COMPUTING MARGINAL EFFECTS IN THE BOX–COX MODEL

ABSTRACT

This paper considers computation of fitted values and marginal effects in the Box–Cox regression model. Two methods, 1 the “smearing” technique suggested by Duan (see Ref. [10] Duan, N. 1983. Smearing Estimate: A Nonparametric Retransformation Method. J. Amer. Statistical Assoc., 78: 605–610. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and 2 direct numerical integration, are examined and compared with the “naive” method often used in econometrics.     

Repeated confidence intervals and prediction intervals using stochastic curtailment under fractional Brownian motion

ABSTRACT

Repeated confidence intervals (RCIs) and prediction intervals (PIs) can be used for the design and monitoring of group sequential trials. Stochastically curtailed tests (SCTs) under fractional Brownian motion (FBM) have been studied for the interim analysis of clinical trials (Zhang et al., 2015 Zhang, Q., Lai, D.J., Davis, B.R. (2015). Stochastically curtailed tests under fractional Brownian motion. Commun. Stat. Theory Methods. 44(5):1053–1064.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). In this article, we derive RCIs and PIs based on SCTs under FBM for one-sided derived tests (Jennison and Turnbull, 2000 Jennison, C., Turnbull, B.W. (2000). Group Sequential Methods with Applications to Clinical Trials. Boca Raton, London: Chapman and Hall. [Google Scholar]). Comparisons of RCI width and sample size requirement are made to those under Brownian motion (BM) and to those of Pocock and O'Brien-Fleming design types for various type I, type II error rates, and number of interim analyses. Interim data from Beta-Blocker Heart Attack Trial are used to illustrate how to design and monitor clinical trials using these RCIs and PIs under FBM. Results show that these one-sided derived tests based on SCTs have narrower final confidence intervals and require smaller sample sizes than those using classical group sequential designs. The Hurst parameter has more impact on the RCI width than on the sample size requirements for the proposed designs.     

Construction of peculiar diffusion process having Gaussian marginals

ABSTRACT

In this note, we construct a new strange diffusion process whose drift coefficient does not satisfy the usual Lipschitz and linear-growth conditions. Beginning with the necessary characteristic equation between drift and diffusion coefficients established in Mudakkar (2012 Mudakkar, S.R. (2012). A necessary characteristic equation of diffusion processes having gaussian marginals. Abstr. Appl. Anal. 2012, Article ID 598590:9. doi:10.1155/2012/598590.[Crossref] , [Google Scholar]) and using the Foster–Lyapunov criterion and truncation technique, the existence is established.     

Convergence of weighted sums for sequences of pairwise NQD random variables

ABSTRACT

The authors discuss the convergence for weighted sums of pairwise negatively quadrant dependent (NQD) random variables and obtain some new results which extend and improve the result of Bai and Cheng (2000) Bai, Z.D., Cheng, P.E. (2000). Marcinkiewicz strong laws for linear statistics. Stat. Probab. Lett. 46:105–112.[Crossref], [Web of Science ®] , [Google Scholar]. In addition, we relax some restrictions of the conditions in their result. Some new methods are used in this article which differ from that of Bai and Cheng (2000) Bai, Z.D., Cheng, P.E. (2000). Marcinkiewicz strong laws for linear statistics. Stat. Probab. Lett. 46:105–112.[Crossref], [Web of Science ®] , [Google Scholar].     

On the probability distribution of economic growth

Three important and significantly heteroscedastic gross domestic product series are studied. Omnipresent heteroscedasticity is removed and the distributions of the series are then compared to normal, normal mixture and normal–asymmetric Laplace (NAL) distributions. NAL represents a skewed and leptokurtic distribution, which is in line with the Aghion and Howitt [1 Aghion, P. and Howitt, P. 1992. A model of growth through creative destruction. Econometrica, 60: 323–351. [Crossref], [Web of Science ®] [Google Scholar]] model for economic growth, based on Schumpeter's idea of creative destruction. Statistical properties of the NAL distributions are provided and it is shown that NAL fits the data better than the alternatives.     

