Has the national minimum wage reduced UK wage inequality?

Summary.  The paper investigates the effect on the wage distribution of the introduction, in April 1999, of the national minimum wage (NMW) in the UK. Because of the structure of UK earnings statistics, it is not straightforward to investigate this and various methods for adjusting the published statistics are discussed. The main conclusions are that the NMW does have a detectable effect on the wage distribution and that compliance with the NMW is widespread but the effect is limited because the NMW has been set at a level such that only 6–7% of workers are directly affected and the NMW has had virtually no effect on the pay of workers who are not directly affected. Furthermore, virtually all the changes occurred within 2 months of the introduction in April 1999 and its effect declined over time from April 1999 to September 2001 as the minimum wage was not uprated in line with the increase in average earnings. The more substantial increase in the NMW in October 2001 partially, but not wholly, restored some of this decline in impact.     

Reliability estimation of generalized inverted exponential distribution

A generalized version of inverted exponential distribution (IED) is introduced in this paper. This lifetime distribution is capable of modelling various shapes of failure rates, and hence various shapes of ageing criteria. The model can be considered as another useful two-parameter generalization of the IED. Statistical and reliability properties of the generalized inverted exponential distribution are derived. Maximum likelihood estimation and least square estimation are used to evaluate the parameters and the reliability of the distribution. Properties of the estimates are also studied.     

Importance resampling for the smoothed bootstrap

Alternative methods of estimating properties of unknown distributions include the bootstrap and the smoothed bootstrap. In the standard bootstrap setting, Johns (1988) introduced an importance resam¬pling procedure that results in more accurate approximation to the bootstrap estimate of a distribution function or a quantile. With a suitable “exponential tilting” similar to that used by Johns, we derived a smoothed version of importance resampling in the framework of the smoothed bootstrap. Smoothed importance resampling procedures were developed for the estimation of distribution functions of the Studentized mean, the Studentized variance, and the correlation coefficient. Implementation of these procedures are presented via simulation results which concentrate on the problem of estimation of distribution functions of the Studentized mean and Studentized variance for different sample sizes and various pre-specified smoothing bandwidths for the normal data; additional simulations were conducted for the estimation of quantiles of the distribution of the Studentized mean under an optimal smoothing bandwidth when the original data were simulated from three different parent populations: lognormal, t(3) and t(10). These results suggest that in cases where it is advantageous to use the smoothed bootstrap rather than the standard bootstrap, the amount of resampling necessary might be substantially reduced by the use of importance resampling methods and the efficiency gains depend on the bandwidth used in the kernel density estimation.     

A New Discrete Distribution Related to Generalized Gamma Distribution and Its Properties

In this article, a new discrete distribution related to the generalized gamma distribution (Stacy, 1962) is derived from a statistical mechanical setup. This new distribution can be seen as generalization of two-parameter discrete gamma distribution (Chakraborty and Chakravarty, 2012) and encompasses discrete version of many important continuous distributions. Some basic distributional and reliability properties, parameter estimation by different methods, and their comparative performances using simulation are investigated. Two-real life data sets are considered for data modeling and likelihood ratio test for illustrating the advantages of the proposed distribution over two-parameter discrete gamma distribution.     

A Modified Version of Logarithmic Series Distribution and Its Applications

In this article, we consider a modified version of logarithmic series distribution and study some of its properties. The maximum likelihood estimation of the parameters of the modified distribution is discussed and the distribution has been fitted to certain real life data sets. Tests are also carried out for justifying the significance of the additional parameter of the modified distribution.     

A discrete analogue of the Laplace distribution

Following Kemp (J. Statist. Plann. Inference 63 (1997) 223) who defined a discrete analogue of the normal distribution, we derive a discrete version of the Laplace (double exponential) distribution. In contrast with the discrete normal case, here closed-form expressions are available for the probability density function, the distribution function, the characteristic function, the mean, and the variance. We show that this discrete distribution on integers shares many properties of the classical Laplace distribution on the real line, including unimodality, infinite divisibility, closure properties with respect to geometric compounding, and a maximum entropy property. We also discuss statistical issues of estimation under the discrete Laplace model.     

