Explicit formulas for the joint third and fourth central moments of the multinomial distribution
aa r X i v : . [ s t a t . O T ] J un Explicit formula for the joint third and fourth central momentsof the multinomial distribution
Fr´ed´eric Ouimet a,1, ∗ a California Institute of Technology, Pasadena, USA.
Abstract
We give the first explicit formula for the joint third and fourth central moments of the multinomialdistribution, by differentiating the moment generating function. A general formula for the joint factorialmoments was previously given in Mosimann (1962).
Keywords: multinomial distribution, simplex, central moments, third moment, fourth moment
Primary : 60E05 Secondary : 62E15
1. The multinomial distribution
For any d ∈ N , the d -dimensional (unit) simplex is defined by S := (cid:8) x ∈ [0 , d : P di =1 x i ≤ (cid:9) , andthe probability mass function for ξ := ( ξ , ξ , . . . , ξ d ) ∼ Multinomial( m, x ) is defined by p ξ ( x ) := m !( m − P di =1 k i )! Q di =1 k i ! · (1 − d X i =1 x i ) m − P di =1 k i d Y i =1 x k i i , k ∈ N d ∩ m S . (1.1)In this paper, our goal is to compute, for all i, j, ℓ, p ∈ { , , . . . , d } , E (cid:2) ( ξ i − E [ ξ i ])( ξ j − E [ ξ j ])( ξ ℓ − E [ ξ ℓ ]) (cid:3) and E (cid:2) ( ξ i − E [ ξ i ])( ξ j − E [ ξ j ])( ξ ℓ − E [ ξ ℓ ])( ξ p − E [ ξ p ]) (cid:3) . (1.2)
2. Motivation
To the best of our knowledge, an explicit formula for the joint third and fourth central moments ofthe multinomial distribution has never been derived in the literature. These central moments can arisenaturally, for example, when studying asymptotic properties, via Taylor expansions, of statistical estima-tors involving the multinomial distribution. Two examples of such estimators are, for a given sequenceof observations X , X , . . . , X n , the Bernstein estimator for the cumulative distribution function F ⋆n,m ( x ) := X k ∈ N d ∩ m S n n X i =1 ( − ∞ , k m ] ( X i ) P k ,m ( x ) , x ∈ S , m, n ∈ N , (2.1)and the Bernstein estimator for the density function (also called smoothed histogram)ˆ f n,m ( x ) := X k ∈ N d ∩ ( m − S m d n n X i =1 ( k m , k +1 m ] ( X i ) P k ,m − ( x ) , x ∈ S , m, n ∈ N , (2.2)over the d -dimensional simplex. Their asymptotic properties were investigated by Vitale (1975); Gawronski & Stadtm¨uller(1981); Stadtm¨uller (1983); Gawronski (1985); Stadtm¨uller (1986); Tenbusch (1997); Petrone (1999a,b);Ghosal (2001); Petrone & Wasserman (2002); Babu et al. (2002); Kakizawa (2004); Bouezmarni & Rolin(2007); Bouezmarni et al. (2007); Leblanc (2009, 2010); Curtis & Ghosh (2011); Leblanc (2012a,b);Igarashi & Kakizawa (2014); Turnbull & Ghosh (2014); Lu (2015); Guan (2016, 2017); Belalia et al. (2017, 2019) when d = 1, by Tenbusch (1994) when d = 2, and by Ouimet (2020a) for all d ≥
1, usinga local limit theorem from Ouimet (2020c) for the multinomial density (see also Arenbaev (1976)). Theestimator (2.2) is a discrete analogue of the Dirichlet kernel estimator introduced by Aitchison & Lauder(1985) and studied theoretically in Brown & Chen (1999); Chen (1999, 2000); Bouezmarni & Rolin (2003)when d = 1, and in Ouimet (2020b) for all d ≥ ∗ Corresponding author
Email address: [email protected] (Fr´ed´eric Ouimet) F. O. is supported by a postdoctoral fellowship from the NSERC (PDF) and the FRQNT (B3X supplement).
