Minimax Estimation of a Bounded Discrete Parameter

Éric Marchand^{a,}^{1} , Ahmad Parsian^{b,} ^{2},

a University of New Brunswick and Université de Sherbrooke, Département de mathématiques, Sherbrooke Qc, CANADA, J1K 2R1 (e-mail: eric.marchand@usherbrooke.ca)

b School of Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156, IRAN (e-mail: ahmad_{−}p@cc.iut.ac.ir)

ABSTRACT

For a vast class of discrete model families with cdf’s F_{θ}, and for estimating θ under squared error loss under a constraint of the type θ ∈ [0, m], we present a general and unified development concerning the minimaxity of a boundary supported prior Bayes estimator. While the sufficient conditions obtained are of the expected form m ≤ m(F), the approach presented leads, in many instances, to both necessary and sufficient conditions, and/or explicit values for m(F). Finally, the scope of the results is illustrated with various examples that, not only include several common distributions (e.g., Poisson, Binomial, Negative Binomial), but many others as well.

Key words and phrases: Minimax estimation, restricted parameter space, discrete distributions, squared error loss

1research supported by NSERC of Canada and by the University of New Brunswick Academic Fund

2research supported by a grant of the Research Council of the Isfahan University of Technology

1 Introduction

In many statistical problems, there exists bounds on the values that unknown parameters can take. Correspondingly, a large body of work concerned with estimation problems in restricted parameter spaces has emerged, as reviewed for instance by van Eeden (1996) or Marchand and Strawderman (2004). One influential, much studied, and useful criterion to select or to evaluate a procedure is minimaxity (e.g., Brown, 1994, Strawderman, 2000).

We are concerned here with minimax estimation, under squared error loss, of an unknown parameter θ for certain kinds of discrete families of distributions under a constraint of the type θ ∈ [ 0, m]. More specifically, we focus on explicit conditions for the minimax estimator to be Bayes with respect to a boundary supported prior. It follows here from the work of DasGupta (1986) that the least favourable prior is quite generally supported on the boundary {0, m} of the parameter space; and that the corresponding Bayes estimator is minimax; for small enough m, say m ≤ m^{∗} with m^{∗} > 0. Since the phenomenon also extends in various univariate and multivariate settings (DasGupta, 1986), and to other strictly convex loss functions (e.g., Bader and Bischoff, 2003), it has lead researchers to analytical investigations of m^{∗} in many specific settings. In particular for discrete models, analytical investigations of m^{∗} appeared in: (i) Marchand and MacGibbon (2000) for the Binomial model under both scaled and unscaled squared error losses, in (ii) Wan et al (2000) for a Poisson model and Linex loss, and in (iii) Johnstone and MacGibbon (1992) for a Poisson model and scaled squared error loss.

But, such analytical investigations often focus on either a single model and/or sufficient conditions for the least favourable prior to be boundary supported, in other words lower bounds for m^{∗}. Our work provides a general and unified development applicable to a large class of discrete models. Members of this class include Poisson and Consul’s (1989) Generalized Poisson, Binomial, Negative Binomial, Waring, many members of the “Power Series” family of distributions, and various types of mixtures and “Stopped Sum” distributions. Moreover, we are able to specify not only sufficient conditions, but also in many cases necessary and sufficient conditions for least favourable priors to be supported on the boundary of the parameter space. Finally, in Section 3, we give explicitly these conditions for various cases.

2 Preliminaries

Our results apply to observable random vectors X = (X_{1}, …, X_{n}); n ≥ 1, where: (i) the X_{i}’s are identically, but not necessarily independently distributed, discrete random variables with joint probability function p_{θ}(x) = P_{θ}(X = x), and (ii) to situations where the support of P_{θ}(X_{1} = x_{1}) is lower bounded (say by \underline{s}). We will be concerned with minimax estimation of θ, under squared error loss, in situations where θ is constrained to a known, and small enough interval [a, b], and where the distribution of X under θ = a is degenerate at (\underline{s}, …, \underline{s}). Without loss of generality, given that we can reparametrize and translate, we assume hereafter that [a, b] = [0, m] and \underline{s} = 0. The degenerate property of X at θ = 0 renders possible the minimaxity results below, but it is hardly an unusual or exceptional property as it is shared by many common discrete distributions.

2.1 Distributional assumptions

It will be convenient to set A = {x ∈ ℜ^{n} : ∑_{i=1}^{n} x_{i} = 0}, and denote G(n, θ) as follows.

