Exponentiated models preserve stochastic orderings of parallel and series systems

Abstract

In this paper, we establish that the usual stochastic, hazard rate, reversed hazard rate, likelihood ratio, dispersive and star orders are all preserved for parallel systems under exponentiated models for lifetimes of components. We then use the multiple-outlier exponentiated gamma models to illustrate this result. Finally, we consider the dual family with exponentiated survival function and establish similar results for series systems. The results established here extend some well-known results for series and parallel systems arising from different exponentiated distributions such as generalized exponential and exponentiated Weibull, established previously in the literature.     

Precedence-type test based on Kaplan–Meier estimator of cumulative distribution function

In this paper, we introduce a precedence-type test based on Kaplan–Meier estimator of cumulative distribution function (CDF) for testing the hypothesis that two distribution functions are equal against a stochastically ordered hypothesis. This test is an alternative to the precedence life-test proposed first by Nelson (1963). After deriving the null distribution of the test statistic, we present its exact power function under the Lehmann alternative, and compare the exact power as well as simulated power (under location-shift) of the proposed test with other precedence-type tests. Next, we extend this test to the case of progressively Type-II censored data. Critical values for some combination of sample sizes and progressive censoring schemes are presented. We then examine the power properties of this test procedure and compare them to those of the weighted precedence and weighted maximal precedence tests under a location-shift alternative by means of Monte Carlo simulations. Finally, we present two examples to illustrate all the test procedures discussed here, and then make some concluding remarks.     

A CHARACTERIZATION OF GEOMETRIC DISTRIBUTIONS THROUGH CONDITIONAL INDEPENDENCE

Let X₁,…, X_n be mutually independent non-negative integer-valued random variables with probability mass functions f_i(x) > 0 for z= 0,1,…. Let E denote the event that {X₁≥X₂≥…≥X_n}. This note shows that, conditional on the event E, X_i-X_i+ 1 and X_i+ 1 are independent for all t = 1,…, k if and only if X_i (i= 1,…, k) are geometric random variables, where 1 ≤k≤n-1. The k geometric distributions can have different parameters θ_i, i= 1,…, k.     

Impeding the Juggernaut of Innovation Diffusion: A Production‐Constrained Model

Models of innovation diffusion typically depict an inexorable momentum once the process begins to roll. Limited production capacity, however, can place a cap on this process, leading to waiting lines of potential customers, thus diminishing overall service quality and the speed of diffusion. Identifying the minimum production capacity needed for unimpeded and unimpaired diffusion can ensure that there are no customers waiting to adopt the product. We propose a production‐capacity‐constrained diffusion model that considers an exogenous industry production capacity and accounts for word‐of‐mouth effects from adopters as well as waiting customers. We derive analytical expressions for minimum capacity needed under multiple production scenarios. We present a dual‐objective non‐linear least squares procedure with large‐scale grid search for estimating the parameters. We apply our model to several new product innovation data sets, ranging from vacuum cleaners to sports utility vehicles in the United States to iPhones globally. Our estimates show that product shortages exist, ranging from mild to severe, in all of these product markets. We are able to corroborate some of our findings with independent external sources of evidence. We find that information on industry capacity can be recovered with as few as 5 years of sales data. Our model has practical implications for policy makers and can help equity analysts triangulate industry capacity better, particularly when such information is closely held.     

Browse‐and‐Switch: Retail‐Online Competition under Value Uncertainty

Although online shopping is becoming popular, consumers who are unsure about whether to buy a product may find it advantageous to visit a brick‐and‐mortar retail store to first examine the product before purchasing it. But, after browsing at the store, consumers have the option of switching to an e‐tailer to purchase the item at a cheaper price rather than buying at the store. Recent business press refers to this browse‐and‐switch behavior as “showrooming,” and attributes to it the declining profits of brick‐and‐mortar retailers. To study the effect of the browse‐and‐switch option on retail and online pricing strategies and profits, we analyze a stylized economic model that incorporates uncertainty in consumers' valuation of the product, captures the heterogeneity among consumers in their inclination to purchase online, and permits product returns. We consider various equilibrium scenarios for different combinations of consumer shopping behaviors, characterize the parameter ranges for each scenario, and demonstrate that browse‐and‐switch behavior can indeed occur under equilibrium. Our analysis further shows that the option for consumers to browse‐and‐switch intensifies competition, reducing the profits for both firms.     

On the distribution-free confidence intervals and universal bounds for quantiles based on joint records

In this paper, we consider three distribution-free confidence intervals for quantiles given joint records from two independent sequences of continuous random variables with a common continuous distribution function. The coverage probabilities of these intervals are compared. We then compute the universal bounds of the expected widths of the proposed confidence intervals. These results naturally extend to any number of independent sequences instead of just two. Finally, the proposed confidence intervals are applied for a real data set to illustrate the practical usefulness of the procedures developed here.     

Inference on constant stress accelerated life tests for log-location-scale lifetime distributions with type-I hybrid censoring

In this paper, we consider a constant stress accelerated life test terminated by a hybrid Type-I censoring at the first stress level. The model is based on a general log-location-scale lifetime distribution with mean life being a linear function of stress and with constant scale. We obtain the maximum likelihood estimators (MLE) and the approximate maximum likelihood estimators (AMLE) of the model parameters. Approximate confidence intervals, likelihood ratio tests and two bootstrap methods are used to construct confidence intervals for the unknown parameters of the Weibull and lognormal distributions using the MLEs. Finally, a simulation study and two illustrative examples are provided to demonstrate the performance of the developed inferential methods.     

Maximum likelihood estimation of the parameters of a multiple step-stress model from the Birnbaum-Saunders distribution under time-constraint: A comparative study

The cumulative exposure model (CEM) is a commonly used statistical model utilized to analyze data from a step-stress accelerated life testing which is a special class of accelerated life testing (ALT). In practice, researchers conduct ALT to: (1) determine the effects of extreme levels of stress factors (e.g., temperature) on the life distribution, and (2) to gain information on the parameters of the life distribution more rapidly than under normal operating (or environmental) conditions. In literature, researchers assume that the CEM is from well-known distributions, such as the Weibull family. This study, on the other hand, considers a p-step-stress model with q stress factors from the two-parameter Birnbaum-Saunders distribution when there is a time constraint on the duration of the experiment. In this comparison paper, we consider different frameworks to numerically compute the point estimation for the unknown parameters of the CEM using the maximum likelihood theory. Each framework implements at least one optimization method; therefore, numerical examples and extensive Monte Carlo simulations are considered to compare and numerically examine the performance of the considered estimation frameworks.     

A generally weighted moving average exceedance chart

Distribution-free control charts gained momentum in recent years as they are more efficient in detecting a shift when there is a lack of information regarding the underlying process distribution. However, a distribution-free control chart for monitoring the process location often requires information on the in-control process median. This is somewhat challenging because, in practice, any information on the location parameter might not be known in advance and estimation of the parameter is therefore required. In view of this, a time-weighted control chart, labelled as the Generally Weighted Moving Average (GWMA) exceedance (EX) chart (in short GWMA-EX chart), is proposed for detection of a shift in the unknown process location; this chart is based on exceedance statistic when there is no information available on the process distribution. An extensive performance analysis shows that the proposed GWMA-EX control chart is, in many cases, better than its contenders.