Preprint submitted to Elsevier June 17, 2020 . Results
First, we compute the non-central moments of the multinomial distribution from (1.1).
Theorem 1 (Non-central moments) . Let m ( k ) := m ( m − . . . ( m − k +1) and let ξ ∼ Multinomial( m, x ) .Then, for all i, j, ℓ, p ∈ { , , . . . , d } , E [ ξ i ] = m x i , (3.1) E [ ξ i ξ j ] = m (2) x i x j + { i = j } m x i , (3.2) E [ ξ i ξ j ξ ℓ ] = m (3) x i x j x ℓ + { i = j = ℓ } [3 m (2) x i + m x i ] (3.3)+ { i = j = ℓ = i } m (2) x i x ℓ + { i = j = ℓ = i } m (2) x i x j + { i = j = ℓ = i } m (2) x j x ℓ , E [ ξ i ξ j ξ ℓ ξ p ] = m (4) x i x j x ℓ x p (3.4)+ { i = j = ℓ = i }∩{ i = p = ℓ } m (3) x i x ℓ x p + { i = j = ℓ = i }∩{ i = p = j } m (3) x i x j x p + { i = j = ℓ = i }∩{ j = p = ℓ } m (3) x j x ℓ x p + { i = j = ℓ = i }∩{ i = p = ℓ } [3 m (3) x i x ℓ + m (2) x i x ℓ ]+ { i = j = ℓ = i }∩{ i = p = j } [3 m (3) x i x j + m (2) x i x j ]+ { i = j = ℓ = i }∩{ j = p = ℓ } [3 m (3) x j x ℓ + m (2) x j x ℓ ]+ { p = i }∩{ j = ℓ = p = j } m (3) x i x j x ℓ + { p = j }∩{ i = ℓ = p = i } m (3) x i x j x ℓ + { p = ℓ }∩{ i = j = p = i } m (3) x i x j x ℓ + { i = j = ℓ = i }∩{ i = p = ℓ } [ m (3) x i x ℓ + m (3) x i x ℓ + m (2) x i x ℓ ]+ { i = j = ℓ = i }∩{ i = p = j } [ m (3) x i x j + m (3) x i x j + m (2) x i x j ]+ { i = j = ℓ = i }∩{ j = p = ℓ } [ m (3) x j x ℓ + m (3) x j x ℓ + m (2) x j x ℓ ]+ { i = j = ℓ = p } [3 m (3) x i x p + m (2) x i x p ]+ { i = j = ℓ = p } [6 m (3) x i + 7 m (2) x i + mx i ] . Proof.
The moment generating function of ξ is M m ( t ) = (cid:18) − d X i =1 x i + d X i =1 x i e t i (cid:19) m . (3.5)We have ∂∂t i M m ( t ) = mM m − ( t ) x i e t i , (3.6) ∂ ∂t i ∂t j M m ( t ) = m (2) M m − ( t ) x i e t i x j e t j + { i = j } mM m − ( t ) x i e t i , (3.7) ∂ ∂t i ∂t j ∂t ℓ M m ( t ) = m (3) M m − ( t ) x i e t i x j e t j x ℓ e t ℓ (3.8)+ { i = j = ℓ = i } m (2) M m − ( t ) x i e t i x ℓ e t ℓ + { i = j = ℓ = i } m (2) M m − ( t ) x i e t i x j e t j + { i = j = ℓ = i } m (2) M m − ( t ) x j e t j x ℓ e t ℓ + { i = j = ℓ } [3 m (2) M m − ( t ) x i e t i + mM m − ( t ) x i e t i ] ,∂ ∂t i ∂t j ∂t ℓ ∂t p M m ( t ) = m (4) M m − ( t ) x i e t i x j e t j x ℓ e t ℓ x p e t p (3.9)+ { i = j = ℓ = i }∩{ i = p = ℓ } m (3) M m − ( t ) x i e t i x ℓ e t ℓ x p e t p { i = j = ℓ = i }∩{ i = p = j } m (3) M m − ( t ) x i e t i x j e t j x p e t p + { i = j = ℓ = i }∩{ j = p = ℓ } m (3) M m − ( t ) x j e t j x ℓ e t ℓ x p e t p + { i = j = ℓ = i }∩{ i = p = ℓ } (cid:20) m (3) M m − ( t ) x i e t i x ℓ e t ℓ + m (2) M m − ( t ) x i e t i x ℓ e t ℓ (cid:21) + { i = j = ℓ = i }∩{ i = p = j } (cid:20) m (3) M m − ( t ) x i e t i x j e t j + m (2) M m − ( t ) x i e t i x j e t j (cid:21) + { i = j = ℓ = i }∩{ j = p = ℓ } (cid:20) m (3) M m − ( t ) x j e t j x ℓ e t ℓ + m (2) M m − ( t ) x j e t j x ℓ e t ℓ (cid:21) + { p = i }∩{ j = ℓ = p = j } m (3) M m − ( t ) x i e t i x j e t j x ℓ e t ℓ + { p = j }∩{ i = ℓ = p = i } m (3) M m − ( t ) x i e t i x j e t j x ℓ e t ℓ + { p = ℓ }∩{ i = j = p = i } m (3) M m − ( t ) x i e t i x j e t j x ℓ e t ℓ + { i = j = ℓ = i }∩{ i = p = ℓ } m (3) M m − ( t ) x i e t i x ℓ e t ℓ + m (3) M m − ( t ) x i e t i x ℓ e t ℓ + m (2) M m − ( t ) x i e t i x ℓ e t ℓ + { i = j = ℓ = i }∩{ i = p = j } m (3) M m − ( t ) x i e t i x j e t j + m (3) M m − ( t ) x i e t i x j e t j + m (2) M m − ( t ) x i e t i x j e t j + { i = j = ℓ = i }∩{ j = p = ℓ } m (3) M m − ( t ) x j e t j x ℓ e t ℓ + m (3) M m − ( t ) x j e t j x ℓ e t ℓ + m (2) M m − ( t ) x j e t j x ℓ e t ℓ + { i = j = ℓ = p } (cid:20) m (3) M m − ( t ) x i e t i x p e t p + m (2) M m − ( t ) x i e t i x p e t p (cid:21) + { i = j = ℓ = p } m (3) M m − ( t ) x i e t i +7 m (2) M m − ( t ) x i e t i + mM m − ( t ) x i e t i . By taking t = , we get the conclusion.With some algebraic manipulations and a careful analysis, we can now obtain the central moments. Theorem 2 (Central moments) . Let ξ ∼ Multinomial( m, x ) , then, for all i, j, ℓ, p ∈ { , , . . . , d } , E (cid:2) ξ i − E [ ξ i ] (cid:3) = 0 , (3.10) E (cid:2) ( ξ i − E [ ξ i ])( ξ j − E [ ξ j ]) (cid:3) = m ( x i { i = j } − x i x j ) , (3.11) E (cid:2) ( ξ i − E [ ξ i ])( ξ j − E [ ξ j ])( ξ ℓ − E [ ξ ℓ ]) (cid:3) = m (cid:0) x i x j x ℓ − { i = j } x i x ℓ − { j = ℓ } x i x j − { i = ℓ } x j x ℓ + { i = j = ℓ } x i (cid:1) , (3.12) E (cid:2) ( ξ i − E [ ξ i ])( ξ j − E [ ξ j ])( ξ ℓ − E [ ξ ℓ ])( ξ p − E [ ξ p ]) (cid:3) = (3 m − m ) x i x j x ℓ x p − (12 m − m ) { i = j = ℓ = p } x i (3.13)+ m (cid:26) { i = j } x i x ℓ x p + { i = ℓ } x j x ℓ x p + { i = p } x i x j x ℓ { j = ℓ } x i x j x p + { j = p } x i x j x ℓ + { ℓ = p } x i x j x ℓ (cid:27) + (3 m − m ) { i = j = ℓ = p } x i + m { i = j = ℓ = p } x i − (6 m − m ) (cid:26) { i = j = ℓ = p } x i x p + { i = j = p = ℓ } x i x ℓ + { i = ℓ = p = j } x j x ℓ + { j = ℓ = p = i } x i x j (cid:27) − (2 m − m ) (cid:26) { ℓ = i = j = p } x i x ℓ x p + { j = i = ℓ = p } x j x ℓ x p + { j = i = p = ℓ } x i x j x ℓ { i = j = ℓ = p } x i x j x p + { i = j = p = ℓ } x i x j x ℓ + { i = ℓ = p = j } x i x j x ℓ (cid:27) + ( m − m ) (cid:8) { i = j = ℓ = p } x i x ℓ + { i = p = j = ℓ } x i x j + { i = ℓ = j = p } x j x ℓ (cid:9) − m (cid:8) { i = j = ℓ = p } x i x p + { i = j = p = ℓ } x i x ℓ + { i = ℓ = p = j } x j x ℓ + { j = ℓ = p = i } x i x j (cid:9) . roof. The expression (3.11) for the covariance follows directly from (3.1) and (3.2). For the third centralmoments, (3.1), (3.2) and (3.3) yield E (cid:2) ( ξ i − E [ ξ i ])( ξ j − E [ ξ j ])( ξ ℓ − E [ ξ ℓ ]) (cid:3) = E [ ξ i ξ j ξ ℓ ] − m E [ ξ i ξ j ] x ℓ − m E [ ξ i ξ ℓ ] x j − m E [ ξ j ξ ℓ ] x i + m E [ ξ i ] x j x ℓ + m E [ ξ j ] x i x ℓ + m E [ ξ ℓ ] x i x j − m x i x j x ℓ = (cid:26) m (3) x i x j x ℓ + { i = j = ℓ } [3 m (2) x i + m x i ]+ { i = j = ℓ = i } m (2) x i x ℓ + { i = j = ℓ = i } m (2) x i x j + { i = j = ℓ = i } m (2) x j x ℓ (cid:27) − m (cid:8) m (2) x i x j + { i = j } m x i (cid:9) x ℓ − m (cid:8) m (2) x i x ℓ + { i = ℓ } m x i (cid:9) x j − m (cid:8) m (2) x j x ℓ + { j = ℓ } m x j (cid:9) x i + 2 m x i x j x ℓ = 2 m x i x j x ℓ − m { i = j } x i x ℓ − m { j = ℓ } x i x j − m { i = ℓ } x j x ℓ + m { i = j = ℓ } x i = m (cid:0) x i x j x ℓ − { i = j } x i x ℓ − { j = ℓ } x i x j − { i = ℓ } x j x ℓ + { i = j = ℓ } x i (cid:1) . Similarly, by (3.1), (3.2), (3.3) and (3.4), we have E (cid:2) ( ξ i − E [ ξ i ])( ξ j − E [ ξ j ])( ξ ℓ − E [ ξ ℓ ])( ξ p − E [ ξ p ]) (cid:3) = E [ ξ i ξ j ξ ℓ ξ p ] − m E [ ξ i ξ j ξ ℓ ] x p − m E [ ξ i ξ j ξ p ] x ℓ − m E [ ξ i ξ ℓ ξ p ] x j − m E [ ξ j ξ ℓ ξ p ] x i + m E [ ξ i ξ j ] x ℓ x p + m E [ ξ i ξ ℓ ] x j x p + m E [ ξ i ξ p ] x j x ℓ + m E [ ξ j ξ ℓ ] x i x p + m E [ ξ j ξ p ] x i x ℓ + m E [ ξ ℓ ξ p ] x i x j − m E [ ξ i ] x i x j x ℓ − m E [ ξ j ] x i x ℓ x p − m E [ ξ ℓ ] x i x j x p − m E [ ξ p ] x i x j x ℓ + m x i x j x ℓ x p = m (4) x i x j x ℓ x p + { i = j = ℓ = i }∩{ i = p = ℓ } m (3) x i x ℓ x p + { i = j = ℓ = i }∩{ i = p = j } m (3) x i x j x p + { i = j = ℓ = i }∩{ j = p = ℓ } m (3) x j x ℓ x p + { i = j = ℓ = i }∩{ i = p = ℓ } [3 m (3) x i x ℓ + m (2) x i x ℓ ]+ { i = j = ℓ = i }∩{ i = p = j } [3 m (3) x i x j + m (2) x i x j ]+ { i = j = ℓ = i }∩{ j = p = ℓ } [3 m (3) x j x ℓ + m (2) x j x ℓ ]+ { p = i }∩{ j = ℓ = p = j } m (3) x i x j x ℓ + { p = j }∩{ i = ℓ = p = i } m (3) x i x j x ℓ + { p = ℓ }∩{ i = j = p = i } m (3) x i x j x ℓ + { i = j = ℓ = i }∩{ i = p = ℓ } [ m (3) x i x ℓ + m (3) x i x ℓ + m (2) x i x ℓ ]+ { i = j = ℓ = i }∩{ i = p = j } [ m (3) x i x j + m (3) x i x j + m (2) x i x j ]+ { i = j = ℓ = i }∩{ j = p = ℓ } [ m (3) x j x ℓ + m (3) x j x ℓ + m (2) x j x ℓ ]+ { i = j = ℓ = p } [3 m (3) x i x p + m (2) x i x p ]+ { i = j = ℓ = p } [6 m (3) x i + 7 m (2) x i + mx i ] − m (cid:26) m (3) x i x j x ℓ + { i = j = ℓ } [3 m (2) x i + m x i ]+ { i = j = ℓ = i } m (2) x i x ℓ + { i = j = ℓ = i } m (2) x i x j + { i = j = ℓ = i } m (2) x j x ℓ (cid:27) x p − m (cid:26) m (3) x i x j x p + { i = j = p } [3 m (2) x i + m x i ]+ { i = j = p = i } m (2) x i x p + { i = j = p = i } m (2) x i x j + { i = j = p = i } m (2) x j x p (cid:27) x ℓ − m (cid:26) m (3) x i x ℓ x p + { i = ℓ = p } [3 m (2) x i + m x i ]+ { i = ℓ = p = i } m (2) x i x p + { i = ℓ = p = i } m (2) x i x ℓ + { i = ℓ = p = i } m (2) x ℓ x p (cid:27) x j − m (cid:26) m (3) x j x ℓ x p + { j = ℓ = p } [3 m (2) x j + m x j ]+ { j = ℓ = p = j } m (2) x j x p + { j = ℓ = p = j } m (2) x j x ℓ + { j = ℓ = p = j } m (2) x ℓ x p (cid:27) x i + m (cid:8) m (2) x i x j + { i = j } m x i (cid:9) x ℓ x p + m (cid:8) m (2) x i x ℓ + { i = ℓ } m x i (cid:9) x j x p + m (cid:8) m (2) x i x p + { i = p } m x i (cid:9) x j x ℓ + m (cid:8) m (2) x j x ℓ + { j = ℓ } m x j (cid:9) x i x p + m (cid:8) m (2) x j x p + { j = p } m x j (cid:9) x i x ℓ + m (cid:8) m (2) x ℓ x p + { ℓ = p } m x ℓ (cid:9) x i x j − m x i x j x ℓ x p .