Definition 1 We define G(n, θ) = P_{θ}(X ∈ A); and in cases where the X_{i}’s are independent with common cdf F, we will sometimes denote G(n, θ) by G_{F} (n, θ); (recall that our above conditions imply that G(n, 0) = 1)

We will assume throughout this paper that G(n, θ) is three times differentiable for θ ∈ [0, m]. We now describe the class C of families of distributions for which the main result of this paper, Theorem ??, applies.

Definition 2 We define C as the class of families p_{θ} for X such that G(n, 0) = 1, and

(−1)^{k} \frac{∂^{k}}{∂θ^{k}} G(n, θ) ≥ 0, on [0, m], for k = 1, 2, 3.

Here are some immediate examples of families p_{θ} that are members of C (other examples follow later in this section). Various properties of these families, as well as additional examples, can be found in the excellent reference book on discrete distributions by Johnson, Kotz and Kemp (1993).

Example 1 (a) X_{i}’s are independently distributed Poisson(θ), with G(n, θ) = e^{−nθ}.

(b) X_{i}’s are independently distributed Bernoulli(θ) with G(n, θ) = (1 − θ)^{n}.

(c) X_{i}’s are independently distributed as Negative Binomial (NBI(α, p)) with parameters (α, p); α > 0 (known), 0 < p ≤ 1; reparametrized such that E_{θ}(X_{i}) = θ = α(\frac{1}{p} − 1), and P_{θ}(X_{i} = x) = \frac{Γ(α+x)}{x!Γ(α)}(\frac{α}{θ+α})^{α}(\frac{θ}{θ+α})^{x}I_{\{0,1,...\}}(x). In such a case, we have G(n, θ) = (\frac{α}{α+θ})^{nα}. Observe here that the constraint θ ≤ m is equivalent to a lower bound constraint for p (i.e., p ≥ \frac{α}{α+m}).

(d) X_{i}’s are independently distributed “Generalized Poisson” (as denoted by Consul, 1989; or Lagrangian Poisson as denoted by others including Johnson, Kotz, and Kemp, 1993) with parameters (θ, λ); λ ≥ 0, and θ ≥ 0. Here P_{θ}(X_{i} = x) = \frac{θ(θ+xλ)^{x−1}e^{−θ−xλ}}{x!} I_{\{0,1,...\}}(x), and G(n, θ) = e^{−nθ} as in the above Poisson case.

(e) X_{i}’s are independently distributed Waring(k, ρ), with P_{θ}(X_{i} = x) = ρ\frac{(k)_{x}}{(k+ρ)_{x+1}} I_{\{0,1,...\}}(x), k > 0, ρ > 0 and θ = k/ρ. Whenever (i) k is known and ρ is unknown, (which includes the Yule distribution for k = 1), or (ii) k is unknown and ρ is known, the families of p_{θ}’s belong to C with G(n, θ) = (\frac{1}{1+θ})^{n}. Observe that this expression matches the Negative Binomial G(n, θ) above for α = 1.

Observe that the above G(n, θ) are also examples of completely monotone (CM) functions, which possess alternating derivative signs of all positive order. Now, completely monotone functions (see Feller, 1966, section XIII.4) are perhaps best known for their usefulness in characterizing functions which are Laplace transforms, but we draw the connection here mostly because they possess rather interesting properties; for instance that the product of completely monotone functions is completely monotone. The next lemma provides a springboard, or a method of identifying or generating easily new members in C from others.

Lemma 1 (a) If X_{1}, …, X_{n} are independent (with common cdf F), then

G_{F} (1, θ) is CM ⇒ G_{F} (n, θ) is CM;

(b) For the hierarchical (or mixture) model such that,

P (X ∈ A|λ) = G_{0}(n, λ), λ|θ ∼ f_{θ}(λ) = \frac{1}{θ}f_{1}(\frac{λ}{θ}),

with E_{1}[λ^{k}] = ∫_{0}^{∞} λ^{k}f_{1}(λ)dλ < ∞; k = 1, 2, …; ^{3} then

G_{0}(n, λ) is CM in λ ⇒ G(n, θ) is CM in θ;

(c) If X_{1}, …, X_{n} are independent with

P (X_{i} = 0|λ) = G_{0}(1, λ), and λ|θ ∼ f_{θ}(λ) = \frac{1}{θ}f_{1}(\frac{λ}{θ}),

with E_{1}[λ^{k}] = ∫_{0}^{∞} λ^{k}f_{1}(λ)dλ < ∞; k = 1, 2, …; then
G_{0}(1, λ) is CM in λ ⇒ G(n, θ) is CM in θ.

