NON-EXISTENCE OF AN EXACT β-CONFIDENCE INTERVAL FOR A SCALE PARAMETER*

Abstract There are given k (≥22) independent distributions with c.d.f.'s F(x;θ_j) indexed by a scale parameter θ_j, j = 1,…, k. Let θ_[i] (i = 1,…, k) denote the ith smallest one of θ₁,…, θ_k. In this paper we wish to show that, under some regularity conditions, there does not exist an exact β-level (0≤β1) confidence interval for the ith smallest scale parameter θ_i based on k independent samples. Since the log transformation method may not yield the desired results for the scale parameter problem, we will treat the scale parameter case directly without transformation. Application is considered for normal variances. Two conservative one-sided confidence intervals for the ith smallest normal variance and the percentage points needed to actually apply the intervals are provided.     

Simultaneous selection of extreme populations:a bayesian approach

The problem of simultaneously selecting two non-empty subsets, S_Land S_U, of k populations which contain the lower extreme population (LEP) and the upper extreme population (UEP), respectively, is considered. Unknown parameters θ₁,…,θ_kcharacterize the populations π₁,…,π_kand the populations associated with θ_[1]=min θ_i. and θ_[k]= max θ_i. are called the LEP and the UEP, respectively. It is assumed that the underlying distributions possess the monotone likelihood ratio property and that the prior distribution of θ= (θ₁,…,θ_k) is exchangeable. The Bayes rule with respect to a general loss function is obtained. Bayes rule with respect to a semi-additive and non-negative loss function is also determined and it is shown that it is minimax and admissible. When the selected subsets are required to be disjoint, it shown that the Bayes rule with respect to a specific loss function can be obtained by comparing certain computable integrals, Application to normal distributions with unknown means θ₁,…,θ_kand a common known variance is also considered.     

Some subset selection problems

Consider that we have a collection of k populations π₁, π₂…,π_k. The quality of the ith population is characterized by a real parameter θ_i and the population is to be designated as superior or inferior depending on how much the θ_i differs from θ_max = max{θ₁, θ₂,…,θ_k}. From the set {π₁, π₂,…,π_k}, we wish to select the subset of superior populations. In this paper we devise rules of selection which have the property that their selected set excludes all the inferior populations with probability at least 1?α, where a is a specified number.     

Distributions of certain variables in samples from two-parameter exponential distribution

Cumulative distribution function of the variable Y=(U+c)/(Z/2ν)) is given. Here U and Z are independent random variables, U has the exponential distribution (1.1) with θ=0, σ=1, Z has the distribution χ² (2ν) and c is a real quantity. The variable Y with U and Z given by (2.2) and (2.3) is used for inference about the parametric functions ?=θ?kσ of a two-parameter exponential distribution (1.1) with k or ? known. Special cases of ? or k are: the parameter θ, the Pth quantile Xp, the mean θ+σ and the value of the cumulative distribution function or of the reliability function at given point a. Also one-sided tolerance limits for a two-parameter exponential distribution can be derived from the distribution of the variable Y. The results are also applied to the Pareto distribution.     

Estimating the parameter of selected uniform population under the squared log error loss function

Let π₁, …, π_k be k (? 2) independent populations, where π_i denotes the uniform distribution over the interval (0, θ_i) and θ_i > 0 (i = 1, …, k) is an unknown scale parameter. The population associated with the largest scale parameter is called the best population. For selecting the best population, We use a selection rule based on the natural estimators of θ_i, i = 1, …, k, for the case of unequal sample sizes. Consider the problem of estimating the scale parameter θ_L of the selected uniform population when sample sizes are unequal and the loss is measured by the squared log error (SLE) loss function. We derive the uniformly minimum risk unbiased (UMRU) estimator of θ_L under the SLE loss function and two natural estimators of θ_L are also studied. For k = 2, we derive a sufficient condition for inadmissibility of an estimator of θ_L. Using these condition, we conclude that the UMRU estimator and natural estimator are inadmissible. Finally, the risk functions of various competing estimators of θ_L are compared through simulation.     