4e notice that all the terms with powers m and m cancel out with each other. The above is thus= (11 m − m ) x i x j x ℓ x p +( − m + 2 m ) { i = j = ℓ = i }∩{ i = p = ℓ } x i x ℓ x p +( − m + 2 m ) { i = j = ℓ = i }∩{ i = p = j } x i x j x p +( − m + 2 m ) { i = j = ℓ = i }∩{ j = p = ℓ } x j x ℓ x p +( − m + 6 m ) { i = j = ℓ = i }∩{ i = p = ℓ } x i x ℓ +( − m + 6 m ) { i = j = ℓ = i }∩{ i = p = j } x i x j +( − m + 6 m ) { i = j = ℓ = i }∩{ j = p = ℓ } x j x ℓ +(1 m − m ) { i = j = ℓ = i }∩{ i = p = ℓ } x i x ℓ +(1 m − m ) { i = j = ℓ = i }∩{ i = p = j } x i x j +(1 m − m ) { i = j = ℓ = i }∩{ j = p = ℓ } x j x ℓ +( − m + 2 m ) { p = i }∩{ j = ℓ = p = j } x i x j x ℓ +( − m + 2 m ) { p = j }∩{ i = ℓ = p = i } x i x j x ℓ +( − m + 2 m ) { p = ℓ }∩{ i = j = p = i } x i x j x ℓ +( − m + 2 m ) { i = j = ℓ = i }∩{ i = p = ℓ } x i x ℓ +( − m + 2 m ) { i = j = ℓ = i }∩{ i = p = j } x i x j +( − m + 2 m ) { i = j = ℓ = i }∩{ j = p = ℓ } x j x ℓ +( − m + 2 m ) { i = j = ℓ = i }∩{ i = p = ℓ } x i x ℓ +( − m + 2 m ) { i = j = ℓ = i }∩{ i = p = j } x i x j +( − m + 2 m ) { i = j = ℓ = i }∩{ j = p = ℓ } x j x ℓ +(1 m − m ) { i = j = ℓ = i }∩{ i = p = ℓ } x i x ℓ +(1 m − m ) { i = j = ℓ = i }∩{ i = p = j } x i x j +(1 m − m ) { i = j = ℓ = i }∩{ j = p = ℓ } x j x ℓ +( − m + 6 m ) { i = j = ℓ = p } x i x p +(1 m − m ) { i = j = ℓ = p } x i x p +( − m + 12 m ) { i = j = ℓ = p } x i +(7 m − m ) { i = j = ℓ = p } x i +1 m { i = j = ℓ = p } x i + (cid:26) − m x i x j x ℓ x p + 3 m { i = j = ℓ } x i x p − m { i = j = ℓ } x i x p +1 m { i = j = ℓ = i } x i x ℓ x p + 1 m { i = j = ℓ = i } x i x j x p + 1 m { i = j = ℓ = i } x j x ℓ x p (cid:27) + (cid:26) − m x i x j x ℓ x p + 3 m { i = j = p } x i x ℓ − m { i = j = p } x i x ℓ +1 m { i = j = p = i } x i x ℓ x p + 1 m { i = j = p = i } x i x j x ℓ + 1 m { i = j = p = i } x j x ℓ x p (cid:27) + (cid:26) − m x i x j x ℓ x p + 3 m { i = ℓ = p } x i x j − m { i = ℓ = p } x i x j +1 m { i = ℓ = p = i } x i x j x p + 1 m { i = ℓ = p = i } x i x j x ℓ + 1 m { i = ℓ = p = i } x j x ℓ x p (cid:27) + (cid:26) − m x i x j x ℓ x p + 3 m { j = ℓ = p } x i x j − m { j = ℓ = p } x i x j +1 m { j = ℓ = p = j } x i x j x p + 1 m { j = ℓ = p = j } x i x j x ℓ + 1 m { j = ℓ = p = j } x i x ℓ x p (cid:27) = (3 m − m ) x i x j x ℓ x p − (12 m − m ) { i = j = ℓ = p } x i + m (cid:26) { i = j } x i x ℓ x p + { i = ℓ } x j x ℓ x p + { i = p } x i x j x ℓ { j = ℓ } x i x j x p + { j = p } x i x j x ℓ + { ℓ = p } x i x j x ℓ (cid:27) + (3 m − m ) { i = j = ℓ = p } x i + m { i = j = ℓ = p } x i − (6 m − m ) (cid:26) { i = j = ℓ = p } x i x p + { i = j = p = ℓ } x i x ℓ + { i = ℓ = p = j } x j x ℓ + { j = ℓ = p = i } x i x j (cid:27) − (2 m − m ) (cid:26) { ℓ = i = j = p } x i x ℓ x p + { j = i = ℓ = p } x j x ℓ x p + { j = i = p = ℓ } x i x j x ℓ { i = j = ℓ = p } x i x j x p + { i = j = p = ℓ } x i x j x ℓ + { i = ℓ = p = j } x i x j x ℓ (cid:27) + ( m − m ) (cid:8) { i = j = ℓ = p } x i x ℓ + { i = p = j = ℓ } x i x j + { i = ℓ = j = p } x j x ℓ (cid:9) − m (cid:8) { i = j = ℓ = p } x i x p + { i = j = p = ℓ } x i x ℓ + { i = ℓ = p = j } x j x ℓ + { j = ℓ = p = i } x i x j (cid:9) . This ends the proof. 5 eferences
