab/(σ+δ_ab)) with Z_1 and Z_2, we obtain

    𝔼[g(X)]
    	   =
    	𝔼
    	[
    	g(
    	(1-B)T^-1(Z_1)+BT^-1(Z_2)
    	)
    	]
    	
       =
    	𝔼(1-B)
    	𝔼
    	[
    	g(T^-1(Z_1))
    	]
    	+
    	𝔼(B)
    	𝔼
    	[
    	g(T^-1(Z_2))
    	]
    	
       =
    	σσ+δ_ab 𝔼
    	[
    	g(T^-1(Z_1))
    	]
    	+
    	δ_abσ+δ_ab 𝔼
    	[
    	g(T^-1(Z_2))
    	].

This completes the proof.




	If  T(x)=x^-s, x>0 and s≠ 0, then the real moments of X are  given by
	
    𝔼(X^q)
    	    
    		=
    
    
    
    
    
    
    
    
    
    
    
    
    
    		(μσ)^q s Γ(μ-q s)Γ(μ)[
    		σσ+δ_ab
    		+
    		δ_abσ+δ_ab (μ-q s)μ],
    		  where μ-q s>0.

	









By using Proposition <ref>, with g(x)=x^q and T(x)=x^-s, x>0 and s≠ 0, we have	

    𝔼(X^q)
    	=
    	σσ+δ_ab 𝔼
    	(
    	Z_1^-q/s
    	)
    	+
    	δ_abσ+δ_ab 𝔼
    	(
    	Z_2^-q/s
    	).

As Z_j∼ Gamma(μ+j-1,1/(μσ)), j=1,2 and

    𝔼(Z^ν)=θ^ν Γ(k+ν)Γ(k),
       
    Z∼ Gamma(k,θ),  ν>-k,

from (<ref>), the proof follows.




		If  T(x)=x^-s, x>0 and s≠ 0, then
	
    𝔼[X^-plog(X)]
    	=
    	-1 s Γ(μ+p s) (μσ)^p/sΓ(μ){σσ+δ_ab[
    ψ^(0)(μ+p s)-log(μσ)
    ]
    	
    
    +
    δ_abσ+δ_ab μ+p sμ[
    ψ^(0)(μ+p s+1)-log(μσ)
    ]
    },
      where μ+p s>0.



By using Proposition <ref>, with g(x)=log(x)/x^p and T(x)=x^-s, x>0 and s≠ 0, we have	

    𝔼[X^-plog(X)]
    	=
    	-1 s{σσ+δ_ab 𝔼[Z_1^p/slog(Z_1)]
    	+
    	δ_abσ+δ_ab 𝔼[Z_2^p/slog(Z_2)]
    	}.

As Z_j∼ Gamma(μ+j-1,1/(μσ)), j=1,2 and

    𝔼[Z^νlog(Z)]
    =
    Γ(k+ν)θ^-νΓ(k) 
    [ψ^(0)(k+ν)-log(1/θ)],
       
    Z∼ Gamma(k,θ),  
    k>-ν,

from (<ref>), the proof follows.




	If  T(x)=x^-s, x>0 and s≠ 0, then
	
    𝔼[X^-slog(X^-s) 1+δ_abX^-s]
    		=
    		1σ+δ_ab 
    		[ψ^(0)(μ+1)-log(μσ)].
		
	
	
By using the definition (<ref>) with T(x)=x^-s, x>0 and s≠ 0, and by making the change of variable y=x^-s, note that
	
    𝔼[X^-slog(X^-s) 1+δ_abX^-s]
    	   =
    	s(μσ)^μ+1 (σ+δ_ab)Γ(μ+1)∫_0^∞log(x^-s) 
    	(x^-s)^μ+1 s+1exp{-μσ x^-s} dx
    	
       =
    	(μσ)^μ+1 (σ+δ_ab)Γ(μ+1)∫_0^∞log(y) y^μexp{-μσ y} dy
    	
       =
    	1σ+δ_ab 𝔼[log(Y)],

	where Y∼ Gamma(μ+1,1/(μσ)).
	By employing formula in (<ref>) with ν=0, k=μ+1 and θ=1/(μσ), in the last equality,
	the proof readily follows.
	