Proof. Part (a) follows given that the product of completely monotone functions is completely monotone (see Criterion 1, page 417, Feller 1966). Observe for part (b) that the given representation implies:

G(n, θ) = ∫_{0}^{∞} G_{0}(n, θλ)f_{1}(λ)dλ;  (1)

from which the result follows by the assumed completely monotonicity of G_{0} and the finiteness conditions on E_{1}[λ^{k}]. Finally, part (c) is a direct consequence of parts (a) and (b).

Remark 1 The Negative Binomial representation as a Poisson mixture is well known, and its G(n, θ) actually arises as particular instances of both models (b) and (c) of Lemma ??. Indeed, denoting T = ∑_{i=1}^{n} X_{i} and referring to Example 1(c), we have as illustrations of Lemma ??’s part (b) and part (c) respectively:

(i) T ∼ NBI(ϵ, p = (1 + nθ)^{−1}), (with G(n, θ) = (1 + nθ)^{−ϵ}), whenever X_{1}, …, X_{n}|λ are independent Poisson(λ), with λ|θ ∼ Gamma(ϵ, θ);

(ii) T ∼ NBI(nϵ, p = (1 + θ)^{−1}), (with G(n, θ) = (\frac{ϵ}{ϵ+θ})^{nϵ}) , whenever X_{1}, …, X_{n} are independent with X_{i}|λ ∼ Poisson(λ), and λ|θ ∼ Gamma(ϵ, θ).

3Weaker conditions can be given. Moreover, for the specification of members of C, and for applying the results below, we only need to assume E_{1}[λ^{3}] < ∞.

Observe that the X_{i}’s are not independent in (i), while scenario (ii) reduces directly to the situation in part (c) of Example 1 given the independence of the X_{i}’s and their Poisson mixture representation.

Many members of the class of families of distributions C, including those in parts (a), (b), and (c) of Example ??, can be found among the family of power series distributions, where X_{1}, …, X_{n} are independent with a common probability function given by:

P_{θ}(X_{i} = x) = \frac{a_{x} [γ(θ)]^{x}}{c (γ(θ))} I_{\{0,1,...\}}(x),  (2)

with a_{0} = 1 (without loss of generality), γ(θ) being a positive and nondecreasing function of θ, and c(γ) = ∑_{x≥0} a_{x}γ^{x}. Here G(n, θ) = (\frac{1}{c(γ(θ))})^{n}. For such families to belong to C, it will suffice, as just observed with the previous lemma, that \frac{1}{c(γ(θ))} be completely monotone in θ. We note that, although a sufficiency reduction is available here with the statistic ∑_{i=1}^{n} X_{i}, and its family of distributions being power series (as in (??) with the coefficients a_{x} being the coefficients in the power series representation of (c(γ(θ))^{n}, we will not (and it is not necessary for our purposes) make use of it, which will be make it somewhat easier to highlight the role of the sample size n in some of our results below. We conclude this section with two more interesting examples of classes of families of distributions which belong to C, and which greatly increase the number of applications of Theorem ?? below.

Example 2 (Generalized Hypergeometric Factorial Moment Distributions)

Described by Johnson, Kotz, and Kemp (1993, section 4.2), these distributions (which include, although with a possibly different parametrization, the distributions in Example 1, parts (a), (b), and (c) have probability generating functions of the form E_{θ}[t^{X_{1}}] = pF_{q}(a_{1}, …, a_{p}; b_{1}, …, b_{q}; θ(t − 1)), where_{p}F_{q} is a Generalized Hypergeometric function. Therefore, for X_{1}, …, X_{n} independent and identically distributed arising from one such family, we have:

G(n, θ) = {P_{θ}(X_{1} = 0)}^{n} = {_{p}F_{q}(a_{1}, …, a_{p}; b_{1}, …, b_{q}; −θ)}^{n}

(note that, for our general i.i.d. case, G(n, θ) = (E_{θ}[t^{X_{1}}]|_{t=0})^{n}). Now, for cases where both the a_{i}’s and b_{i}’s are positive, it is easy to see from the above representation and part (a) of Lemma ?? that G(n, θ) is completely monotone and, consequently, the corresponding family of p_{θ}’s belong to C.

