5.1 Uniformly Most Accurate Families of Confidence

Size: px

Start display at page:

Download "5.1 Uniformly Most Accurate Families of Confidence"

Hillary Garrison
5 years ago
Views:

1 Chapter 5 Confidence Estimation Let (X, B X, P ), P {P θ θ Θ} be sample space and {Θ, B Θ, µ} be measure space with some σ finite measure µ. 5.1 Uniformly Most Accurate Families of Confidence Sets Def : Let S : X B Θ be a measurable function. A family {S(x) x X }, where S(x) depends on x but not on θ, is called a family of random sets. If Θ IR and S(x) is an interval (θ(x), θ(x)), where θ and θ are functions of x alone, then we call S(x) a random interval with θ(x) and θ(x) as lower and upper bounds, respectively. θ(x) may be and θ(x) may be +. The problem of confidence estimation consists in finding a function S : X B Θ such that for a given α P θ {S(X) θ} 1 α. (5.1) Def : Let Θ IR and 0 < α < 1. A function θ(x) satisfying P θ {θ(x) θ} 1 α for all θ Θ. (5.2) is called a lower confidence bound for θ at confidence level 1 α. The quantity inf P θ{θ(x) θ} (5.3) θ Θ 49

2 50 CHAPTER 5. CONFIDENCE ESTIMATION is called confidence coefficient. A function θ(x) satisfying P θ {θ(x) θ} 1 α for all θ Θ is called an upper confidence bound for θ at confidence level 1 α and the quantity inf P θ{θ θ(x)} θ Θ confidence coefficient. Def 5.1.3: A function θ(θ) that minimizes P θ {θ(x) θ } P θ {θ θ(x)} for all θ < θ (θ > θ) subject to (5.2) is called a uniformly most accurate (UMA) lower (upper) confidence bound at confidence level 1 α. If S(X) is of the form S(X) = (θ(x, θ(x)) such that P θ {θ(x) θ θ(x)} 1 α for all θ Θ, we call it a confidence interval at confidence level 1 α and inf θ P θ{θ(x) < θ < θ(x)} the confidence coefficient associated with the random interval. Def : A family of 1 α level confidence sets {S(x)} is called a UMA family of confidence sets at level 1 α if P θ {S(X) contains θ} P θ {S (X) contains θ} for all θ, θ Θ and 1 α level family {S (X} of confidence sets. Theorem 5.1.1: Let Θ IR be an interval and T (X; θ) be such that for each θ, T as a measurable function is strictly monotone (antitone) in θ at every x X. Let Λ be the range of T and for every λ Λ and x X let the equation λ = T (x; θ) be solvable. If the distribution of T (X; θ) is independent of θ, then one can construct a confidence interval for θ at any level.

3 5.2. CONFIDENCE INTERVALS OF SHORTEST-LENGTH Confidence Intervals of Shortest-Length Def : A random variable T (X; θ), whose distribution is independent of θ, is called a pivot. Remark: An alternative to minimizing the length of a given confidence interval consists in the minimization of the expected length E θ [θ(x) θ(x)]. In general there does not exist an element in the class of 1 α confidence sets which minimizes E θ [θ(x) θ(x)] for all θ Θ. On the other hand, procedures based on a pivot are also applicable for finding CI s with minimal expected length. 5.3 Relation between Confidence Estimation Hypotheses Testing In the sequel we consider only nonrandomized tests and introduce the short hand notation H 0 (θ 0 ) for H 0 : θ = θ 0 and H 1 (θ 1 ) for the alternative, which may be one or two-sided. Theorem 5.3.1: Let A(θ 0 ), θ 0 Θ, denote the region of acceptance of an α level test of H 0 (θ 0 ), and for each x X let S(x) = {θ x A(θ), θ Θ}, (A(θ 0 ) B X, S(x) B Θ ). Then S(x) is a family of confidence sets for θ at confidence level 1 α. If, moreover, A(θ 0 ) is UMP for the problem (α, H 0 (θ 0 ), H 1 (θ 0 )), then S(X) minimizes P θ {S(X) θ } for all θ H 1 (θ ), among all 1 α level families of confidence sets. Remark: In view of Def , the family of confidence sets associated with a UMP acceptance region is a UMA family at level 1 α.

4 52 CHAPTER 5. CONFIDENCE ESTIMATION 5.4 Unbiased Confidence Sets Def : A family {S(x)} of confidence sets for a parameter θ is said to be unbiased at confidence level 1 α if (i) P θ {S(X) contains θ} 1 α and (ii) P θ{s(x) contains θ} 1 α for all θ, θ Θ. If S(X) is an interval satisfying (i) and (ii), we call it a 1 α level unbiased confidence interval. If a family of unbiased 1 α confidence sets is UMA in the class of all 1 α level unbiased confidence sets, we call it a UMA unbiased (UMAU) family of confidence sets at level 1 α. If S (X) satisfies (i) and (ii) and minimizes P θ {S(X) contains θ } for all θ, θ Θ among all unbiased families of confidence sets at level 1 α, then S (X) is a UMAU family of confidence sets at level 1 α. According to the above definition a family S(X) of confidence sets for a parameter θ is unbiased at level 1 α if S(X) traps the true parameter value with probability at least 1 α and S(X) traps a false parameter value with a probability at most 1 α. Hence un unbiased set traps a true parameter value more often than it does a false one. Theorem 5.4.1: Let A(θ 0 ) be the acceptance region of a UMPU size α test of H 0 (θ 0 ) : θ = θ 0 against H 1 (θ 0 ) : θ θ 0 for each θ 0. Then S(x) = {θ Θ : x A(θ)} is a UMAU family of confidence sets at level 1 α. Remarks (1) The concept of unbiasedness is most suitable in situations where UMP tests do not exist. This is, e.g., the case, if Θ consists of points (θ, τ), both unknown, and one is interested in obtaining confidence sets for θ alone. The parameter τ is usually refered to as a nuisance parameter. (2) Shortest-length confidence intervals do not exist for most commonly used distributions. The restriction to the unbiased family of confidence intervals makes it often possible ot obtain 1 α level confidence intervals

5 5.4. UNBIASED CONFIDENCE SETS 53 that have uniformly minimum expected length among all 1 α level unbiased confidence intervals. Theorem 5.4.2: Let Θ IR be an interval, let f θ be a µ density for P = {P θ θ Θ} µ and let S(X) = (θ(x), θ(x)) be a family of 1 α level confidence intervals with stochastically finite length, i.e. P θ (θ(x) θ(x) < ) = 1. Then [(θ(x) θ(x)]f θ (x)µ(dx) = P θ {S(X) contains θ }dθ θ θ for all θ Θ. The Theorem says that the expected length of the confidence interval is the probability that S(X) includes θ, averaged over all false values of θ.

Interval Estimation. Chapter 9

Interval Estimation. Chapter 9 Chapter 9 Interval Estimation 9.1 Introduction Definition 9.1.1 An interval estimate of a real-values parameter θ is any pair of functions, L(x 1,..., x n ) and U(x 1,..., x n ), of a sample that satisfy