 §.§ Main formulas

By using Proposition <ref> with q=-s, we have
	
    𝔼(X^-s)
    	=
    1σ(
    1
    +
    δ_abσ+δ_ab 1μ).


By using Proposition <ref> with p=0, we have

    𝔼[log(X)]
    	   =
    	-1 s{σσ+δ_ab  
    	[
    	ψ^(0)(μ)-log(μσ)
    	]
    	+
    	δ_abσ+δ_ab  
    	[
    	ψ^(0)(μ+1)-log(μσ)
    	]
    	}
       =
    	-1 s[
    	ψ^(0)(μ+1)-log(μσ)
    	-
    	1μ(
    	σσ+δ_ab)
    	],

where in the last line the identity ψ^(0)(z+1)=ψ^(0)(z)+1/z has been used.


By using Proposition <ref> with p=s, we have

    𝔼[X^-slog(X)]
       =
    -1 s 1σ{σσ+δ_ab 
    [
    ψ^(0)(μ+1)-log(μσ)
    ]
    +
    δ_abσ+δ_ab μ+1μ 
    [
    ψ^(0)(μ+2)-log(μσ)
    ]
    }
       =
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    1σ{
    -1 s[
    ψ^(0)(μ+1)-log(μσ)
    	-
    1μ(
    σσ+δ_ab)
    ]
    
       
    -
    1 sμ{δ_abσ+δ_ab  
    [
    ψ^(0)(μ+1)-log(μσ)
    ]
    +
    1
    }},

where in the last equality we have again used the identity ψ^(0)(z+1)=ψ^(0)(z)+1/z. By (<ref>) and Proposition <ref> note that 𝔼[X^-slog(X)] can be written as

    𝔼[X^-slog(X)]
    =
    1σ{𝔼[log(X)]
    -
    1 sμ{δ_ab𝔼[X^-slog(X^-s) 1+δ_abX^-s]
    +
    1
    }},

from which we can express μ as follows:

    μ
    	=
    	δ_ab𝔼[h_1(X)]
    		+
    		1
    		σ𝔼[h_2(X)]
    			-
    			𝔼[h_3(X)]
    			,

where we have adopted the following notations:

    h_1(x)≡x^-slog(x^-s) 1+δ_abx^-s,
    	  
    	h_2(x)≡x^-slog(x^-s),
      
    	h_3(x)≡log(x^-s).


Plugging (<ref>) into (<ref>) and solving for σ gives

    σ   =
    	
    	1
    	-
    	δ_ab𝔼[h_4(X)]
    	+
    	δ_ab𝔼[h_2(X)]δ_ab𝔼[h_1(X)]
    		+
    		1
    		2𝔼[h_4(X)]
    	
       +
    			√({	1
    			-
    			δ_ab𝔼[h_4(X)]
    			+
    			δ_ab𝔼[h_2(X)]δ_ab𝔼[h_1(X)]
    				+
    				1}^2
    				+
    			4𝔼[h_4(X)]
    			{δ_ab-δ_ab𝔼[h_3(X)]δ_ab𝔼[h_1(X)]
    			+
    			1 })
    			2𝔼[h_4(X)]
    	,

where

    h_4(x)≡ T(x)= x^-s.






§ CLOSED-FORM ESTIMATORS


Let {X_i : i = 1,… , n} be a univariate random sample of size n from  X having density in (<ref>). 