Bayesian bridge-randomized penalized quantile regression estimation for linear regression model with AP(q) perturbation

Bridge penalized regression has many desirable statistical properties such as unbiasedness, sparseness as well as ‘oracle’. In Bayesian framework, bridge regularized penalty can be implemented based on generalized Gaussian distribution (GGD) prior. In this paper, we incorporate Bayesian bridge-randomized penalty and its adaptive version into the quantile regression (QR) models with autoregressive perturbations to conduct Bayesian penalization estimation. Employing the working likelihood of the asymmetric Laplace distribution (ALD) perturbations, the Bayesian joint hierarchical models are established. Based on the mixture representations of the ALD and generalized Gaussian distribution (GGD) priors of coefficients, the hybrid algorithms based on Gibbs sampler and Metropolis-Hasting sampler are provided to conduct fully Bayesian posterior estimation. Finally, the proposed Bayesian procedures are illustrated by some simulation examples and applied to a real data application of the electricity consumption.     

Bayes Risk Consistency of Nonparametric Bayes Density Estimates2

Even though the literature on nonparametric density estimation is large, the literature on Bayesian estimation of the density function is relatively small. The reason is the lack of a suitable prior over the space of probability density functions. There have been attempts to define priors over the space of probability measures, but they have not yielded any workable prior for the purpose of density estimation. Dubins & Freedman (1963) have denned random distribution functions which are singular with probability one. Kraft (1964) has denned a class of distribution functions which have derivatives but not continuous derivatives and hence are not suitable for density estimation. The only really convenient prior is the Dirichlet process prior due to Ferguson (1973), but unfortunately this prior concentrates all its mass over the discrete distribution with a dense set of jumps. Recently Lo (1978) has overcome this difficulty by taking convolution of the Dirichlet process with a fixed continuous kernel. In Section 2, the existence of a version of the posterior distribution and the conditional expectation for arbitrary prior over the space of continuous density functions are discussed. The Bayes risk consistency of the Bayes estimator is discussed in Section 3. The Bayes estimator and its properties with respect to two specific prior distributions are discussed in Section 4. In Section 5 some negative results are presented. Finally a numerical example is given in Section 6.     

A generalized two-piece skew normal distribution and some of its properties

In this paper, we develop a generalized version of the two-piece skew normal distribution of Kim [On a class of two-piece skew-normal distributions, Statistics 39(6) (2005), pp. 537–553] and derive explicit expressions for its distribution function and characteristic function and discuss some of its important properties. Further estimation of the parameters of the generalized distribution is carried out.     

On a modified version of intervened negative binomial distribution

In this paper, a generalized version of the intervened negative binomial distribution of Kumar and Sreeja [On intervened negative binomial distribution and some of its properties. Statistica. 2012;72:395–404] is considered and studied some of its properties. Certain methods of estimation of the parameters of the distribution are discussed and illustrated with the help of real life data sets. A test procedure is suggested for testing the intervention parameter and a simulation study is conducted for examining the performance of the estimators.     

A new weighted distribution as an extension of the generalized half-normal distribution with applications

In this study, a new extension of generalized half-normal (GHN) distribution is introduced. Since this new distribution can be viewed as weighted version of GHN distribution, it is called as weighted generalized half-normal (WGHN) distribution. It is shown that WGHN distribution can be observed as a single constrained and hidden truncation model. Therefore, the new distribution is more flexible than the GHN distribution. Some statistical properties of the WGHN distribution are studied, i.e. moments, cumulative distribution function, hazard rate function are derived. Furthermore, maximum likelihood estimation of the parameters is considered. Some real-life data sets taken from the literature are modelled using the WGHN distribution. It is seen that for these data sets the WGHN distribution provides better fitting than the GHN and slashed generalized half-normal (SGHN) distributions.     

Bootstrap-assisted tests of symmetry for dependent data*

The paper considers the problem of testing for symmetry (about an unknown centre) of the marginal distribution of a strictly stationary and weakly dependent stochastic process. The possibility of using the autoregressive sieve bootstrap and stationary bootstrap procedures to obtain critical values and P-values for symmetry tests is explored. Bootstrap-assisted tests for symmetry are straightforward to implement and require no prior estimation of asymptotic variances. The small-sample properties of a wide variety of tests are investigated using Monte Carlo experiments. A bootstrap-assisted version of the triples test is found to have the best overall performance.     

On Discrete Gamma Distribution

A new discrete counterpart of gamma distribution for modelling discrete life data is defined based on similar mathematical form and properties of the continuous version. The main statistical and reliability properties of this distribution are derived and it is shown that this model can deal with both over and under-dispersed data. Geometric variables and finite sum of geometric variables, i.e., negative binomial are shown to be special cases of the proposed discrete gamma. Also, the size-biased discrete gamma distribution is derived and discussed. Moreover, different estimation methods of the underlying parameters of this distribution are utilized and comparisons of their performance have been made. Finally, an application in real-life data is used to elucidate the earlier results of this article.     

