Estimation of the mean of the selected gamma population

Let л₁ and л₂ denote two independent gamma populations G(α₁, p) and G(α₂, p) respectively. Assume α(i=1,2)are unknown and the common shape parameter p is a known positive integer. Let Y_i denote the sample mean based on a random sample of size n from the i-th population. For selecting the population with the larger mean, we consider, the natural rule according to which the population corresponding to the larger Y_i is selected. We consider? in this paper, the estimation of M, the mean of the selected population. It is shown that the natural estimator is positively biased. We obtain the uniformly minimum variance unbiased estimator(UMVE) of M. We also consider certain subclasses of estikmators of the form c₁x₍₁₎ +c₁x₍₂₎ and derive admissible estimators in these classes. The minimazity of certain estimators of interest is investigated. Itis shown that p(p+1)^-1x₍₁₎ is minimax and dominates the UMVUE. Also UMVUE is not minimax.     

Estimating the mean of the selected uniform population

Let π_i(i=1,2,…K) be independent U(0,?_i) populations. Let Y_i denote the largest observation based on a random sample of size n from the i-th population. for selecting the best populaton, that is the one associated with the largest ?_i, we consider the natural selection rule, according to which the population corresponding to the largest Y_i is selected. In this paper, the estimation of M. the mean of the selected population is considered. The natural estimator is positively biased. The UMVUE (uniformly minimum variance unbiased estimator) of M is derived using the (U,V)-method of Robbins (1987) and its asymptotic distribution is found. We obtain a minimax estimator of M for K≤4 and a class of admissible estimators among those of the form cY_max. For the case K = 2, the UMVUE is improved using the Brewster-Zidek (1974) Technique with respect to the squared error loss function L₁ and the scale-invariant loss function L₂. For the case K = 2, the MSE'S of all the estimators are compared for selected values of n and ρ=?₁/(?₁+?₂).     

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.     

Estimating Average Worth of the Selected Subset from Two-Parameter Exponential Populations

ABSTRACT

Suppose independent random samples are available from k(k ≥ 2) exponential populations ∏₁,…,∏ _k with a common location θ and scale parameters σ₁,…,σ _k , respectively. Let X _i and Y _i denote the minimum and the mean, respectively, of the ith sample, and further let X = min{X ₁,…, X _k } and T _i  = Y _i  ? X; i = 1,…, k. For selecting a nonempty subset of {∏₁,…,∏ _k } containing the best population (the one associated with max{σ₁,…,σ _k }), we use the decision rule which selects ∏ _i if T _i  ≥ c max{T ₁,…,T _k }, i = 1,…, k. Here 0 < c ≤ 1 is chosen so that the probability of including the best population in the selected subset is at least P* (1/k ≤ P* < 1), a pre-assigned level. The problem is to estimate the average worth W of the selected subset, the arithmetic average of means of selected populations. In this article, we derive the uniformly minimum variance unbiased estimator (UMVUE) of W. The bias and risk function of the UMVUE are compared numerically with those of analogs of the best affine equivariant estimator (BAEE) and the maximum likelihood estimator (MLE).     

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.     

Estimating parameters of a selected Pareto population

Let Π₁,…,Π_k be k populations with Π_i being Pareto with unknown scale parameter α_i and known shape parameter β_i;i=1,…,k. Suppose independent random samples (X_i1,…,X_in), i=1,…,k of equal size are drawn from each of k populations and let X_i denote the smallest observation of the ith sample. The population corresponding to the largest X_i is selected. We consider the problem of estimating the scale parameter of the selected population and obtain the uniformly minimum variance unbiased estimator (UMVUE) when the shape parameters are assumed to be equal. An admissible class of linear estimators is derived. Further, a general inadmissibility result for the scale equivariant estimators is proved.     

Estimating a positive normal mean

Let X ₁, X ₂, ..., X _n be a random sample from a normal population with mean μ and variance σ ². In many real life situations, specially in lifetime or reliability estimation, the parameter μ is known a priori to lie in an interval [a, ∞). This makes the usual maximum likelihood estimator (MLE) ̄ an inadmissible estimator of μ with respect to the squared error loss. This is due to the fact that it may take values outside the parameter space. Katz (1961) and Gupta and Rohatgi (1980) proposed estimators which lie completely in the given interval. In this paper we derive some new estimators for μ and present a comparative study of the risk performance of these estimators. Both the known and unknown variance cases have been explored. The new estimators are shown to have superior risk performance over the existing ones over large portions of the parameter space.     