By using the method of moments in (<ref>), the corresponding sample moment to obtain the estimator of σ is

    σ
    	=
    	
    		1
    		-
    		δ_abX_4
    		+
    		δ_abX_2δ_abX_1
    			+
    			1
    		+
    		√({	1
    			-
    			δ_abX_4
    			+
    			δ_abX_2δ_abX_1
    				+
    				1}^2
    			+
    			4X_4
    			{δ_ab-δ_abX_3δ_abX_1
    				+
    				1 })
    		2X_4
    	,

where we have defined

    X≡[ X_1; X_2; X_3; X_4 ]
    	=
    	1 n∑_i=1^n[ h_1(X_i); h_2(X_i); h_3(X_i); h_4(X_i) ],

with h_1,h_2,h_3 and h_4 being as in (<ref>) and (<ref>), respectively.

Plugging (<ref>) in (<ref>) and using the method of moments, the sample moment to obtain the estimator of μ is

    μ
    	=
    	δ_abX_1
    		+
    		1
    	σX_2
    		-
    		X_3
    	.





 §.§ Case a≠ b

In this case, from (<ref>) and (<ref>), we obtain the new estimators for σ and μ as

    σ
    	=
    	
    1
    	X_4

and 

    μ
    	=
    	
    		1
    	σX_2
    		-
    		X_3
    	.

respectively, where X_2, X_3 and X_4 are as given in (<ref>). Note that, in this case, σ coincides with the maximum likelihood (ML) estimator <cit.>. 




 §.§ Case a= b


In this case, from (<ref>) and (<ref>), we obtain the new estimators for σ and μ as

    σ
    	=
    	
    		1
    		-
    		X_4
    		+
    		X_2X_1
    			+
    			1
    		+
    		√({	1
    			-
    			X_4
    			+
    			X_2X_1
    				+
    				1}^2
    			+
    			4X_4
    			{1-X_3X_1
    				+
    				1 })
    		2X_4

and

    μ
    	=
    	X_1
    		+
    		1
    	σX_2
    		-
    		X_3
    	,

respectively, where X_1 (with a=b), X_2, X_3 and X_4 are as given in (<ref>).




It is simple to observe that the weighted probability distributions in (<ref>) belongs to the exponential family with vector of sufficient statistics given by (T(x),log(T(x)))^⊤. Furthermore, for exponential families with sufficient statistics  (T(x),log(T(x)))^⊤ it is well-known that the moment-type  estimators are in fact the maximum likelihood estimators <cit.>. The estimators μ and σ (in (<ref>) and (<ref>) for case a≠ b, and (<ref>) and (<ref>) for case a= b) were derived using the moment-type method (modification of the method of moments) with vector of statistics

    (T(x),log(T(x)),T(x)log(T(x)),T(x)log(T(x)) 1+δ_abT(x))^⊤,
      for 
    T(x)=x^-s,   x>0  and  s≠ 0.

By using the maximum likelihood equations corresponding to an extension of the weighted probability model in (<ref>), simple closed-forms for the estimators of μ and σ, denoted by μ_∙ and σ_∙, respectively, were derived in reference <cit.> (case a≠ b) and <cit.> (case a= b). Note that these estimators are not, in fact, the maximum likelihood estimators corresponding to the probability distribution in (<ref>), so we cannot say with certainty that the estimators provided by the moment-type method, μ and σ, are the same as μ_∙ and σ_∙ in the special case T(x)=x^-s, x>0 and s≠ 0. In other words, we cannot apply the results obtained in <cit.>. However, to our surprise, through a simple but laborious calculation and some  simulations, we found that estimators μ and σ (in (<ref>) and (<ref>) for case a≠ b, and (<ref>) and (<ref>) for case a= b) coincide with the estimators μ_∙ and σ_∙ obtained in <cit.> (case a≠ b) and <cit.> (case a= b).