Convergence rates in the law of large numbers for END linear processes with random coefficients

Abstract

In this paper, we consider convergence rates in the Marcinkiewicz–Zygmund law of the large numbers for the END linear processes with random coefficients. We extend some results of Baum and Katz (1965 Baum, L. E., and M. Katz. 1965. Convergence rates in the law of large numbers. Transactions of the American Mathematical Society 120 (1):108–23. doi: 10.2307/1994170.[Crossref], [Web of Science ®] , [Google Scholar]) to the case of dependent linear processes with the random coefficients.     

Tests for Multiple Outliers in an Exponential Sample

By applying the recursion of Huffer (1988 Huffer, F. 1988. Divided differences and the joint distribution of linear combinations of spacings. Journal of Applied Probability, 25: 346–354. [Crossref], [Web of Science ®] , [Google Scholar]) repeatedly, we propose an algorithm for evaluating the null joint distribution of Dixon-type test statistics for testing discordancy of k upper outliers in exponential samples. By using the critical values of Dixon-type test statistics determined from the proposed algorithm and those of Cochran-type test statistics presented earlier by Lin and Balakrishnan (2009 Lin, C. T. and Balakrishnan, N. 2009. Exact computation of the null distribution of a test for multiple outliers in an exponential sample. Computational Statistics & Data Analysis, 53: 3281–3290. [Crossref], [Web of Science ®] , [Google Scholar]), we carry out an extensive Monte Carlo study to investigate the powers and the error probabilities for the effects of masking and swamping when the number of outliers k = 2 and 3. Based on our empirical findings, we recommend Rosner’s (1975 Rosner, B. 1975. On the detection of many outliers. Technometrics, 17: 221–227. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) sequential test procedure based on Dixon-type test statistics for testing multiple outliers from an exponential distribution.     

A MONTE CARLO COMPARISON OF VARIOUS ASYMPTOTIC APPROXIMATIONS TO THE DISTRIBUTION OF INSTRUMENTAL VARIABLES ESTIMATORS

ABSTRACT

We examine empirical relevance of three alternative asymptotic approximations to the distribution of instrumental variables estimators by Monte Carlo experiments. We find that conventional asymptotics provides a reasonable approximation to the actual distribution of instrumental variables estimators when the sample size is reasonably large. For most sample sizes, we find Bekker[11] Bekker, P. A. 1994. Alternative Approximations to the Distributions of Instrumental Variable Estimators. Econometrica, 62: 657–681. [Crossref], [Web of Science ®] , [Google Scholar] asymptotics provides reasonably good approximation even when the first stage R ² is very small. We conclude that reporting Bekker[11] Bekker, P. A. 1994. Alternative Approximations to the Distributions of Instrumental Variable Estimators. Econometrica, 62: 657–681. [Crossref], [Web of Science ®] , [Google Scholar] confidence interval would suffice for most microeconometric (cross-sectional) applications, and the comparative advantage of Staiger and Stock[5] Staiger, D. and Stock, J. H. 1997. Instrumental Variables Regression with Weak Instruments. Econometrica, 65: 556–586. [Crossref], [Web of Science ®] , [Google Scholar] asymptotic approximation is in applications with sample sizes typical in macroeconometric (time series) applications.     

Evaluating the relative merits of competing models based on empirical likelihood ratio test

Competing models arise naturally in many research fields, such as survival analysis and economics, when the same phenomenon of interest is explained by different researcher using different theories or according to different experiences. The model selection problem is therefore remarkably important because of its great importance to the subsequent inference; Inference under a misspecified or inappropriate model will be risky. Existing model selection tests such as Vuong's tests [26 Q.H. Vuong, Likelihood ratio test for model selection and non-nested hypothesis, Econometrica 57 (1989), pp. 307–333. doi: 10.2307/1912557[Crossref], [Web of Science ®] , [Google Scholar]] and Shi's non-degenerate tests [21 X. Shi, A non-degenerate Vuong test, Quant. Econ. 6 (2015), pp. 85–121. doi: 10.3982/QE382[Crossref], [Web of Science ®] , [Google Scholar]] suffer from the variance estimation and the departure of the normality of the likelihood ratios. To circumvent these dilemmas, we propose in this paper an empirical likelihood ratio (ELR) tests for model selection. Following Shi [21 X. Shi, A non-degenerate Vuong test, Quant. Econ. 6 (2015), pp. 85–121. doi: 10.3982/QE382[Crossref], [Web of Science ®] , [Google Scholar]], a bias correction method is proposed for the ELR tests to enhance its performance. A simulation study and a real-data analysis are provided to illustrate the performance of the proposed ELR tests.     