Example 3 (Stopped-Sum Distributions) Closely related to mixture models, such as those in Lemma ??, are Stopped-Sum distributions which admit the representation: S = ∑_{i=1}^{N} X_{i}, with N, X_{1}, X_{2}, … being independent random variables such that the X_{i}’s are identically distributed with P_{θ}(X_{i} = 0) = G_{0}(1, θ) (for some G_{0}); the distribution of N is known with P (N ∈ {1, 2, …}) = 1, E[N^{k}] < ∞ for k = 1, 2, …; and P (S < ∞) = 1. Now, suppose G_{0}(1, θ) is completely monotone in θ. Then, it is easy to see that G_{S}(1, θ) = P_{θ}(S = 0) is also completely monotone in θ given the representation P_{θ}(S = 0) = E[(G_{0}(1, θ))^{N}], and part (a) of Lemma ??. Hence, such a family of probability functions P_{θ}(S = s); θ ∈ [0, m]; belongs to the class C. The function G_{0} is general, with Poisson and Bernoulli G_{0}’s being perhaps of particular importance.

2.2 Minimaxity

Our results rely on a following well-known criteria for minimaxity applied to boundary twopoint priors.

Lemma 2 A two-point boundary prior π on {0, m} is least favourable, and the corresponding Bayes estimator δ_{π}(X) is minimax, iff

R(0, δ_{π}) = R(m, δ_{π}) = sup{R(θ, δ_{π}); 0 ≤ θ ≤ m}.  (3)

Now, for our discrete models, Bayes estimators δ_{π}(X) corresponding to two-point boundary supported priors π are of a simple form; and, as we now show, exactly one of these is an “equalizer” rule (i.e., R(0, δ_{π}) = R(m, δ_{π})).

Lemma 3 Among two-point boundary priors π exactly one of them leads to an equalizer Bayes rule under squared error loss, and it is given by

δ^{∗}(x) = y^{∗}[x ∈ A] + m[x ∉ A],  (4)

with

y^{∗} = m\frac{\sqrt{G(n, m)}}{1 + \sqrt{G(n, m)}}.  (5)

Proof. Given that P_{0}(X ∈ A) = 1 and P_{m}(X ∈ A) = G(n, m), the expected posterior loss E[(d − θ)^{2}|x] becomes, for the prior with π(0) = 1 − π(m):

(d − m)^{2} if x ∉ A;

and

π(0|x)d^{2} + (1 − π(0|x))(d − m)^{2} if x ∈ A,  (6)

where π(0|x) = (1 + \frac{π(m)}{π(0)}G(n, m))^{−1}. From this, it follows that Bayes estimators δ_{π}(X) corrresponding to two-point boundary priors π are of the form

δ_{π}(x) = y[x ∈ A] + m[x ∉ A],

with y minimizing (??) in d. Furthermore, observe that y is a continuous and strictly decreasing function of π(0|x), and hence of π(0), taking values on [0, m] as π(0|x) (or π(0)) varies on [0, 1]. Now, evaluating the difference in risks of δ_{π}(X) at θ = 0 and θ = m, we have R(0, δ_{π}) − R(m, δ_{π}) = y^{2} − (y − m)^{2}G(n, m), which clearly admits the unique root y^{∗} in y.

Remark 2 It is not too difficult to see that the above development giving the uniqueness of an “equalizer” two-point boundary Bayes rule also remains valid for general losses ρ(d − θ) with strictly convex ρ. This was observed for a Linex loss ρ by Wan, Zou, and Lee (2000, Theorem 2.1).

3 Main Results

The general development that follows with Lemma ?? and Theorem ?? capitalizes on:

(i) the necessary condition:

\frac{∂}{∂θ}R(θ, δ^{∗})|_{θ=0} ≤ 0,  (7)

for (??) to hold with δ(X) = δ^{∗}(X); and

(ii) the sufficient condition:

\frac{∂^{2}}{∂^{2}θ}R(θ, δ^{∗}) nondecreasing in θ; θ ∈ [0, m],  (8)

for (??) to hold with δ(X) = δ^{∗}(X) in cases where (??) is satisfied.