§ ASYMPTOTIC BEHAVIOR OF ESTIMATORS


Let X_1,…, X_n be a random sample of size n from the  variable X with PDF (<ref>). If we further let  X=(X_1,X_2,X_3,X_4)^⊤ and 

    X≡[ h_1(X); h_2(X); h_3(X); h_4(X) ],

with X_i, i=1,…,6, as given in (<ref>), and  h_1,h_2,h_3 and h_4 being as in (<ref>) and (<ref>), respectively.
By applying strong law of large numbers, we have

    X a.s.⟶𝔼(X),

where “a.s.⟶” denotes almost sure convergence.
Hence, continuous-mapping theorem <cit.>  gives

    σ=g_1(X) a.s.⟶
    g_1(𝔼(X))
      and  μ=g_2(X) a.s.⟶
    g_2(𝔼(X)),

with

    g_1(x_1,x_2,x_3,x_4)
    ≡
    	1
    	-
    	δ_abx_4
    	+
    	δ_abx_2δ_ab
    		x_1
    		+
    		1
    	+
    	√({	1
    		-
    		δ_abx_4
    		+
    		δ_abxδ_ab
    			x_1
    			+
    			1}^2
    		+
    		4x_4
    		{δ_ab-δ_abx_3δ_ab
    			x_1
    			+
    			1 })
    	2x_4

and

    g_2(x_1,x_2,x_3,x_4)
    ≡δ_ab
    	x_1
    	+
    	1
    
    	g_1(x_1,x_2,x_3,x_4) x_2
    	-
    	x_3
    .


Furthermore, by Central limit theorem,

    √(n)[X-𝔼(X)]𝒟⟶ N_4( 0, Σ),

where Σ denotes the covariance matrix of X and “𝒟⟶” means convergence in distribution.
So, delta method provides

    √(n)[
    	[ μ; σ ]
    	-
    	[ g_2(𝔼( X)); g_1(𝔼( X)) ]]
    	(<ref>)=√(n)[
    	[ g_2(X); g_1(X) ]
    	-
    	[ g_2(𝔼( X)); g_1(𝔼( X)) ]]
    	𝒟⟶
    	N_2( 0,  AΣ A^⊤),

with A being the partial derivatives matrix defined as

    A
    	=
    	. 
    	[ ∂ g_2( x)∂ x_1 ∂ g_2( x)∂ x_2 ∂ g_2( x)∂ x_3 ∂ g_2( x)∂ x_4; ∂ g_1( x)∂ x_1 ∂ g_1( x)∂ x_2 ∂ g_1( x)∂ x_3 ∂ g_1( x)∂ x_4 ] |_ x=𝔼( X).

For simplicity of presentation, we do not present the partial derivatives of g_j, j=1,2, here. Analogously to the calculation of 𝔼( X), the second moments of the components of X can be determined which is sufficient to guarantee the existence of the matrix Σ.

The following result shows that for generators of type T(x)=x^-s, x>0 and s 0, the strong consistency property and a Central limit type theorem for the estimators σ and μ are satisfied. 


	If T(x)=x^-s, x>0 and s 0, then
	g_1(𝔼( X))=σ and g_2(𝔼( X))=μ, where g_1 and g_2 are given in (<ref>) and (<ref>), respectively. Moreover, from (<ref>),

    √(n)[
    	[ μ; σ ]
    	-
    	[ μ; σ ]]
    	𝒟⟶
    	N_2( 0,  AΣ A^⊤),
	
	where A was given lines above and Σ is the covariance matrix of X.


	The proof follows immediately from (<ref>) and (<ref>), because
	
    g_2(𝔼( X))
    	=
    	δ_ab𝔼[h_1(X)]
    		+
    		1
    	σ𝔼[h_2(X)]
    		-
    		𝔼[h_3(X)]
    	(<ref>)=μ

	and
		
    g_1(𝔼( X))
    		   =
    			
    			1
    			-
    			δ_ab𝔼[h_4(X)]
    			+
    			δ_ab𝔼[h_2(X)]δ_ab𝔼[h_1(X)]
    				+
    				1
    				2𝔼[h_4(X)]
    			
       +
    			√({	1
    				-
    				δ_ab𝔼[h_4(X)]
    				+
    				δ_ab𝔼[h_2(X)]δ_ab𝔼[h_1(X)]
    					+
    					1}^2
    				+
    				4𝔼[h_4(X)]
    				{δ_ab-δ_ab𝔼[h_3(X)]δ_ab𝔼[h_1(X)]
    					+
    					1 })
    			2𝔼[h_4(X)]
    		(<ref>)=σ.
	