Model selection for stock prices data

The geometric Brownian motion (GBM) is very popular in modeling the dynamics of stock prices. However, the constant volatility assumption is questionable and many models with nonconstant volatility have been developed. In the papers [7 M.L. Esquível and P.P. Mota, On some auto-induced regime switching double-threshold glued diffusions, J. Stat. Theory Pract. 8 (2014), pp. 760–771. doi: 10.1080/15598608.2013.854184.[Taylor &; Francis Online] , [Google Scholar],12 P. P. Mota and M.L. Esquível, On a continuous time stock price model with regime switching, delay, and threshold, Quant. Financ. 14 (2014), pp. 1479–1488. doi: 10.1080/14697688.2013.879990.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] the authors introduce a regime switching process where in each regime the process is driven by GBM and the change in regime is defined by the crossing of a threshold. In this paper we used Akaike's and Bayesian information criteria to show that the GBM with regimes provides a better fit than the GBM. We also perform a forecasting comparison of the models for two selected companies.     

General results of complete convergence and complete moment convergence for weighted sums of some class of random variables

ABSTRACT

In the article, the complete convergence and complete moment convergence for weighted sums of sequences of random variables satisfying a maximal Rosenthal type inequality are studied. As an application, the Marcinkiewicz–Zygmund type strong law of large numbers is obtained. Our partial results generalize and improve the corresponding ones of Shen (2013 Shen, A.T. (2013). On strong convergence for weighted sums of a class of random variables. Abstr. Appl. Anal.2013, Article ID 216236: 1–7. [Google Scholar]).     

Confidence Intervals from Stochastic Approximation

We propose a nonparametric method of constructing confidence interval for a scalar parameter from stochastic approximation through the efficient Robbins–Monro procedure proposed by Joseph (2004 Joseph, V.R. (2004). Efficient Robbins–Monro procedure for binary data. Biometrika 91:461–470.[Crossref], [Web of Science ®] , [Google Scholar]). Unlike the bootstrap method where the number of resampling is fixed in advance, the proposed procedure iteratively searches the endpoints in an optimal way such that the convergence is fast and the coverage is obtained accurately. Simulation and real data application illustrate its superiority over the usual Robbins–Monro procedure and common bootstrap methods.     

LIMIT THEOREMS FOR ASYMPTOTICALLY MINIMAX ESTIMATION OF A DISTRIBUTION WITH INCREASING FAILURE RATE UNDER A RANDOM MIXED CENSORSHIP/TRUNCATION MODEL

ABSTRACT

The search for optimal non-parametric estimates of the cumulative distribution and hazard functions under order constraints inspired at least two earlier classic papers in mathematical statistics: those of Kiefer and Wolfowitz[1] Kiefer, J. and Wolfowitz, J. 1976. Asymptotically Minimax Estimation of Concave and Convex Distribution Functions. Z. Wahrsch. Verw. Gebiete, 34: 73–85. [Crossref], [Web of Science ®] , [Google Scholar] and Grenander[2] Grenander, U. 1956. On the Theory of Mortality Measurement. Part II. Scand. Aktuarietidskrift J., 39: 125–153.  [Google Scholar] respectively. In both cases, either the greatest convex minorant or the least concave majorant played a fundamental role. Based on Kiefer and Wolfowitz's work, Wang3-4 Wang, J.L. 1986. Asymptotically Minimax Estimators for Distributions with Increasing Failure Rate. Ann. Statist., 14: 1113–1131. Wang, J.L. 1987. Estimators of a Distribution Function with Increasing Failure Rate Average. J. Statist. Plann. Inference, 16: 415–427.   found asymptotically minimax estimates of the distribution function F and its cumulative hazard function Λ in the class of all increasing failure rate (IFR) and all increasing failure rate average (IFRA) distributions. In this paper, we will prove limit theorems which extend Wang's asymptotic results to the mixed censorship/truncation model as well as provide some other relevant results. The methods are illustrated on the Channing House data, originally analysed by Hyde.5-6 Hyde, J. 1977. Testing Survival Under Right Censoring and Left Truncation. Biometrika, 64: 225–230. Hyde, J. 1980. “Survival Analysis with Incomplete Observations”. In Biostatistics Casebook 31–46. New York: Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics.       