Lemma 4 For a family p_{θ} ∈ C, a necessary condition for (??) to be satisfied with δ_{π}(X) = δ^{∗}(X) is m ≤ m_{0}, where m_{0} is the unique (positive) solution in m of the equation T (m) = 0 with

T (m) = m + ( \frac{2}{\frac{∂}{∂θ}G(n, θ)|_{θ=0}} ) \frac{(\sqrt{G(n, m)})(1 + \sqrt{G(n, m)})}{1 + 2\sqrt{G(n, m)}}.  (9)

Proof. Given representation (??) of δ^{∗}(X), we obtain directly

R(θ, δ^{∗}) = (m − θ)^{2} + G(n, θ)[(y^{∗} − θ)^{2} − (m − θ)^{2}]
\frac{∂}{∂θ}R(θ, δ^{∗}) = 2(θ − m) + 2G(n, θ)(m − y^{∗}) + \frac{∂}{∂θ}G(n, θ)\{(y^{∗} − θ)^{2} − (m − θ)^{2}\} (10)
and \frac{∂}{∂θ}R(θ, δ^{∗})|_{θ=0} = \frac{∂}{∂θ}G(n, θ)|_{θ=0}(y^{∗}^{2} − m^{2}) − 2y^{∗}, (11)

since G(n, 0) = 1. Now, substituting y^{∗} as in (??), and solving (??) with the help of (??) yields the necessary condition

m ≤  ( \frac{−2}{\frac{∂}{∂θ} G(n,θ)|_{θ=0}} )\frac{\frac{y^{∗}}{m}}{1−( \frac{y^{∗}}{m})^{2}}
⟺ m ≤ ( \frac{−2}{\frac{∂}{∂θ} G(n,θ)|_{θ=0}} ) ^{\frac{(\sqrt{G(n,m)})(1+\sqrt{G(n,m)})}{1+2\sqrt{G(n,m)}}}.  (12)

Finally, the result follows since, for p_{θ} ∈ C, G(n, m) is decreasing in m implying that the rhs of (??) is decreasing in m.

Before pursuing, we define z(θ) = θ\frac{∂^{2}}{∂θ^{2}} G(n, θ) and m_{1} = inf\{θ : z′(θ) < 0\}. Here, the infimum is taken over the set of all θ such that p_{θ} is a valid probability measure. Also, observe that for p_{θ} ∈ C, we must have m_{1} > 0.

Theorem 1 For squared error loss and p_{θ} ∈ C,

(a) \frac{∂^{2}}{∂θ^{2}} R(θ, δ^{∗}) is nondecreasing for θ ∈ [0, m_{1}]; (b) δ^{∗}(X) is minimax whenever m ≤ m_{0} ∧ m_{1};

(c) Whenever m_{0} ≤ m_{1}; or equivalently T (m_{1}) ≥ 0; δ^{∗}(X) is minimax iff m ≤ m_{0}.

Proof. Part (b) follows part (a), criterion (??), and Lemma ?? while part (c) follows from part (b) and Lemma ??. Hence, there remains only to prove (a). From (??), we obtain

\frac{∂^{2}}{∂θ^{2}} R(θ, δ^{∗}) = 2 + 4(m − y^{∗})\frac{∂}{∂θ}G(n, θ) + (m^{2} − (y^{∗})^{2})(−\frac{∂^{2}}{∂θ^{2}} G(n, θ)) + 2(m − y^{∗})z(θ).

The result follows since p_{θ} ∈ C, y^{∗} ≤ m, and z(θ) is nondecreasing by definition for θ ≤ m_{1}.

We now turn to various illustrations and implications of Theorem ??. Given the generality of Theorem ??, remaining interest lies in the specification (and properties) of m_{0} and m_{1} for various members p_{θ} ∈ C. We do this below for the families p_{θ} given in Example ??. Also, observe that whenever Theorem ??’s δ^{∗}(X) is minimax, the minimax risk is given by R(0, δ^{∗}(X)) = y^{∗}^{2}, with y^{∗} given in (??). In what follows, we will sometimes denote m_{0} and m_{1} as m_{0}(n) and m_{1}(n) to emphasize the dependence on the sample size n.