Aitchison, J., & Lauder, I. J. 1985. Kernel Density Estimation for Compositional Data.
Journal of the Royal StatisticalSociety. Series C (Applied Statistics) , (2), 129–137. doi:10.2307/2347365.Arenbaev, N. K. 1976. Asymptotic behavior of the multinomial distribution. Teor. Veroyatnost. i Primenen. , (4),826–831. MR0478288.Babu, G. J., Canty, A. J., & Chaubey, Y. P. 2002. Application of Bernstein polynomials for smooth estimation of adistribution and density function. J. Statist. Plann. Inference , (2), 377–392. MR1910059.Belalia, M., Bouezmarni, T., & Leblanc, A. 2017. Smooth conditional distribution estimators using Bernstein polynomials. Comput. Statist. Data Anal. , , 166–182. MR3630225.Belalia, M., Bouezmarni, T., & Leblanc, A. 2019. Bernstein conditional density estimation with application to conditionaldistribution and regression functions. J. Korean Statist. Soc. , (3), 356–383. MR3983257.Bouezmarni, T., & Rolin, J.-M. 2003. Consistency of the beta kernel density function estimator. Canad. J. Statist. , (1),89–98. MR1985506.Bouezmarni, T., & Rolin, J.-M. 2007. Bernstein estimator for unbounded density function. J. Nonparametr. Stat. , (3),145–161. MR2351744.Bouezmarni, T., M., Mesfioui, & Rolin, J. M. 2007. L -rate of convergence of smoothed histogram. Statist. Probab. Lett. , (14), 1497–1504. MR2395599.Brown, B. M., & Chen, S. X. 1999. Beta-Bernstein smoothing for regression curves with compact support. Scand. J.Statist. , (1), 47–59. MR1685301.Chen, S. X. 1999. Beta kernel estimators for density functions. Comput. Statist. Data Anal. , (2), 131–145. MR1718494.Chen, S. X. 2000. Beta kernel smoothers for regression curves. Statist. Sinica , (1), 73–91. MR1742101.Curtis, S. M., & Ghosh, S. K. 2011. A variable selection approach to monotonic regression with Bernstein polynomials. J.Appl. Stat. , (5), 961–976. MR2782409.Gawronski, W. 1985. Strong laws for density estimators of Bernstein type. Period. Math. Hungar , (1), 23–43. MR0791719.Gawronski, W., & Stadtm¨uller, U. 1981. Smoothing histograms by means of lattice and continuous distributions. Metrika , (3), 155–164. MR0638651.Ghosal, S. 2001. Convergence rates for density estimation with Bernstein polynomials. Ann. Statist. , (5), 1264–1280.MR1873330.Guan, Z. 2016. Efficient and robust density estimation using Bernstein type polynomials. J. Nonparametr. Stat. , (2),250–271. MR3488598.Guan, Z. 2017. Bernstein polynomial model for grouped continuous data. J. Nonparametr. Stat. , (4), 831–848.MR3740722.Igarashi, G., & Kakizawa, Y. 2014. On improving convergence rate of Bernstein polynomial density estimator. J. Non-parametr. Stat. , (1), 