Estimation of a parametric function associated with the lognormal distribution

Estimation of the mean of the lognormal distribution has received much attention in the literature beginning with Finney (1941 Finney, D.J. (1941). On the distribution of a variate whose logarithm is normally distributed. Supp. J. Royal Stat. Soc. 7(2):155–161.[Crossref] , [Google Scholar]). The problem is of significant practical importance because of the ubiquitous use of log-transformation. In this article, we consider the estimation of a parametric function associated with the lognormal distribution of which the mean, median, and moments are special cases. We generalize various estimators from the literature for the mean to this parametric function and propose a new simple estimator. We present the estimators in a unified framework and use this framework to derive asymptotic expressions for their biases and mean square errors (MSEs). Next, we make asymptotic and small-sample comparisons via simulations between them in terms of their MSEs. Our proposed estimator outperforms many of the previously proposed estimators. A numerical example is given to illustrate the various estimators.     

Estimation with the lognormal distribution on the basis of records

Record scheme is a method to reduce the total time on test of an experiment. In this scheme, items are sequentially observed and only values smaller than all previous ones are recorded. In some situations, when the experiments are time-consuming and sometimes the items are lost during the experiment, the record scheme dominates the usual random sample scheme [M. Doostparast and N. Balakrishnan, Optimal sample size for record data and associated cost analysis for exponential distribution, J. Statist. Comput. Simul. 80(12) (2010), pp. 1389–1401]. Estimation of the mean of an exponential distribution based on record data has been treated by Samaniego and Whitaker [On estimating population characteristics from record breaking observations I. Parametric results, Naval Res. Logist. Q. 33 (1986), pp. 531–543] and Doostparast [A note on estimation based on record data, Metrika 69 (2009), pp. 69–80]. The lognormal distribution is used in a wide range of applications when the multiplicative scale is appropriate and the log-transformation removes the skew and brings about symmetry of the data distribution [N.T. Longford, Inference with the lognormal distribution, J. Statist. Plann. Inference 139 (2009), pp. 2329–2340]. In this paper, point estimates as well as confidence intervals for the unknown parameters are obtained. This will also be addressed by the Bayesian point of view. To carry out the performance of the estimators obtained, a simulation study is conducted. For illustration proposes, a real data set, due to Lawless [Statistical Models and Methods for Lifetime Data, 2nd ed., John Wiley & Sons, New York, 2003], is analysed using the procedures obtained.     

Estimation of the common scale of several shifted exponential distributions with unknown locations

We consider the estimation of the common scale parameter of two or more independent shifted exponential distributions with unknown locations. Under a large class of bowl-shaped loss functions, the best location-scale in-variant estimator is shown to be inadmissible. A class of improved estimators is derived. Some numerical results are presented to show the magnitude of risk reduction.     

Parameter and quantile estimation for the three-parameter lognormal distribution based on statistics invariant to unknown location

Lognormal distribution is one of the popular distributions used for modelling positively skewed data, especially those encountered in economic and financial data. In this paper, we propose an efficient method for the estimation of parameters and quantiles of the three-parameter lognormal distribution, which avoids the problem of unbounded likelihood, by using statistics that are invariant to unknown location. Through a Monte Carlo simulation study, we then show that the proposed method performs well compared to other prominent methods in terms of both bias and mean-squared error. Finally, we present two illustrative examples.     

Tests of fit for a lognormal distribution

The problem of goodness of fit of a lognormal distribution is usually reduced to testing goodness of fit of the logarithmic data to a normal distribution. In this paper, new goodness-of-fit tests for a lognormal distribution are proposed. The new procedures make use of a characterization property of the lognormal distribution which states that the Kullback–Leibler measure of divergence between a probability density function (p.d.f) and its r-size weighted p.d.f is symmetric only for the lognormal distribution [Tzavelas G, Economou P. Characterization properties of the log-normal distribution obtained with the help of divergence measures. Stat Probab Lett. 2012;82(10):1837–1840]. A simulation study examines the performance of the new procedures in comparison with existing goodness-of-fit tests for the lognormal distribution. Finally, two well-known data sets are used to illustrate the methods developed.     

Estimation in the singly truncated weibull distribution with an unknown truncation point

When truncation points are unknown, they must be treated as additional parameters to be estimated from the sample data. In this article, estimators are derived for the truncation parameter in addition to basic parameters of both 1eft and riqht sing1y truncated Weibull distributions, Maximum likelihood estimators and estimators involving expected values of appropriate order statistics are derived, Asymptotic sampling errors of estimates are also given, Ill ustrative examples are inc1uded.     

Estimation of Weibull distribution parameters for irregular interval group failure data with unknown failure times

SUMMARY This paper presents three methods for estimating Weibull distribution parameters for the case of irregular interval group failure data with unknown failure times. The methods are based on the concepts of the piecewise linear distribution function (PLDF), an average interval failure rate (AIFR) and sequential updating of the distribution function (SUDF), and use an analytical approach similar to that of Ackoff and Sasieni for regular interval group data. Results from a large number of simulated case problems generated with specified values of Weibull distribution parameters have been presented, which clearly indicate that the SUDF method produces near-perfect parameter estimates for all types of failure pattern. The performances of the PLDF and AIFR methods have been evaluated by goodness-of-fit testing and statistical confidence limits on the shape parameter. It has been found that, while the PLDF method produces acceptable parameter estimates, the AIFR method may fail for low and high shape parameter values that represent the cases of random and wear-out types of failure. A real-life application of the proposed methods is also presented, by analyzing failures of hydrogen make-up compressor valves in a petroleum refinery.     

