The Problem of the Greater Mean

Let π₁, π₂ be normal populations with means m₁, m₂ respectively and a common variance σ², the parameter point ω = (m₁, m₂:σ) which characterizes the two populations being unknown, and let Ω be an arbitrary given set of possible points ω. Random samples of fixed sizes n₁, n₂ are drawn from π₁, π₂ respectively, giving the combined sample point ν = (x₁₁, x₁₂, ⋯, x1n₁; x₂₁, x₂₂, ⋯, x2n₂). For reasons which will be made clear later in connection with practical examples, any function f(ν) such that 0 ≤ f(ν) ≤ 1 is called a decision function, and for any such f(ν) the risk function is defined to be {equation*}{(1)} r(f\|ω) = max m_1, m_2 - m_1E f ω - m_2E 1 - f\|ω ≥ 0{equation*} where E denotes the expectation operator. A decision function f̄(ν) is said to be (a) uniformly better than f(ν) if r(f̄ \| ω) ≤ r(f\| ω) for all ω in Ω, the strict inequality holding for at least one ω, (b) admissible if no decision function is uniformly better than f̄(ν), and (c) minimax if ωεΩ r (f̄ \| ω) = inff ωεΩ r(f\| ω). The "problem of the greater mean" is, for any given Ω, to determine the minimax decision functions, particularly those which are also admissible. Special interest attaches to the case in which there exists a unique minimax decision function f̄(ν) (in the sense that if f(ν) is any minimax decision function then f(ν) = f̄(ν) for almost every ν in the sample space); such an f̄(ν) is automatically admissible. The problem of the greater mean is, of course, a special problem in Wald's general theory of statistical decision functions [1]. Our results will, however, be derived by very simple direct methods which make no use of Wald's general theorems. We cite without proofs a few examples in order to show how strongly the solution of the problem of the greater mean depends on the structure of Ω. In each case the minimax decision function is a function only of the two sample means x̄₁, x̄₂. (i) Let Ω' consist of the two points (a, b: σ) and (b, a: σ), with $a < b$. Then {equation*}{(2)} f^(ν) = {cases}1 if n_1x̄_1 - n_2x̄_2 > (n_1 - n_2)(a + b)/2,\\0 otherwise{cases}{equation*} is the unique minimax decision function. (ii) Let Ω" consist of the two points (c + h, c: σ) and (c - h, c: σ), with $h > 0$. Then {equation*}{(3)} f^0_c(ν) = {cases}1if x̄_1 > c\\0otherwise,{cases}{equation*} is the unique minimax decision function. (iii) Let Ω"' consist of the three points (1/2, - 1/2:1), (1/2, 3/2:1), (-3/2, -1/2:1), and let n₁ = n₂ = n. Then {equation*}{(4)}f^{}(ν) = {cases}1if e-2nx̄_1 + e2nx̄_2 < λ,\\0 otherwise{cases}{equation*}, where λ is a certain definite constant, is the unique minimax decision function. The parameter spaces of two or three points specified in these examples are rather trivial, but in fact the corresponding decision functions (2), (3), (4) remain the unique minimax solutions of the decision problem with respect to much more general parameter spaces. Thus, for example, it is clear that f⁽ν) will remain the unique minimax decision function with respect to any Ω which contains Ω' and is such that ωεΩ r(f^ \| ω) = ωεΩ' r(f^ \|ω). Corresponding remarks apply to f⁰c(ν) and f^(ν). When n₁ = n₂, (2) reduces to {equation*}{(5)}f^0(ν) = {cases}1 if x̄_1 > x̄_2,\\0 otherwise{cases}{equation*}. This decision function is of particular interest when both the means m₁, m₂ are unknown. It will be shown that whether or not n₁ = n₂, f⁰(ν) is the unique minimax decision function under certain conditions on Ω which are likely to hold in practice, at least when both n₁ and n₂ are sufficiently large (Theorem 3). Likewise, f⁰c(ν), which is the analogue of f⁰(ν) when one of the means (m₂) is known exactly, is apt to be the unique minimax decision function in such cases, at least when n₁ is sufficiently large (Theorem 4). These results on f⁰(ν) and f⁰c(ν) form the main results of the present paper. So much by way of a general summary. We shall now give a practical illustration (another is given in Section 3) to show how the problem of the greater mean arises in applications. Suppose that a consumer requires a certain number of manufactured articles which can be supplied at the same cost by each of two sources π₁ and π₂. The quality of an article is measured by a numerical characteristic x, and it is known that in the product of πᵢ, x is normally distributed with mean mᵢ and variance σ², but the values of these parameters are unknown. The consumer has obtained a random sample of n₁, and n₂ articles from π₁ and π₂ respectively, and has found the values of x to be (x₁₁, x₁₂, ⋯, x1n₁; x₂₁, x₂₂, ⋯, x2n₂) = ν. What is the best way of ordering a total of N articles from the two sources? The usual statistical theory, which confines itself to estimating the unknown parameters and to testing hypotheses of the form H₀(m₁ = m₂), has at best an indirect bearing on the problem at hand. We therefore adopt Wald's point of view and investigate the consequences of any