Improved estimation of scale parameters of morgenstern type bivariate uniform distribution using ranked set sampling

ABSTRACT

In this article we suggest some improved version of estimators of scale parameter of Morgenstern-type bivariate uniform distribution (MTBUD) based on the observations made on the units of the ranked set sampling regarding the study variable Y which is correlated with the auxiliary variable X, when (X, Y) follows a MTBUD. We also suggest some linear shrinkage estimators of scale parameter of Morgenstern type bivariate uniform distribution (MTBUD). Efficiency comparisons are also made in this work.     

Estimation of parameters of a pareto distribution by generalized order statistics

In this paper we address the problem of estimating the parameters of Pareto II distribution based on generalized order statistics. The estimators based on order statistics and record values are shown to be special cases of these estimators.     

Estimation of location and scale parameters of a distribution by U-statistics based on best linear functions of order statistics

In this work we propose a technique of estimating the location parameter μ$\mu$ and scale parameter σ$\sigma$ of a distribution by U-statistics constructed by taking best linear functions of order statistics as kernels. The method has been illustrated for estimating the location and scale parameters of type-I extreme value distribution. We have computed the asymptotic relative efficiencies of the proposed U-statistics with the appropriate maximum likelihood estimators based on samples drawn from each of type-I extreme value, logistic and normal distributions. In all cases very high asymptotic relative efficiencies are obtained.     

Estimation of the parameters of Downton's bivariate exponential distribution using ranked set sampling scheme

The parameters of Downton's bivariate exponential distribution are estimated based on a ranked set sample. Parametric and nonparametric methods are considered. The suggested estimators are compared to the corresponding ones based on simple random sampling. It turns out that some of the suggested estimators are significantly more efficient than the ones based on simple random sampling.     

Linear estimation of location and scale parameters in the generalized gamma distribution

The expressions for moments of order statistics from the generalized gamma distribution are derived. Coefficients to get the BLUEs of location and scale parameters in the generalized gamma distribution are computed. Some simple alternative linear unbiased estimates of location and scale parameters are also proposed and their relative efficiencies compared to the BLUEs are studied.     

Distribution of concomitants of order statistics and their order statistics

For a random sample of size n$n$ from an absolutely continuous random vector (X,Y)$(X,Y)$, let _Yi:n$Y_{i:n}$ be i$i$th Y$Y$-order statistic and _Y[j:n]$Y_{[j:n]}$ be the Y$Y$-concomitant of _Xj:n$X_{j:n}$. We determine the joint pdf of _Yi:n$Y_{i:n}$ and _Y[j:n]$Y_{[j:n]}$ for all i,j=1$i,j=1$ to n$n$, and establish some symmetry properties of the joint distribution for symmetric populations. We discuss the uses of the joint distribution in the computation of moments and probabilities of various ranks for _Y[j:n]$Y_{[j:n]}$. We also show how our results can be used to determine the expected cost of mismatch in broken bivariate samples and approximate the first two moments of the ratios of linear functions of _Yi:n$Y_{i:n}$ and _Y[j:n]$Y_{[j:n]}$. For the bivariate normal case, we compute the expectations of the product of _Yi:n$Y_{i:n}$ and _Y[i:n]$Y_{[i:n]}$ for n=2$n=2$ to 8 for selected values of the correlation coefficient and illustrate their uses.     

Weak convergence of bivariate sequence to bivariate normal

It is shown that under certain conditions the distributions of a bivariate sequence of random vectors converge weakly to that of a bivariate normal distribution.     

Concomitant record ranked set sampling

In this work, we define a new method of ranked set sampling (RSS) which is suitable when the characteristic (variable) Y of primary interest on the units is jointly distributed with an auxiliary characteristic X on which one can take its measurement on any number of units, so that units having record values on X alone are ranked and retained for making measurement on Y. We name this RSS as concomitant record ranked set sampling (CRRSS). We propose estimators of the parameters associated with the variable Y of primary interest based on observations of the proposed CRRSS which are applicable to a very large class of distributions viz. Morgenstern family of distributions. We illustrate the application of CRRSS and our estimation technique of parameters, when the basic distribution is Morgenstern-type bivariate logistic distribution. A primary data collected by CRRSS method is demonstrated and the obtained data used to illustrate the results developed in this work.     

Prediction intervals for the future record values from exponential distribution: comparative study

!n this paper we consider the predicf an problem of the future nth record value based an the first m (m < n) observed record values from one-parameter exponential distribution. We introduce four procedures for obtaining prediction intervals for the nth record value. The performance of the so obtained intervals is assessed through numerical and simulation studies. In these studies, we provide the means and standard errors of lower limits. upper limits and lengths of prediction intervals. Further, we check the validation of these intervals based on some point predictors.     