61–84. MR3174309.Kakizawa, Y. 2004. Bernstein polynomial probability density estimation. J. Nonparametr. Stat. , (5), 709–729.MR2068610.Leblanc, A. 2009. Chung-Smirnov property for Bernstein estimators of distribution functions. J. Nonparametr. Stat. , (2), 133–142. MR2488150.Leblanc, A. 2010. A bias-reduced approach to density estimation using Bernstein polynomials. J. Nonparametr. Stat. , (3-4), 459–475. MR2662607.Leblanc, A. 2012a. On estimating distribution functions using Bernstein polynomials. Ann. Inst. Statist. Math. , (5),919–943. MR2960952.Leblanc, A. 2012b. On the boundary properties of Bernstein polynomial estimators of density and distribution functions. J. Statist. Plann. Inference , (10), 2762–2778. MR2925964.Lu, L. 2015. On the uniform consistency of the Bernstein density estimator. Statist. Probab. Lett. , , 52–61. MR3412755.Mosimann, J. E. 1962. On the compound multinomial distribution, the multivariate β -distribution, and correlations amongproportions. Biometrika , , 65–82. MR143299.Ouimet, F. 2018. Complete monotonicity of multinomial probabilities and its application to Bernstein estimators on thesimplex. J. Math. Anal. Appl. , (2), 1609–1617. MR3825458.Ouimet, F. 2019. Extremes of log-correlated random fields and the Riemann-zeta function, and some asymptotic resultsfor various estimators in statistics . PhD thesis, Universit´e de Montr´eal. http://hdl.handle.net/1866/22667 .Ouimet, F. 2020a. Asymptotic properties of Bernstein estimators on the simplex.
Preprint , 1–15. arXiv:2002.07758.Ouimet, F. 2020b. Density estimation using Dirichlet kernels.
Preprint , 1–14. arXiv:2002.06956.Ouimet, F. 2020c. A precise local limit theorem for the multinomial distribution.
Preprint , 1–7. arXiv:2001.08512.Petrone, S. 1999a. Bayesian density estimation using Bernstein polynomials.
Canad. J. Statist. , (1), 105–126.MR1703623.Petrone, S. 1999b. Random Bernstein polynomials. Scand. J. Statist. , (3), 373–393. MR1712051.Petrone, S., & Wasserman, L. 2002. Consistency of Bernstein polynomial posteriors. J. Roy. Statist. Soc. Ser. B , (1),79–100. MR1881846.Stadtm¨uller, U. 1983. Asymptotic distributions of smoothed histograms. Metrika , (3), 145–158. MR0726014.Stadtm¨uller, U. 1986. Asymptotic properties of nonparametric curve estimates. Period. Math. Hungar. , (2), 83–108.MR0858109.Tenbusch, A. 1994. Two-dimensional Bernstein polynomial density estimators. Metrika , (3-4), 233–253. MR1293514.Tenbusch, A. 1997. Nonparametric curve estimation with Bernstein estimates. Metrika , (1), 1–30. MR1437794.Turnbull, B. C., & Ghosh, S. K. 2014. Unimodal density estimation using Bernstein polynomials. Comput. Statist. DataAnal. , , 13–29. MR3139345.Vitale, R. A. 1975. Bernstein polynomial approach to density function estimation. Pages 87–99 of: Statistical Inferenceand Related Topics . Academic Press, New York. MR0397977.. Academic Press, New York. MR0397977.