Thus completes the proof.














§ SIMULATION STUDY



In this section, we carry out a Monte Carlo simulation study for evaluating the performance of the proposed estimators. Particularly, we evaluate a bias-reduced version of the proposed moment-type estimators, as they are biased <cit.>. For illustrative purposes, we only present the results for the weighted inverse Lindley distribution. Then, by considering the parameters μ=ϕ, σ=λ/ϕ and generator T(x)=x^-1 of the weighted inverse Lindley distribution, given in Table <ref>, from (<ref>) and (<ref>) the bootstrap biased-reduced moment-type estimators for λ and ϕ are given by


    λ^* = 2λ - 1/B∑_b=1^Bλ^(b),
    ϕ^* = 2ϕ - 1/B∑_b=1^Bϕ^(b),

where λ^(b) and ϕ^(b) are the b-th bootstrap replicates from the b-th bootstrap sample,

    λ
    		=
    		
    			1
    			-
    			X_4
    			+
    			X_2X_1
    				+
    				1
    			+
    			√({	1
    				-
    				X_4
    				+
    				X_2X_1
    					+
    					1}^2
    				+
    				4X_4
    				{1-X_3X_1
    					+
    					1 })
    			2X_4
    		 ϕ

	and
	
    ϕ
    		=
    		X_1
    			+
    			1
    		
    			1
    			-
    			X_4
    			+
    			X_2X_1
    				+
    				1
    			+
    			√({	1
    				-
    				X_4
    				+
    				X_2X_1
    					+
    					1}^2
    				+
    				4X_4
    				{1-X_3X_1
    					+
    					1 })
    			2X_4
    		 X_2
    			-
    			X_3
    		,

with
	
    X_1
    	   =
    	1 n∑_i=1^nX_i^-1log(X_i^-1) 1+X_i^-1,
    	
    X_2   =1 n∑_i=1^nX_i^-1log(X_i^-1),
    
    X_3   =1 n∑_i=1^nlog(X_i^-1),
    
    X_4   =1 n∑_i=1^nX_i^-1.



For assessing the performance of the proposed bootstrap biased-reduced moment-type estimators, we calculated the relative bias (RB) and root mean square error (RMSE), given by

    RB(θ^*) =  |1/N∑_i = 1^Nθ^*(i) - θ/θ| ,    RMSE(θ^*) = √(1/N∑_i = 1^N (θ^*(i) - θ)^2),

where θ∈{λ,ϕ} and θ^*(i)∈{λ^*(i),ϕ^*(i)} are the true parameter value and its i-th bootstrap bias-reduced estimate, and N is the number of Monte Carlo replications.

The simulation scenario considers the following setting: sample size n ∈{20,50,100,200,400,1000}, ϕ∈{0.5,1,3,5,9}, and λ=1.  The number of Monte Carlo replications was N=1,000 and the number of bootstrap replications was B=200. The numerical evaluations were implemented using the  software; see <http://cran.r-project.org>.

Figures <ref> and <ref> show the results of Monte Carlo simulation study to assess the performance of the proposed bootstrap biased-reduced moment-type (MOM) estimators.  For comparison purposes, we also considered the results of the classical maximum likelihood estimator (MLE) discussed by <cit.>. Figures <ref> and <ref> show that, as expected, both biases and RMSEs of the estimators approach zero as the sample size increases. However, the bias is much lower for smaller samples for the proposed estimator. Moreover, the RMSE is similar for both estimators. Finally, the results do not seem to be affected by the parameter ϕ.





































































































































































	











































































	

  
*Acknowledgements
The research was supported in part by CNPq and CAPES grants from the Brazilian government.
	
	

  
*Disclosure statement
	There are no conflicts of interest to disclose.
	
	
	










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