Sequential experimental design and response optimisation

We consider the situation where one wants to maximise a functionf(θ,x) with respect tox, with θ unknown and estimated from observationsy _k . This may correspond to the case of a regression model, where one observesy _k =f(θ,x _k )+ε _k , with ε _k some random error, or to the Bernoulli case wherey _k ∈{0, 1}, with Pr[y _k =1|θ,x _k |=f(θ,x _k ). Special attention is given to sequences given by , with an estimated value of θ obtained from (x₁, y₁),...,(x _k ,y _k ) andd _k (x) a penalty for poor estimation. Approximately optimal rules are suggested in the linear regression case with a finite horizon, where one wants to maximize ∑ _i=1 ^N w _i f(θ, x _i ) with {w _i } a weighting sequence. Various examples are presented, with a comparison with a Polya urn design and an up-and-down method for a binary response problem.     

Estimation of the Scale Parameter of the Selected Gamma Population Under the Entropy Loss Function

Let X ₁, X ₂,…, X _k be k (≥2) independent random variables from gamma populations Π₁, Π₂,…, Π _k with common known shape parameter α and unknown scale parameter θ _i , i = 1,2,…,k, respectively. Let X _(i) denotes the ith order statistics of X ₁,X ₂,…,X _k . Suppose the population corresponding to largest X _(k) (or the smallest X ₍₁₎) observation is selected. We consider the problem of estimating the scale parameter θ _M (or θ _J ) of the selected population under the entropy loss function. For k ≥ 2, we obtain the Unique Minimum Risk Unbiased (UMRU) estimator of θ _M (and θ _J ). For k = 2, we derive the class of all linear admissible estimators of the form cX ₍₂₎ (and cX ₍₁₎) and show that the UMRU estimator of θ _M is inadmissible. The results are extended to some subclass of exponential family.     

A note on the admissibility of hammersley's estimator of an integer mean

Let X₁, X₂, …, X_n be identically, independently distributed N(i,1) random variables, where i = 0, ±1, ±2, … Hammersley (1950) showed that d = [X?_n], the nearest integer to the sample mean, is the maximum likelihood estimator of i. Khan (1973) showed that d is minimax and admissible with respect to zero-one loss. This note now proves a conjecture of Stein to the effect that in the class of integer-valued estimators d is minimax and admissible under squared-error loss.     

Average worth estimation of the selected subset of Poisson populations

Let Π₁, …, Π _p be p(p≥2) independent Poisson populations with unknown parameters θ₁, …, θ _p , respectively. Let X _i denote an observation from the population Π _i , 1≤i≤p. Suppose a subset of random size, which includes the best population corresponding to the largest (smallest) θ _i , is selected using Gupta and Huang [On subset selection procedures for Poisson populations and some applications to the multinomial selection problems, in Applied Statistics, R.P. Gupta, ed., North-Holland, Amsterdam, 1975, pp. 97–109] and (Gupta et al. [On subset selection procedures for Poisson populations, Bull. Malaysian Math. Soc. 2 (1979), pp. 89–110]) selection rule. In this paper, the problem of estimating the average worth of the selected subset is considered under the squared error loss function. The natural estimator is shown to be biased and the UMVUE is obtained using Robbins [The UV method of estimation, in Statistical Decision Theory and Related Topics-IV, S.S. Gupta and J.O. Berger, eds., Springer, New York, vol. 1, 1988, pp. 265–270] UV method of estimation. The natural estimator is shown to be inadmissible, by constructing a class of dominating estimators. Using Monte Carlo simulations, the bias and risk of the natural, dominated and UMVU estimators are computed and compared.     

Comparison of estimation methods for the finite population mean in simple random sampling: symmetric super-populations

In this paper, a new estimator combined estimator (CE) is proposed for estimating the finite population mean ¯ Y _N in simple random sampling assuming a long-tailed symmetric super-population model. The efficiency and robustness properties of the CE is compared with the widely used and well-known estimators of the finite population mean ¯ Y _N by Monte Carlo simulation. The parameter estimators considered in this study are the classical least squares estimator, trimmed mean, winsorized mean, trimmed L-mean, modified maximum-likelihood estimator, Huber estimator (W24) and the non-parametric Hodges–Lehmann estimator. The mean square error criteria are used to compare the performance of the estimators. We show that the CE is overall more efficient than the other estimators. The CE is also shown to be more robust for estimating the finite population mean ¯ Y _N , since it is insensitive to outliers and to misspecification of the distribution. We give a real life example.     

Bayesian and Robust Bayesian analysis under a general class of balanced loss functions

For estimating an unknown parameter θ, we introduce and motivate the use of balanced loss functions of the form L_{r, w, d0}(q, d)=wr(d₀, d)+ (1-w) r(q, d){L_{\rho, \omega, \delta_0}(\theta, \delta)=\omega \rho(\delta_0, \delta)+ (1-\omega) \rho(\theta, \delta)}, as well as the weighted version q(q) L_{r, w, d0}(q, d){q(\theta) L_{\rho, \omega, \delta_0}(\theta, \delta)}, where ρ(θ, δ) is an arbitrary loss function, δ ₀ is a chosen a priori “target” estimator of q, w ? [0,1){\theta, \omega \in[0,1)}, and q(·) is a positive weight function. we develop Bayesian estimators under L_{r, w, d0}{L_{\rho, \omega, \delta_0}} with ω > 0 by relating such estimators to Bayesian solutions under L_{r, w, d0}{L_{\rho, \omega, \delta_0}} with ω = 0. Illustrations are given for various choices of ρ, such as absolute value, entropy, linex, and squared error type losses. Finally, under various robust Bayesian analysis criteria including posterior regret gamma-minimaxity, conditional gamma-minimaxity, and most stable, we establish explicit connections between optimal actions derived under balanced and unbalanced losses.     