Subset selection based on likelihood from uniform and related popupulations

Let π_i (i=1,2,…, k) be charceterized by the uniform distribution on (a_i;b_i), where exactly one of a_i and b_i is unknown. With unequal sample sizes, suppose that from the k (>=2) given populations, we wish to select a random-size subset containing the one with the smllest value of θ_i= b_i - a_i. RuleR_i selects π if a likelihood-based k-dimensional confidence region for the unknown (θ₁,… θ_k) contains at least one point having θ_i as its smallest component. A second rule, R , is derived through a likelihood ratio and turns out to be that of Barr and prabhu whenthe sample sizes are equal. Numerical comparisons are made. The results apply to the larger class of densities g ( z ; θ_i) =M(z)Q(θ_i) if a(θ_i) < z <b(θ_i). Extensions to the cases when both a_i and b_i are unknown and when θ_j isof interest are indicated. 1<=j<=k     

Complete ranking procedures with quadratic loss

Suppose that one wishes to rank k normal populations, each with common variance σ² and unknown means θ_i (i=1,2,…,k). Independent samples of size n are taken from each population, and the sample averages are used to rank the populations. In this paper, we investigate what sample sizes, n, are necessary to attain “good” rankings under various loss functions. Section discusses various loss functions and their interpretation. Section 2 gives the solution for a reasonable non-parametric loss function. Section 3 gives the solution for a reasonable parameteric loss function.     

Improving on truncated linear estimates of exponential and gamma scale parameters

We consider the problem of estimating the scale parameter of an exponential or a gamma distribution under squared error loss when the scale parameter θ is known to be greater than some fixed value θ₀. Natural estimators in this setting include truncated linear functions of the sufficient statistic. Such estimators are typically inadmissible, but explicit improvements seem difficult to find. Some are presented here. A particularly interesting finding is that estimators which are admissible in the untruncated problem which take values only in the interior of the truncated parameter space are found to be inadmissible for the truncated problem.     

Non-parametric estimation of an acceleration parameter

Let X₁,… X_m be a random sample of m failure times under normal conditions with the underlying distribution F(x) and Y₁,…,Y_n a random sample of n failure times under accelerated condititons with underlying distribution G(x);G(x)=1?[1?F(x)]^θ with θ being the unknown parameter under study.Define:U_ij=1 otherwise.The joint distribution of _ijdoes not involve the distribution F and thus can be used to estimate the acceleration parameter θ.The second approach for estimating θ is to use the ranks of the Y-observations in the combined X- and Y-samples.In this paper we establish that the rank of the Y-observations in the pooled sample form a sufficient statistic for the information contained in the U_ii 's about the parameter θ and that there does not exist an unbiassed estimator for the parameter θ.We also construct several estimators and confidence interavals for the parameter θ.     

Some multiple decision problems in analysis of variance

In most practical situations to which the analysis of variance tests are applied, they do not supply the information that the experimenter aims at. If, for example, in one-way ANOVA the hypothesis is rejected in actual application of the F-test, the resulting conclusion that the true means θ₁,…,θ_k are not all equal, would by itself usually be insufficient to satisfy the experimenter. In fact his problems would begin at this stage. The experimenter may desire to select the “best” population or a subset of the “good” populations; he may like to rank the populations in order of “goodness” or he may like to draw some other inferences about the parameters of interest.

The extensive literature on selection and ranking procedures depends heavily on the use of independence between populations (block, treatments, etc.) in the analysis of variance. In practical applications, it is desirable to drop this assumption or independence and consider cases more general than the normal.

In the present paper, we derive a method to construct optimal (in some sense) selection procedures to select a nonempty subset of the k populations containing the best population as ranked in terms of θ_i’s which control the size of the selected subset and which maximizes the minimum average probability of selecting the best. We also consider the usual selection procedures in one-way ANOVA based on the generalized least squares estimates and apply the method to two-way layout case. Some examples are discussed and some results on comparisons with other procedures are also obtained.     

Model Selection Using Conditional Densities

In this article we propose a new method to select a discrete model f(x; θ), based on the conditional density of a sample given the value of a sufficient statistic for θ. The main idea is to work with a broad family of discrete distributions, called the family of power series distribution, for which there is a common sufficient statistic for the parameter of interest. The proposed method uses the maximum conditional density in order to select the best model.

We compare our proposal with the usual methodology based on Bayes factors. We provide several examples that show that our proposal works fine in most instances. Bayes factors are strongly dependent on the prior information about the parameters. Since our method does not require the specification of a prior distribution, it provides a useful alternative to Bayes factors.     

A one-sided test for testino homogeneity of scale parameters against simple ordered alternative

In an earlier paper the authors (1997) extended the results of Hayter (1990) to the two parameter exponential probability model. This paper addressee the extention to the scale parameter case under location-scale probability model. Consider k (k≧3) treatments or competing firms such that an observation from with treatment or firm follows a distribution with cumulative distribution function (cdf) F_i(x)=F[(x-μ_i)/Q_i], where F(·) is any absolutely continuous cdf, i=1,…,k. We propose a test to test the null hypothesis H₀:θ₁=…=θ_k against the simple ordered alternative H₁:θ₁≦…≦θ_k, with at least one strict inequality, using the data X_i,j, i=1,…k; j=1,…,n₁. Two methods to compute the critical points of the proposed test have been demonstrated by talking k two parameter exponential distributions. The test procedure also allows us to construct simultaneous one sided confidence intervals (SOCIs) for the ordered pairwise ratios θ_j/θ_i, 1≦i<j≦k. Statistical simulation revealed that: 9i) actual sizes of the critical points are almost conservative and (ii) power of the proposed test relative to some existing tests is higher.     