given course of action. If the consumer orders $fN$ articles from π₁ and $(1 - f)N$ from π₂, where 0 ≤ f ≤ 1, then the expectation of the sum of the x-values in the articles he obtains will be N(m₁f + m₂(1 - f)). The maximum possible value of this quantity is N max m₁, m₂, and the "loss" per article which he sustains may therefore be taken as W(ω, f) = max m₁, m₂ - m₁f - m₂(1 - f) ≥ 0, where ω = (m₁, m₂: σ) is the true parameter point. The consumer wants to choose f so as to make W as small as possible. If he knew m₁ to be greater, or to be less, than m₂, then by choosing $f = 1$ or 0 respectively he could make $W = 0$. But since he does not know which m₁ is the greater he will presumably choose f as some function of the sample point ν. Suppose, therefore, that a "decision function" f(ν), such that 0 ≤ f(ν) leq 1 but not necessarily taking on only the values 0 and 1, is defined for all points ν in the sample space and that the consumer sets f = f(ν). In repeated applications of this procedure, the "risk" or expected loss (a double expectation is involved: the expected loss for a given f and the expected value of f in using the decision function f(ν)) per article is given by (1), and the consumer will try to find an f(ν) which minimizes this risk. Since the value of the risk depends on ω it is necessary to specify which values of ω are to be regarded as possible in the given problem; let the set of all such ω be denoted by Ω. If the consumer agrees to adopt the "conservative" criterion of minimizing the maximum possible risk, then the statistician's problem is to find the minimax decision functions in the sense defined above. We have given the solutions of this problem for certain types of parameter spaces. The reader will observe that each of the minimax decision functions (2), (3), (4) was of the "all or nothing" type, with values 0 and 1 only. (Whether this remains true for every Ω we do not know.) By using one of these decision functions in a given instance one arrives at either the best possible decision or the worst. The attitudes of doubt sometimes associated with the non-rejection of the hypothesis H₀(m₁ = m₂) are therefore irrelevant to the problem of the greater mean in the examples cited. (Cf. footnote 2; also Example 1 in Section 3.) The risk function (1) is but one of a general class R of risk functions, to be defined in Section 2, which are associated with the problem of the greater mean. The most important members of R are (1) and {equation*}{(6)}r̄(f\| ω) = P(incorrect decision using f(ν)\|ω),{equation*} where "m₁ ≤ m₂" and "m₁ ≥ m₂" are the two possible decisions. The risk function (6) is relevant to applications of a purely "scientific" nature in which the statistician is asked merely to give his opinion as to which population has the greater mean. Although the problem of constructing a suitable decision function for (6) is akin in spirit to the problems considered in the now classical Neyman-Pearson theory of statistical tests, no satisfactory solutions seem to be available. It is easy to see, however, that (1) and (6) are quite similar. Of course, in the case of (1) a decision function f(ν) may take on any value between 0 and 1 inclusive, while for (6) we allow only functions which take on only the values 0 and 1, corresponding respectively to the decisions "m₁ ≤ m₂" and "m₁ ≥ m₂". We then have for any such f(ν), {equation*}{(6')}r̄(fω) = {cases}P(f(ν) = 1 \| ω) = E f\| ω if m_1 < m_2,\(f(ν) = 0 \| ω) = E 1 - f \|ω if m_1 > m_2,\\0 if m_1 = m_2,{cases}{equation*} and by comparison with (1) we see that r(f\| ω) = \| m₁ - m₂\| r̄(f\| ω) for all ω. Now, in the three examples (i), (ii), (iii) cited above the unique minimax decision functions happen to take on only the values 0 and 1, and \|m₁ - m₂\| is constant on each of the respective parameter sets. It follows that (2), (3), (4) are also the unique minimax decision functions relative to (6) and to Ω', Ω", Ω''' respectively. The remarks above following Example (iii) also remain valid for the risk function (6). We conclude this section with a remark on the methods of this paper. Any decision function relevant to (6) is equivalent to a test of the hypothesis H₀(m₁ < m₂) against the alternative H₁(m₁ > m₂), the region \ν: f(ν) = 1\ being the "critical region." Hence the Neyman-Pearson probability ratio method can be used to obtain the unique minimax decision function with respect to (6) and an Ω consisting of two (or more) points, and the result carries over to more general types of Ω in the manner already indicated. It turns out, however, that the dominant properties of the probability ratio tests are not confined to the class of tests alone, but extend to the class of all functions f(ν) such that 0 ≤ f(ν) ≤ 1. This result (Theorem 1) enables us to solve the problem of the greater mean for the risk function (1) as well as for (6). The reader who is interested in applications may turn to Section 3.