Simultaneous estimation of ordered parameters

The problem of estimating ordered parameters is encountered in biological, agricultural, reliability and various other experiments. Consider two populations with densities f₁(x₁-ω₁) and f₂(x₂-ω₂) where ω₁#ω₂. The estimation of ω₁,ω₂) with the loss function, the sum of squared errors, is studied. when f_i is the f_i(,_i,,_i ²) density with ,_i known, i=1,2; we obtain a class of minimax estimators. When ω₁ #ω₂ we show some of these estimators are improved by the maximum likelihood estimator. For a general f_i we give sufficient conditions for the minimaxity of the analogue of the Pitman estimator.     

A distribution function estimator for the difference of order statistics from two independent samples

For two independent populations X and Y we develop the empirical distribution function estimator for the difference of order statistics of the form X _(i)−Y _(j). The key practical application for this estimator pertains to inference between quantiles from two independent populations.     

Estimation of the derivative of a density and related functions

Let Y₁,…,Y _n, (Y₁ <Y₂<…<Y _n) be the order statistics of a random sample from a distribution F with density f on the realline. This paper gives a class of estimators of the derivativef'(x) of the density f at points x for which f has

a continuoussecond derivative. These estimators are based on spacings inthe order statistics Y_j+kn -y _j j = 1,…,n-k_n,k_n<n.     

Inference on finite population categorical response: nonparametric regression-based predictive approach

Suppose that a finite population consists of N distinct units. Associated with the ith unit is a polychotomous response vector, d _i , and a vector of auxiliary variable x _i . The values x _i ’s are known for the entire population but d _i ’s are known only for the units selected in the sample. The problem is to estimate the finite population proportion vector P. One of the fundamental questions in finite population sampling is how to make use of the complete auxiliary information effectively at the estimation stage. In this article a predictive estimator is proposed which incorporates the auxiliary information at the estimation stage by invoking a superpopulation model. However, the use of such estimators is often criticized since the working superpopulation model may not be correct. To protect the predictive estimator from the possible model failure, a nonparametric regression model is considered in the superpopulation. The asymptotic properties of the proposed estimator are derived and also a bootstrap-based hybrid re-sampling method for estimating the variance of the proposed estimator is developed. Results of a simulation study are reported on the performances of the predictive estimator and its re-sampling-based variance estimator from the model-based viewpoint. Finally, a data survey related to the opinions of 686 individuals on the cause of addiction is used for an empirical study to investigate the performance of the nonparametric predictive estimator from the design-based viewpoint.     

VARIANCE ESTIMATION IN TWO-PHASE SAMPLING

Two‐phase sampling is often used for estimating a population total or mean when the cost per unit of collecting auxiliary variables, x, is much smaller than the cost per unit of measuring a characteristic of interest, y. In the first phase, a large sample s₁ is drawn according to a specific sampling design p(s₁) , and auxiliary data x are observed for the units i∈s₁ . Given the first‐phase sample s₁ , a second‐phase sample s₂ is selected from s₁ according to a specified sampling design {p(s₂∣s₁) } , and (y, x) is observed for the units i∈s₂ . In some cases, the population totals of some components of x may also be known. Two‐phase sampling is used for stratification at the second phase or both phases and for regression estimation. Horvitz–Thompson‐type variance estimators are used for variance estimation. However, the Horvitz–Thompson ( Horvitz & Thompson, J. Amer. Statist. Assoc. 1952 ) variance estimator in uni‐phase sampling is known to be highly unstable and may take negative values when the units are selected with unequal probabilities. On the other hand, the Sen–Yates–Grundy variance estimator is relatively stable and non‐negative for several unequal probability sampling designs with fixed sample sizes. In this paper, we extend the Sen–Yates–Grundy ( Sen , J. Ind. Soc. Agric. Statist. 1953; Yates & Grundy , J. Roy. Statist. Soc. Ser. B 1953) variance estimator to two‐phase sampling, assuming fixed first‐phase sample size and fixed second‐phase sample size given the first‐phase sample. We apply the new variance estimators to two‐phase sampling designs with stratification at the second phase or both phases. We also develop Sen–Yates–Grundy‐type variance estimators of the two‐phase regression estimators that make use of the first‐phase auxiliary data and known population totals of some of the auxiliary variables.