Estimation of the location parameter and the average worth of the selected subset of two parameter exponential populations under LINEX loss function

Suppose a subset of populations is selected from k exponential populations with unknown location parameters θ₁, θ₂, …, θ_k and common known scale parameter σ. We consider the estimation of the location parameter of the selected population and the average worth of the selected subset under an asymmetric LINEX loss function. We show that the natural estimator of these parameters is biased and find the uniformly minimum risk-unbiased (UMRU) estimator of these parameters. In the case of k = 2, we find the minimax estimator of the location parameter of the smallest selected population. Furthermore, we compare numerically the risk of UMRU, minimax, and the natural estimators.     

Estimation After Selection Under Reflected Normal Loss Function

Let X ₁ and X ₂ be two independent random variables from normal populations Π₁, Π₂ with different unknown location parameters θ₁ and θ₂, respectively and common known scale parameter σ. Let X ₍₂₎ = max (X ₁, X ₂) and X ₍₁₎ = min (X ₁, X ₂). We consider the problem of estimating the location parameter θ _M (or θ _J ) of the selected population under the reflected normal loss function. We obtain minimax estimators of θ _M and θ _J . Also, we provide sufficient conditions for the inadmissibility of invariant estimators of θ _M and θ _J .     

On a conjecture concerning the least favorable configuration of certain two-stage selection procedures

Given k normal populations with unknown means and a common known variance a two-stage procedure p₁ with screening in the first stage to find the population with the largest mean using the indifference-zone approach is under concern. It was proposed and studied previously by Cohen (1959), Alam (1970) and Tamhane and Bechhofer (1977, 1979). But up to now a conjecture concerning the least favorable parameter configuration of p₁remained unproved for k ≥ 3. In this paper we give a non-standard proof of the conjecture in case of k = 3 for p₁. which (under minor changes) works also for a simplified version of p₁,. Besides, a counterexample is provided to show that another (more intuitive) method of proof fails to work.     

SOME PROPERTIES OF PROFILE BOOTSTRAP CONFIDENCE INTERVALS

We consider the problem of finding an equi-tailed confidence interval, with coverage probability (1-α), for a scalar parameter θ₀ in the presence of a (possibly infinite dimensional) nuisance parameter ψ₀. It is supposed that the value taken by θ₀ does not restrict the value that ψ₀ may take and vice-versa. Given a sensible estimate ψ_n of ψ₀, profile bootstrap confidence interval for θ₀ is defined to be the exact equi-tailed confidence interval with coverage probability (1-α) assuming that ψ₀=ψ_n. We compare the properties of the profile bootstrap confidence interval and the ordinary bootstrap confidence interval when they are based on studentised and unstudentised quantities. Under mild regularity conditions the profile bootstrap confidence interval is always a subset of the set of allowable values of θ₀ and is transformation-respecting when based on either an unstudentised quantity or a studentised quantity satisfying certain restrictions. As a confidence interval for the autoregressive parameter of an AR(1) process, the profile bootstrap confidence interval has important advantages over the ordinary bootstrap confidence interval based on a studentised quantity.     

Some Pitman closeness properties pertinent to symmetric populations

In this paper, we focus on Pitman closeness probabilities when the estimators are symmetrically distributed about the unknown parameter θ. We first consider two symmetric estimators θ?₁ and θ?₂ and obtain necessary and sufficient conditions for θ?₁ to be Pitman closer to the common median θ than θ?₂. We then establish some properties in the context of estimation under the Pitman closeness criterion. We define Pitman closeness probability which measures the frequency with which an individual order statistic is Pitman closer to θ than some symmetric estimator. We show that, for symmetric populations, the sample median is Pitman closer to the population median than any other independent and symmetrically distributed estimator of θ. Finally, we discuss the use of Pitman closeness probabilities in the determination of an optimal ranked set sampling scheme (denoted by RSS) for the estimation of the population median when the underlying distribution is symmetric. We show that the best RSS scheme from symmetric populations in the sense of Pitman closeness is the median and randomized median RSS for the cases of odd and even sample sizes, respectively.     

A note on a modified double stage shrinkage estimator

A modified double stage shrinkage estimator has been proposed for the single parameter θ of a distribution function _Fθ. It is shown to be locally better in comparison to the usual double stage shrinkage estimator in the sense of smaller mean squared error in a certain neighbourhood of prior estimate θ_o of θ.