Example 4 (Poisson(θ) or Generalized Poisson(θ, λ)) Here G(n, θ) = e^{−nθ}, z(θ) = n^{2}θe^{−nθ}, and z′(θ) = n^{2}e^{−nθ}(1 − nθ), which gives m_{1}(n) = \frac{1}{n}. From (??), we obtain that m_{0}(n) = \frac{c_{0}}{n} with c_{0} = 2\frac{e^{−c_{0}/2}(1+e^{−c_{0}/2})}{1+2e^{−c_{0}/2}}. With the numerical evaluation c_{0} ≈ 0.912955, part (c) of Theorem ?? tells us that δ^{∗}(X), given in (??), is minimax iff m ≤ m_{0}(n) ≈ \frac{0.912955}{n}. Interestingly, the above minimaxity result still holds when λ is unknown. This is so, because otherwise with both P_{(θ,λ)}(X ∈ A) and the risk R((θ, λ), δ^{∗}) being independent of λ, we arrive at a contradiction.

Example 5 (Binomial; Marchand and MacGibbon, 2000 ^{4}) Here G(n, θ) = (1−θ)^{n}, z′(θ) = n(n − 1)(1 − θ)^{n−3}(1 − θ(n − 1)), m_{1}(1) = 1, and m_{1}(n) = \frac{1}{n−1} for n ≥ 2. From (??), we obtain

T (m) = m − \frac{2}{n} \frac{(1 − m)^{n/2}(1 + (1 − m)^{n/2})}{1 + 2(1 − m)^{n/2}},

and, for n ≥ 2,

T (m_{1}(n)) = T (\frac{1}{n − 1}) ≥ T (\frac{1}{n}) = \frac{1 − 2(1 − \frac{1}{n})^{n}}{n(1 + 2(1 − \frac{1}{n})^{n/2})} > 0.

Here again, part (c) of Theorem ?? applies, and implies that the estimator δ^{∗}(X), as given in (??) is minimax iff m ≤ m_{0}(n), with m_{0}(n) = T^{−1}(0). Marchand and MacGibbon (2000) give a graph for m_{0}(n), and show that lim_{n→∞} nm_{0}(n) = 0.912955... This last result can equally be deduced by the Poisson example above given the limiting Poisson distribution for a sequence of {Bi(n, c/n); n ≥ 1} random variables.

Remark 3 The above limiting result and Binomial large sample approximation for m_{0}(n) (which follows from Corollary 1 below with γ(θ) = \frac{θ}{1−θ} and a_{1} = 1) can be generalized as follows for power series families given in (??) which are members of C. This is done in Corollary 1 with the help of Example ?? the following Poisson approximation result.

4Note that their Theorem 2.1 is correct but that some of the derivatives leading up to it are in error.

Lemma 5 (Pérez-Abreu, 1991)

For X_{1}, …, X_{n} distributed as in (??) with parameter θ_{n}, such that nγ(θ_{n}) → λ, as n → ∞, it follows that S_{n} = ∑_{i=1}^{n} X_{i} converges to a Poisson(λa_{1}) distribution. In particular, G(n, θ_{n}) → e^{−λa_{1}} as n → ∞.

Corollary 1 For power series families as in (??) which are members of C, we have

nm_{1}(n) → a_{1}, nm_{0}(n) → c_{0}a_{1} ≈ 0.912955a_{1}, as n → ∞.

Moreover, for sufficiently large n, the estimator δ^{∗}(X), given in (??) is minimax iff m ≤ m_{0}(n) ≈ \frac{0.912955a_{1}}{n}.

Example 6 (Negative Binomial) Here G(n, θ) = (\frac{α}{α+θ})^{nα} (as in Example 1 c), z(θ) ∝ \frac{θ}{(θ+α)^{nα+2}} , z′(θ) ∝ \frac{1}{(θ+α)^{nα+3}} (α − θ(nα + 1)), which gives m_{1}(n) = \frac{α}{1+nα}. Now, we obtain from (??),

T (m) = m − \frac{2}{n} \frac{(\frac{α}{α+m})^{nα/2}(1 + (\frac{α}{α+m})^{nα/2})}{1 + 2(\frac{α}{α+m})^{nα/2}},  (13)

and

T (m_{1}(n)) = \frac{1}{n}\{\frac{nα}{1 + nα} − 2v[(\frac{1 + nα}{2 + nα})^{\frac{nα}{2}} ]\},

with v(y) = \frac{y(1+y)}{1+2y}. Now, observe that (\frac{1+s}{2+s})^{\frac{s}{2}} is decreasing in s; s ≥ 0; so that

m_{0}(n) ≥ m_{1}(n) ⟺ T (m_{1}(n)) ≥ 0 ⟺ nα ≥ s_{0}, .

where \frac{s_{0}}{1+s_{0}} = 2v((\frac{1+s_{0}}{2+s_{0}} )^{s_{0}/2}). With the numerical evaluation s_{0} ≈ 11.876904 we have the following summary for the Negative Binomial case.

