Journal of the Kerala Statistical Association
Journal of the Kerala Statistical Association is the official peer-reviewed Journal of Kerala Statistical Association (KSA) founded in 1978. The Journal has been published since 1980 and is published annually. The Journal is intended to publish papers that make significant original contributions in both theory and methodological advances in Probability and Statistics. New applications of statistical and probabilistic methods will also be considered for publication. The Journal is open to critical debate in the objective, wide-ranging and free spirit of research. The international editorial board of the Journal is comprised of scientists with interests in applied, computational, methodological and theoretical aspects of Probability and Statistics. The editorial history of the journal includes, Prof K Ramakrishna Pillai (St. Thomas College Pala), Prof R N Pillai (University of Kerala), Prof P Yageen Thomas (University of Kerala) and currently Prof K Jayakumar (University of Calicut) as Journal editors. The journal has already published 32 volume and cumulatively more than 100 research papers. Aim of the Journal: The Journal’s emphasis is on publishing papers developing and analyzing new methods for any active field of statistics and probability theory with direct or potential real-life applications. The Journal seeks papers making significant contributions of interest to a broad group of readers than to a highly specialized group. Original contributions in the interface of statistics and other fields are also published.
Bayesian Prediction of Future Order Statistics and k-Record Values based on Progressive Type-II Censored Sample for Generalized Exponential distribution
In this paper, we consider the problem of two-sample Bayesian prediction of order statistics and lower k-record values from a future sample arising from a generalized exponential distribution based on progressively type-II censored sample. We obtain the Bayesian point predictors based on squared error and linex loss functions. It is observed that the predictors cannot be obtained in closed forms. So we propose importance sampling method to obtain the estimators for both point and interval predictors.
A Versatile Probabilistic Model based on Yun-G Family of Distributions and its Applications in Engineering Sector
The present paper discusses a versatile three -parameter distribution from the Yun-G family. The important statistical properties like moments, stochastic ordering, and entropy are studied in this paper. Two characterizations of the distribution are obtained using the hazard rate function and truncated moments. The statistical inference of the distribution is studied by executing five different methods of parameter estimation, such as maximum likelihood estimation, ordinary least square method, weighted least square method, Cramer-von Mises method, and Anderson–Darling method. To study the fitting and applicability of the proposed distribution, two real life data sets from the engineering sector were analyzed and the proposed distribution is found to be more appropriate than the other competitive distributions. A comprehensive simulation study is also conducted and it showed the accuracy and consistency of the estimation techniques.
A Mixture Regression Approach for Modelling Early Post Operative Hypocalcemia
In this paper, we show that a two component Laplace mixture model is an appropriate distribution to model postoperative calcium levels of patients undergoing thyroidectomy or surgery for thyroid diseases. A mixture regression model is constructed to predict postoperative calcium levels based on a pre-determined set of potential predictors. The parameters of the model are estimated using EM algorithm. Our study based on a real data set shows that two component Laplace mixture regression model is suitable for prediction and interpretation compared to the usual Gaussian mixture regression model.
Generalized T – X Family of Distributions and Applications
In this paper, a new method of generating families of distributions using any p.d.f. as a generator which is called Transformed – Transformer ( T – X ) family of distributions is considered. Some members of this family including Gumbel – Uniform, Gumbel – Pareto, Gumbel – Frechet and their mathematical properties are explored. The method of maximum likelihood is used for estimating the distribution parameters. The flexibility of the new model is illustrated using real life data sets. Also industrial reliability test plans for acceptance or rejection of a lot of products submitted for inspection are developed. The acceptance sampling plan proposed here can save the test time in practical situations. Marshall – Olkin generalization of T – X family is also introduced. Stress-strength analysis is discussed. A simulation study is also conducted to study the behaviour of estimate of stress-strength reliability. Auto-regressive models are discussed based on Marshall – Olkin Gumbel uniform distribution and sample path properties are explored. Transmutation technique is also considered in T – X family. A real data is used to explain the flexibility of Gumbel Transmuted Uniform model, a member of Gumbel Transmuted X family.