Estimation of Stable-Law Parameters: A Comparative Study

The stable distribution has many desirable properties and is applicable in many areas of scientific pursuit (e.g., the study of stock-return behavior). Despite this, little is known about the properties of the various extant estimation techniques for the parameters of the stable laws. This article compares the iterative regression technique with the latest version of the fractile technique, using both simulated and actual data.     

Some simple estimators for the two-parameter gamma distribution

ABSTRACT

In this paper, we propose two new simple estimation methods for the two-parameter gamma distribution. The first one is a modified version of the method of moments, whereas the second one makes use of some key properties of the distribution. We then derive the asymptotic distributions of these estimators. Also, bias-reduction methods are suggested to reduce the bias of these estimators. The performance of the estimators are evaluated through a Monte Carlo simulation study. The probability coverages of confidence intervals are also discussed. Finally, two examples are used to illustrate the proposed methods.     

Modal non‐linear regression in the presence of Laplace measurement error

In this paper, we propose a robust estimation procedure for a class of non‐linear regression models when the covariates are contaminated with Laplace measurement error, aiming at constructing an estimation procedure for the regression parameters which are less affected by the possible outliers, and heavy‐tailed underlying distribution, as well as reducing the bias introduced by the measurement error. Starting with the modal regression procedure developed for the measurement error‐free case, a non‐trivial modification is made so that the modified version can effectively correct the potential bias caused by measurement error. Large sample properties of the proposed estimate, such as the convergence rate and the asymptotic normality, are thoroughly investigated. A simulation study and real data application are conducted to illustrate the satisfying finite sample performance of the proposed estimation procedure.     

ON THE ROBUST ANALYSIS OF VARIANCE COMPONENTS MODELS FOR PEDIGREE DATA

Quantitative traits measured over pedigrees of individuals may be analysed using maximum likelihood estimation, assuming that the trait has a multivariate normal distribution. This approach is often used in the analysis of mixed linear models. In this paper a robust version of the log likelihood for multivariate normal data is used to construct M-estimators which are resistant to contamination by outliers. The robust estimators are found using a minimisation routine which retains the flexible parameterisations of the multivariate normal approach. Asymptotic properties of the estimators are derived, computation of the estimates and their use in outlier detection tests are discussed, and a small simulation study is conducted.     

The exponentiated reduced Kies distribution: Properties and applications

Here we consider an exponentiated version of the reduced Kies distribution and discuss some of its properties. The parameters of the distribution are estimated by the method of maximum likelihood and illustrated with the help of certain real-life data sets. Asymptotic behavior of the maximum likelihood estimators of the parameters of the distribution is also studied by using certain simulated data sets.     

On improved estimation of a gamma shape parameter

This paper deals with improved estimation of a gamma shape parameter from a decision-theoretic point of view. First we study the second-order properties of three estimators – (i) the maximum-likelihood estimator (MLE), (ii) a bias corrected version of the MLE, and (iii) an improved version (in terms of mean squared error) of the MLE. It is shown that all the three estimators mentioned above are second-order inadmissible. Next, we obtain superior estimators which are second order better than the above three estimators. Simulation results are provided to study the relative risk improvement of each improved estimator over the MLE.     

A Conway Maxwell Poisson type generalization of the negative hypergeometric distribution

Abstract

Negative hypergeometric distribution arises as a waiting time distribution when we sample without replacement from a finite population. It has applications in many areas such as inspection sampling and estimation of wildlife populations. However, as is well known, the negative hypergeometric distribution is over-dispersed in the sense that its variance is greater than the mean. To make it more flexible and versatile, we propose a modified version of negative hypergeometric distribution called COM-Negative Hypergeometric distribution (COM-NH) by introducing a shape parameter as in the COM-Poisson and COMP-Binomial distributions. It is shown that under some limiting conditions, COM-NH approaches to a distribution that we call the COM-Negative binomial (COMP-NB), which in turn, approaches to the COM Poisson distribution. For the proposed model, we investigate the dispersion characteristics and shape of the probability mass function for different combinations of parameters. We also develop statistical inference for this model including parameter estimation and hypothesis tests. In particular, we investigate some properties such as bias, MSE, and coverage probabilities of the maximum likelihood estimators for its parameters by Monte Carlo simulation and likelihood ratio test to assess shape parameter of the underlying model. We present illustrative data to provide discussion.