Corollary 2 For X_{1}, …, X_{n} independent NBI(α, p) (as in Example 1c) with θ = E(X_{i}); θ ≤ m; the estimator δ^{∗}(X) given by (??) is minimax:

(a) whenever nα < s_{0} ≈ 11.876904, and m ≤ \frac{α}{1+nα};

(b) iff m ≤ m_{0}(n) = T^{−1}(0) whenever nα ≥ s_{0} ≈ 11.876904.

Example 7 (Waring(k, ρ)) In terms of the pair (k, ρ), Theorem ?? and Corollary ?? below apply for estimating k with ρ known, or estimating \frac{1}{ρ} with k known, with G(n, θ) = (\frac{1}{1+θ})^{n} as in Example 1(e). Matching Example ??’s G(n, θ) for α = 1, we immediately have the following summary.

Corollary 3 For X_{1}, …, X_{n} independent Waring(k, ρ) with θ = k/ρ (as in Example 1e); θ ≤ m; the estimator δ^{∗}(X) given by (??) is minimax:

(a) for m ≤ \frac{1}{1+n} whenever n ≤ 11 (i.e., n ≤ s_{0});

(b) iff m ≤ m_{0}(n) = T^{−1}(0) whenever n ≥ 12, with T (m) given in (??) with α = 1.

We consider next the hierarchical model given in part (b) of Lemma ?? for the particular Poisson case where G_{0}(n, λ) = e^{−nλ}. As witnessed in the Poisson case above where both nm_{0}(n) and nm_{1}(n) were constants independent of n, we show in Lemma ?? that this property holds in general for such Poisson mixtures. Consequently, the comparison of m_{0}(n) and m_{1}(n), as required for applications of Theorem ??, does not depend on n. It will then suffice in applications such as the one given in Example ?? to handle the case n = 1.

Lemma 6 For Poisson mixture cases as in (??) with G_{0}(n, λ) = e^{−nλ} and E_{1}[λ^{3}] < ∞, we have m_{0}(n) = \frac{c_{0}}{n} and m_{1}(n) = \frac{c_{1}}{n}, where c_{0} and c_{1} are independent of n.

Proof. Here G(n, θ) = ∫_{0}^{∞} e^{−nθλ}f_{1}(λ)dλ, and an easy calculation shows that, for a constant c > 0,

z′(\frac{c}{n}) = n^{2} ∫_{0}^{∞} λ^{2}e^{−cλ}(1 − cλ)f_{1}(λ)dλ,

which implies that m_{1}(n) (i.e., the solution of z′(θ) = 0) is of the form \frac{c_{1}}{n} with c_{1} independent of n. Similarly, it is easy to verify by expanding (??) that, for c > 0, nT (\frac{c}{n}) is free of n, implying that Lemma ??’s m_{0}(n) is of the form \frac{c_{0}}{n} with c_{0} independent of n.

Example 8 (The Poisson mixture in Remark ??, part (i)) Referring to Remark ?? (i) with n = 1, we have G(1, θ) = (\frac{1}{1+θ})^{ϵ}. Since this matches Example ??’s G(n, θ) with α = 1 and n = ϵ, we can borrow from that analysis (or even Corollary 2). Along with Lemma ??, we obtain the following.

Corollary 4 For the model: X_{1}, …, X_{n}|λ i.i.d. Poisson(λ), with λ|θ ∼ Gamma(ϵ, θ) (as in Remark ??, part (i)); with θ ≤ m; the estimator δ^{∗}(X) given by (??) is minimax:

(a) whenever ϵ < s_{0} ≈ 11.876904, and m ≤ \frac{1}{n} \frac{α}{1+α};

(b) iff m ≤ \frac{1}{n}m_{0}(1) = \frac{1}{n}T^{−1}(0) whenever ϵ ≥ s_{0} ≈ 11.876904, where T^{−1} is the inverse of the T (m) function given in (??) with α = 1 and n = ϵ.

ACKNOWLEDGEMENTS

The authors benefited from discussions with Yiping Dou and Rokhaya Gueye.

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