Some Problems Connected with the Distributions of Products and Ratios
In this paper, the distributions of products and ratios involving real scalar positive variables, symmetric products and symmetric ratios of real positive definite matrices and symmetric product and symmetric ratios of Hermitian positive definite matrices in the complex domain are considered. Real scalar variable case is taken as a starting point. Here, it is pointed out how distributions of products and ratios are connected to Bessel integrals, reaction-rate probability integrals in nuclear reaction-rate theory, inverse Gaussian density, Kr¨atzel integral and transform, pathway transform, fractional integrals of the first and second kind etc. Only one illustrative example each is given. Then, symmetric product and symmetric ratios of positive definite matrices in the real and complex domain are considered. Here also one illustrative example each is given. Some current research materials are also mentioned in the introduction.
Fractional integrals of first and second kinds
Cumulative Residual Extropy properties of Ranked Set Sample for Cambanis Type Bivariate Distributions
In this paper, we consider the cumulative residual extropy (CREX) of concomitant of order statistic and its properties, when samples are taken from Cambanis family of distributions. By using the expression for CREX of concomitants of order statistics, the CREX of the RSS observations is obtained, when the study variate $ Y $ is difficult to measure but an auxiliary variable $ X $ is used to rank the units in each set, under the assumption that $ (X,Y) $ follows Cambanis type bivariate distributions.
ASYMMETRIC DOUBLE LOMAX DISTRIBUTION: THEORY AND APPLICATIONS
The double Lomax distribution is the ratio of two independent and identically distributed classical Laplace distributions and it can be used to analyse data sets with heavy tails and peakedness. In the present paper, we introduce asymmetric generalization of double Lomax distribution. We derived the probability density function of asymmetric double Lomax distribution and its various properties were studied. The maximum likelihood estimation procedure is employed to estimate the parameters of the proposed distribution and an algorithm in R package is developed to carry out the estimation. To validate the algorithm, simulation studies were conducted with various parameter values. Finally, we fitted the asymmetric double Lomax, asymmetric Laplace, double Lomax and Gaussian distributions to microarray gene expression dataset and financial datasets and compared them.
Asymmetric double Lomax distribution ; asymmetric Laplace distribution ; double Lomax distribution ; Laplace distribution ; microarray gene expression
'c Policy' Batch Service Queueing System under Bernoulli Scheduled Vacation and Manual Service
This paper describes discrete-time single arrival and single-batch service queue with reneging and vacation interruption, under policy 'c'.The system activates when the queue size reaches a predetermined point 'c' and serves all the units in a batch. if there is a queue size less than 'c', either server switches to manual service with the probability 'p', or the server takes a vacation with probability '(1-p)'. Once it starts manual service or takes a vacation, the same continues up to the queue size reaches 'c'. The inter-arrival and service time are considered to be independent and geometrically distributed.
Bivariate Generalized Discrete Modified Weibull Distribution
In this paper we introduce a new generalization of univariate and bivariate modified discrete Weibull distribution. Various properties of univariate generalized modified discrete Weibull distribution such as survival function, probability mass function, hazard rate function, probability generating function, moment generating function are derived. The joint distribution function, joint probability mass function, marginal distributions, moment generating function, conditional distribution of proposed bivariate distribution are derived. Parameters of the distributions are estimated using Maximum likelihood estimation. The use of these distributions are illustrated using real life data sets.
Estimation of the Scale Parameter of Triangular Distribution Using Absolved Order Statistics
In this paper, a new set of ordered random variables called Absolved Order Statistics(AOS) from a scale dependent triangular distribution is considered. The vector of AOS is found to be the minimal sufficient statistic for the triangular distribution with scale parameter σ. Based on AOS, the best linear unbiased estimate ˆσ of σ along with its variance is explicitly derived. It is found that censoring based on AOS is more realistic and the estimate obtained from it for σ is more efficient than the case of censoring with order statistics. In this study, we also obtained the U-statistic estimator based on AOS for the scale parameter σ of the triangular distribution.