Distribution of order statistics from selected subsets of concomitants

For a random sample of size n from an absolutely continuous bivariate population (X, Y), let X_i:n be the i th X-order statistic and Y_[i:n] be its concomitant. We study the joint distribution of (V_s:m, W_t:n−m), where V_s:m is the s th order statistic of the upper subset {Y_[i:n], i=n−m+1,…,n}, and W_t:n−m is the t th order statistic of the lower subset {Y_[j:n], j=1,…,n−m  } of concomitants. When m=⌈_np0⌉$m=\lceil {\mathit{np}}_0\rceil$, s=⌈_mp1⌉$s=\lceil {\mathit{mp}}_1\rceil$, and t=⌈(n−m)_p2⌉$t=\lceil (n-m)p_2\rceil$, 0<_pi<1,i=0,1,2$0<p_i<1,i=0,1,2$, and n→∞$n\to \infty$, we show that the joint distribution is asymptotically bivariate normal and establish the rate of convergence. We propose second order approximations to the joint and marginal distributions with significantly better performance for the bivariate normal and Farlie–Gumbel bivariate exponential parents, even for moderate sample sizes. We discuss implications of our findings to data-snooping and selection problems.     

A further note on the bivariate normal distribution

It is indicated to what extent the conditional normality of the distribution of one comnonent of a bivariate random vector given the value of the other component together with a restricted type of conditional normality or the marginal normality for the other component is equivalent to the bivariate normality of this random vector.     

Estimation of the Parameters of Type-I Generalized Logistic Distribution Using Order Statistics

In this work, we propose a technique of estimating the location parameter μ and scale parameter σ of Type-I generalized logistic distribution by U-statistics constructed by using best linear functions of order statistics as kernels. The efficiency comparison of the proposed estimators with respect to maximum likelihood estimators is also made.     

Estimation and prediction of the Kumaraswamy distribution based on record values and inter-record times

ABSTRACT

The maximum likelihood and Bayesian approaches for estimating the parameters and the prediction of future record values for the Kumaraswamy distribution has been considered when the lower record values along with the number of observations following the record values (inter-record-times) have been observed. The Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. The Bayes and the maximum likelihood estimates are compared by using the estimated risk through Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record values arising from the Kumaraswamy distribution based on record values with their corresponding inter-record times and only record values. The comparison of the derived predictors are carried out by using Monte Carlo simulations. Real data are analysed for an illustration of the findings.     

Shortest tolerance intervals controlling both tails of the exponential distribution based on record values

This paper deals with computing shortest width tolerance intervals controlling both tails of the exponential distribution on the basis of record values. Equal-tailed and shortest tolerance factors are derived. The expected widths of these tolerance intervals are evaluated via a Monte Carlo simulation study. Finally, two illustrative examples are also included.     

Estimation of parameters under a random censorship model

The problem of estimation of parameters of a lifetime distribution is considered under the proportional hazards model of random censorship. Asymptotic variances of several estimators of survival function are compared in the eponential case.     

Estimation with the generalized exponential distribution based on record values and inter-record times

The maximum likelihood and Bayesian approaches have been considered for the two-parameter generalized exponential distribution based on record values with the number of trials following the record values (inter-record times). The maximum likelihood estimates are obtained under the inverse sampling and the random sampling schemes. It is shown that the maximum likelihood estimator of the shape parameter converges in mean square to the true value when the scale parameter is known. The Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo methods due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The confidence intervals for the parameters are constructed based on asymptotic and Bayesian methods. The Bayes and the maximum likelihood estimators are compared in terms of the estimated risk by the Monte Carlo simulations. The comparison of the estimators based on the record values and the record values with their corresponding inter-record times are performed by using Monte Carlo simulations.     

Comparison of two sampling schemes for generating record-breaking data from the proportional hazard rate models

ABSTRACT

In this article, we consider a sampling scheme in record-breaking data set-up, as record ranked set sampling. We compare the proposed sampling with the well-known sampling scheme in record values known as inverse sampling scheme when the underlying distribution follows the proportional hazard rate model. Various point estimators are obtained in each sampling schemes and compared with respect to mean squared error and Pitman measure of closeness criteria. It is observed in most of the situations that the new sampling scheme provides more efficient estimators than their counterparts. Finally, one data set has been analyzed for illustrative purposes.     

On a bivariate Kumaraswamy type exponential distribution

ABSTRACT

This paper considers a class of absolutely continuous bivariate exponential distributions whose univariate margins are the ordinary exponential distributions. We study different mathematical properties of the proposed model. The estimation of the parameters by maximum likelihood is discussed. Application is made to a real data example to illustrate the flexibility of theproposed distribution for data analysis.     

Bayesian inference on extreme value distribution using upper record values

In this paper we address estimation and prediction problems for extreme value distributions under the assumption that the only available data are the record values. We provide some properties and pivotal quantities, and derive unbiased estimators for the location and rate parameters based on these properties and pivotal quantities. In addition, we discuss mean-squared errors of the proposed estimators and exact confidence intervals for the rate parameter. In Bayesian inference, we develop objective Bayesian analysis by deriving non informative priors such as the Jeffrey, reference, and probability matching priors for the location and rate parameters. We examine the validity of the proposed methods through Monte Carlo simulations for various record values of size and present a real data set for illustration purposes.