A 3-parameter Gompertz distribution for survival data with competing risks,with an application to breast cancer data

The cumulative incidence function is of great importance in the analysis of survival data when competing risks are present. Parametric modeling of such functions, which are by nature improper, suggests the use of improper distributions. One frequently used improper distribution is that of Gompertz, which captures only monotone hazard shapes. In some applications, however, subdistribution hazard estimates have been observed with unimodal shapes. An extension to the Gompertz distribution is presented which can capture unimodal as well as monotone hazard shapes. Important properties of the proposed distribution are discussed, and the proposed distribution is used to analyze survival data from a breast cancer clinical trial.     

Statistical control charts for shifted generalized poisson distribution

Summary The use of shifted (or zero-truncated) generalized Poisson distribution to describe the occurrence of events in production processes is considered. The methods of moments and maximum likelihood are proposed for estimating the parameters of shifted generalized Poisson distribution. Control charts for the total number of events and for the average number of events are developed. Finally, a numerical example is used to illustrate the construction of control charts.     

Bayes estimation of reliability for the two-parameter lognormal distribution

Bayes estimators of reliability for the lognormal failure distribution with two parameters (M,∑) are obtained both for informative priors of normal-gamma type and for the vague prior of Jeffreys. The estimators are in terms of the t-distribution function. The Bayes estimators are compared with the maximum likelihood and minimum variance unbiased estimators of reliabil-ity using Monte Carlo simulations.     

Estimation of a finite population distribution function based on a linear model with unknown heteroscedastic errors

The authors consider a finite population ρ = {(Y_k, x_k), k = 1,…,N} conforming to a linear superpopulation model with unknown heteroscedastic errors, the variances of which are values of a smooth enough function of the auxiliary variable X for their nonparametric estimation. They describe a method of the Chambers‐Dunstan type for estimation of the distribution of {Y_k, k = 1,…, N} from a sample drawn from without replacement, and determine the asymptotic distribution of its estimation error. They also consider estimation of its mean squared error in particular cases, evaluating both the analytical estimator derived by “plugging‐in” the asymptotic variance, and a bootstrap approach that is also applicable to estimation of parameters other than mean squared error. These proposed methods are compared with some common competitors in simulation studies.     

Standard and goodness-of-fit parameter estimation methods for the three-parameter lognormal distribution

A class of goodness-of-fit estimators is found to provide a useful alternative in certain situations to the standard maximum likelihood method which has some undesirable estimation characteristics for estimation from the three-parameter lognormal distribution. The class of goodness-of-fit tests considered include the Shapiro-Wilk and Filliben tests which reduce to a weighted linear combination of the order statistics that can be maximized in estimation problems. The weighted order statistic estimators are compared to the standard procedures in Monte Carlo simulations. Robustness of the procedures are examined and example data sets analyzed.     

On umvu estimators for the multivariate lognormal distribution and their variances

Uniformly minimum variance unbiased estimators of several parameters of the multivariate lognormal distribution are expressed by using the hypergeometric functions of matrix argument. And the variances are given in special cases.     

Simulation on probability points for testing of lognormal or weibull distribution with a small sample

Two extensive computer simulated tables of percentage points of the asymptotic test statistics for testing lognormal or Weibull population proposed by Pereira (1978) are discussed. Special attention is given to small sample cases. Some of the most commonly used 16 symmetrical probability points are reported. These points are 0.001, 0.005, 0.01. 002. 0.025. 0.05. 0.10.0.15, 0.85, 0.90, 0.95, 0.975, 0.98, 0.99, 0.995 and 0.999. These empirical Sumulated results can be used to test hypotheses for these two particular populations and are adequate when using a normal approximation.     

Planning life tests with progressively Type-I interval censored data from the lognormal distribution

In this paper, we determine optimally spaced inspection times for the two-parameter lognormal distribution for any given progressive interval censoring plan. We investigate the effect of the number of inspections and the choice of those optimally spaced inspection times based on the asymptotic relative efficiencies of the maximum likelihood estimates of the parameters. We also discuss the optimal progressive Type-I interval censoring plan when the inspection times and the expected proportions of total failures in the experiment are pre-fixed.     

Goodness of fit test for the generalized Rayleigh distribution with unknown parameters

The use of goodness-of-fit test based on Anderson–Darling (AD) statistic is discussed, with reference to the composite hypothesis that a sample of observations comes from a generalized Rayleigh distribution whose parameters are unspecified. Monte Carlo simulation studies were performed to calculate the critical values for AD test. These critical values are then used for testing whether a set of observations follows a generalized Rayleigh distribution when the scale and shape parameters are unspecified and are estimated from the sample. Functional relationship between the critical values of AD is also examined for each shape parameter (α), sample size (n) and significance level (γ). The power study is performed with the hypothesized generalized Rayleigh against alternate distributions.     

Moments and properties of multiplicatively constrained bivariate lognormal distribution with applications to futures hedging

In this paper, we derive explicit expressions for marginal and product moments of a bivariate lognormal distribution when a multiplicative constraint is present. We show that the coefficients of variation always decrease regardless of the multiplicative constraint imposed. We also evaluate the effects of the constraint on the variances and covariance, and present conditions under which the correlation coefficient increases under the presence of such a multiplicative constraint. We finally apply these results to futures hedging analysis and some other financial applications.     

Estimation for the generalized pareto distribution with censored data

We present a methodology for computing the point and interval maximum likelihood parameter estimation for the two-parameter generalized Pareto distribution (GPD) with censored data. The basic idea underlying our method is a reduction of the two-dimensional numerical search for the zeros of the GPD log-likelihood gradient vector to a one-dimensional numerical search. We describe a computationally efficient algorithm which implement this approach. Two illustrative examples are presented. Simulation results indicate that the estimates derived by maximum likelihood estimation are more reliable against those of method of moments. An evaluation of the practical sample size requirements for the asymptotic normality is also included.