Laplace process was introduced in Satheesh (1990) as a possible alternative to Wiener process. Here we discuss some properties of this process viz. correlation, martingales, finite dimensional distributions and sample paths. A review of the rich theoretical developments over Laplace process, driven by data, over the years is also given.
Extended Gamma, Beta and Hypergeometric Functions: Properties and Applications
The main objective of this paper is to study the newly introduced extended gamma function and present some properties of the existing extended beta and hypergeometric functions with their properties such as integral formulas, functional relations, differential formulas, summation formulas, recurrence relations and the Mellin transform. In addition, applications of the extended beta function to statistical distributions have been discussed by providing mean, variance, coefficient of variance, moment generating function, cumulative distribution and reliability function of the new extended beta distribution.
A Class of Tests Based Jointly on Sub-Sample Minimum and Median
In Statistics, nonparametric tests play a very important role when the data sets are not normal. In this paper, we develop a class of nonparametric tests to compare location parameters of two populations. The proposed test statistic is based jointly on minimum and median of the sub-samples. The mean and variance of the test statistic are derived. The test statistic has asymptotic normality under some assumptions. The proposed class of the tests is compared with its existing competitor’s w.r.t. Pitman and Bahadur asymptotic relative efficiency. As an illustration, the proposed test is applied to a real life data set. We carried out the Monte Carlo simulation study to find the power and level of significance of the proposed test.
Measure of Slope Rotatability for Second Order Response Surface Designs Under Tri-Diagonal Correlation Error Structure Using Central Composite Designs
In the design of experiments for estimating the slope of the response surface, slope rotatability is a desirable property. In this paper, measure of slope rotatability for second order response surface designs using central composite designs under tri-diagonal correlation error structure is suggested and illustrated with examples.
tri-diagonal correlation error structure
Comparison of Machine Learning Techniques for Recommender Systems for Financial Data
Recommender Systems are one of the most successful and widespread application of machine learning technologies in business. These are the software tools used to give suggestions to users on the basis of their requirements. Increase in number of options: be it number of online websites or number of products, it has become difficult for the customer to choose from a wide range of products. Today there is no system available for banks to provide financial advisory services to the customers and offer them relevant products as per their preference before they approach the bank. Like any other industries, financial service rarely has any like, feedback and browsing history to record ratings of services. So it becomes a challenge to build recommender systems for financial services. In this research paper, authors propose a collaborative filtering technique to recommend various products to the customer in order to increase the product per customer (PPC) ratio of bank. The advantage of these recommender systems is that it provides better suggestion to the customer based on his needs/requirements for his/her savings, expenditure and investments.
Generalized Lehmann Alternative Type II Family of Distributions and Their Applications
A new generalized family called Generalized Lehmann Alternative Type II (GLA2) family is introduced and studied in this paper. Special cases of this family using Uniform and Kumaraswamy distributions as base are developed and their statistical properties studied. Generalized Lehmann Alternative Type II Exponential (GLA2E) distribution is also developed and its statistical properties are obtained along with application. The new distribution is applied to a real data set to show the effectiveness of the distribution and it is verified that the new model is a better model than the existing exponential model and Marshall-Olkin extended exponential model. A detailed study on the record value theory associated with GLA2E distribution is conducted. Using the mean, variance and covariance of upper record values of the extended model, BLUE’s of location and scale parameters are obtained and future records are predicted which has a number of practical uses. The 95% confidence interval for location and scale parameters are also computed. The result is applied to a real data set to validate the results. Entropy of record values is derived. This result will be useful in characterization of record values based on entropies and a quantification of information contained in each additional record value based on entropy measure.
A General Method of Construction of a Bivariate Lifetime Distribution with a Singular Component
Marshall-Olkin bivariate exponential distribution is the most popular bivariate distribution with a singular component. Since then several other bivariate distributions with a singular component have been introduced in the literature. It is observed that there are mainly two main approaches to construct a bivariate distribution with a singular component. In this paper we have proposed a general method to construct a bivariate distribution with a singular component. All the existing bivariate distributions with a singular component can be obtained using this method. Moreover, more flexible bivariate distributions with a singular component also can be constructed using this method. It is a very simple procedure based on mixing. Using this approach, we have considered one special case, namely bivariate Weibull distribution, in detail. We have derived several properties of the proposed bivariate Weibull distribution and it seems to be more flexible than the popular Marshall-Olkin bivariate Weibull distribution. Maximum likelihood estimators can be obtained quite conveniently in this case. It can be used to model dependent competing risks data and it can be generalized to the multivariate set up also.
Marshall-Olkin bivariate exponential distribution
Inference On Gompertz Distribution Based On Upper k Record Values
In this paper, we consider the inference based on upper k -record values for a two parameter Gompertz distribution. The maximum likelihood estimators of the parameters are obtained. The Bayes estimates of the parameters are also developed by using Markov chain Monte Carlo method under symmetric and asymmetric loss functions. The problem of predicting the future record values based on past record values is also considered. Finally, a simulation study is performed to find the performance of different estimators developed in this paper.
On A Generalized Class Of Weibull Distribution And Its Applications
In this paper, we propose a family of continuous distributions called odds X Weibull(OXW) distributions using the T-X family concept introduced in Alzaatreh et al. (2013). Some of the special models of this family are introduced and one among them, odds exponential-Weibull(OEW) distribution is studied in detail. This distribution is found to be a competitor for the well known lifetime models such as exponential, Weibull, exponentiated exponential, exponentiated Weibull, etc. The method of maximum likelihood is used for estimating the model parameters and a simulation study is carried out to check the performance of the method. Time series models with OEW distribution as marginals are developed. Two real data sets are used to illustrate the applications and flexibility of the OEW distribution.
Marshall-Olkin Extended Rayleigh Distribution and Applications
In this paper a new lifetime distribution as a two parameter generalization of Rayleigh distribution namely Marshall Olkin Extended Rayleigh distribution is introduced and studied. The statistical properties such as quantile function, moments, compounding properties, R´enyi entropy and stochastic ordering etc. are explored. The distribution of order statistics and its asymptotic distribution are derived. As an application of the distribution to the record value theory we evaluate the first and second single moment of nth upper record value. The method of maximum likelihood is used to estimate the unknown parameters. The results are validated using real data sets.
A New Generalization of Inverse Marshall-Olkin Family and Related Inverse Exponential Model
In this paper, we introduce a new family of continuous distributions called inverse truncated discrete Mittag-Leffler family of distributions. In particular, we study inverse truncated discrete Mittag-Leffler exponential (GIE) distribution. The GIE distribution contains inverse Marshall-Olkin exponential distribution, inverse generalized exponential distribution and inverse exponential distribution as special case. The density function of GIE distribution is right skewed and has reverse bathtub hazard rate. We derive expression for the moments, generating function, quantiles and R´enyi entropy. Also the stochastic ordering properties are investigated along with the distributions of order statistics. The method of maximum likelihood is used to estimate the model parameters. The existence and uniqueness of maximum likelihood estimates are proved. Simulation studies are also performed. An application to a real data set is presented to illustrate the potentiality of our proposed model.
On Moments of Order Statistics from Extended Normal Distribution
In this paper, we have introduced a two-parameter distribution, “Extended Normal Distribution”, observed as a bimodal symmetric distribution, of which Normal distribution is a particular case. Its moments, inverse moments, maximum likelihood estimators and other statistical measures have been discussed. For few particular values of unknown parameters, we have obtained the exact moments and the recurrence relation for moments of order statistics arising from Extended Normal Distribution.
A New Bivariate Distribution with Extreme Value Type I and Burr Type XII Distributions as Marginals
In this work, we develop a new bivariate distribution by assuming a specified distribution to one marginal random variable and a specified distribution to the concomitant of the largest order statistic on the other random variable. We also discuss about the role of concomitants of lower record values in the construction of the above bivariate distribution. We assume the form of distribution of one marginal random variable and the distribution of the concomitant of the largest order statistic in such a way that the resulting bivariate distribution is one with extreme value type I and Burr-Type XII distributions as its marginals. Some general characteristics of this newly generated distribution are studied. The method of maximum likelihood is applied for the estimation of the parameters of the generated distribution. We have identified the generated model as a suitable model for two real life bivariate data sets reported in the literature.
Generalization of Gompertz Distribution and its Applications in Reliability and Time series
In this paper, as a generalization of the Gompertz distribution, Marshall- Olkin Gompertz distribution is considered. A three parameter AR(1) process is also considered. When X and Y are two independent random variables following Marshall Olkin Gompertz distribution, then average bias, average mean square error, average confidence length and coverage probability of the of the simulated estimates of reliability R is computed. Data analysis based on a real data set and modeling are also done.
On a Generalization of Marshall-Olkin Weibull Distribution and its Applications
In this article, a new generalized four parameter distribution called Marshall- Olkin Weibull Truncated Negative Binomial (MOWTNB) distribution is introduced and studied. Various structural properties of the new distribution are obtained. The distribution of order statistics is also derived. The method of maximum likelihood for estimating the model parameters is discussed. Applications to two real data sets are provided to show the flexibility and potentiality of the new distribution. We have compared the performance of MOWTNB with ten other competitive models, for modelling the two data sets and showed that MOWTNB performs better compared to these existing models. An AR(1) minification model with this distribution as marginal is developed.
Residual Verma Entropy of k-Record Values
In this paper, we consider a generalized residual entropy known as residual Verma entropy(RVE) of k-record values. A representation of RVE of nth krecord value arising from any continuous distribution is expressed in terms of RVE of nth k-record value arising from uniform distribution. We provide bounds for residual Verma entropy of k-record values. Monotone behaviour of RVE of k-record values in terms of number of observations have also been considered.
Mixtures of Gamma Distributions and Harris Infinite Divisibility
We look at the relation between mixtures of gamma distributions and Harris infinite divisibility. We show that mixtures of gamma distributions with shape parameter 1/k, k > 1 integer, are Harris infinitely divisible. We also prove that Harris infinitely divisible distributions are geometrically infinitely divisible and that all Harris infinitely divisible laws are not mixtures of gamma distributions.
A Discrete Generalization of Marshall-Olkin Scheme and its Application to Geometric Distribution
In this paper, we introduce a new family of discrete distributions and study its properties. The new family is a generalization of discrete Marshall-Olkin family of distributions. In particular, we study discrete generalized Marshall- Olkin exponential (DGMOE) distribution in detail. Generalized geometric distribution and geometric distribution are sub-models of DGMOE distribution. We derive some basic distributional properties such as probability generating function, moments, hazard rate and quantiles of the DGMOE distribution. Estimation of the parameters are done using maximum likelihood method. A real data set is analyzed to illustrate the suitability of the proposed model.
Generalized Marshall-Olkin family of distributions
Estimation of Parameters of Gompertz Distribution Under Progressive Type-II Censoring
In this paper, the problem of estimating unknown parameters of a two parameter Gompertz distribution is considered based on progressively type-II censored sample. The maximum likelihood (ML) estimators of the parameters are obtained. Bayes estimates are also obtained using different loss functions such as squared error, LINEX and general entropy. The Lindley’s approximation method is used to evaluate these Bayes estimates. Monte Carlo simulation is used for numerical comparison between various estimates developed in this paper.
Estimating a Common Parameter of Two Farlie-Gumbel-Morgenstern Bivariate Uniform Distributions by Induced Ranked Set Sampling
The method of ranked set sampling when units are to be induced from several bivariate populations is introduced in this work. The best linear unbiased estimator (BLUE) of the common parameter θ2, based on n ranked set observations, is obtained when n1 observations are drawn from one of the two Farlie-Gumbel-Morgenstern (FGM) bivariate uniform distributions and n2 observations, drawn from the other FGM bivariate uniform distribution such that n = n1 + n2. The BLUE of θ2 based on the upper extreme ranked set sample is also obtained. These estimators are compared with the usual BLUE of the common parameter θ2 based on the concomitants of order statistics of an equivalent sample size.