ESTIMATION OF PARAMETERS IN ARFIMA PROCESSES: A SIMULATION STUDY

It is known that, in the presence of short memory components, the estimation of the fractional parameter d in an Autoregressive Fractionally Integrated Moving Average, ARFIMA(p, d, q), process has some difficulties (see [1] Smith, J., Taylor, N. and Yadav, S. 1997. Comparing the bias and misspecification in ARFIMA models. Journal of Time Series Analysis, 18(5): 507–527. [Crossref] , [Google Scholar]). In this paper, we continue the efforts made by Smith et al. [1] Smith, J., Taylor, N. and Yadav, S. 1997. Comparing the bias and misspecification in ARFIMA models. Journal of Time Series Analysis, 18(5): 507–527. [Crossref] , [Google Scholar] and Beveridge and Oickle [2] Beveridge, S. and Oickle, C. 1993. Estimating fractionally integrated time series models. Economics Letters, 43: 137–142.  [Google Scholar] by conducting a simulation study to evaluate the convergence properties of the iterative estimation procedure suggested by Hosking [3] Hosking, J. 1981. Fractional differencing. Biometrika, 68(1): 165–176. [Crossref], [Web of Science ®] , [Google Scholar]. In this context we consider some semiparametric approaches and a parametric method proposed by Fox-Taqqu[4] Fox, R. and Taqqu, M. S. 1986. Large-sample properties of parameter estimates for strongly dependent stationary gaussian time series. The Annals of Statistics, 14(2): 517–532. [Crossref], [Web of Science ®] , [Google Scholar]. We also investigate the method proposed by Robinson [5] Robinson, P. M. 1995a. Log-periodogram regression of time series with long range dependence. The Annals of Statistics, 23(3): 1048–1072. [Crossref], [Web of Science ®] , [Google Scholar] and a modification using the smoothed periodogram function.     

The Lamperti Transforms of Self-Similar Gaussian Processes and Their Exponentials

We present results on the second order behavior and the expected maximal increments of Lamperti transforms of self-similar Gaussian processes and their exponentials. The Ornstein Uhlenbeck processes driven by fractional Brownian motion (fBM) and its exponentials have been recently studied in Ref.^{[ 20} Matsui , M. ; Shieh , N.-R. On the exponentials of fractional Ornstein-Uhlenbeck processes . Electron. J. Probab. 2009 , 14 , 594 – 611 .[Crossref], [Web of Science ®] , [Google Scholar] ^] and Ref.^{[ 21} Matsui , M. ; Shieh , N.-R. On the exponential process associated with a CARMA-type process. Stochastics , 2012 . doi: 10.1080/17442508.2012.654791 .[Taylor &; Francis Online] , [Google Scholar] ^], where we essentially make use of some particular properties, e.g., stationary increments of fBM. Here, the treated processes are fBM, bi-fBM, and sub-fBM; the latter two are not of stationary increments. We utilize decompositions of self-similar Gaussian processes and effectively evaluate the maxima and correlations of each decomposed process. We also present discussion on the usage of the exponential stationary processes for stochastic modeling.     

Stress–strength models with more than two states under exponential distribution

In the stress–strength models, analysis is based on the reliability of the system where the system is either in operational state or in failure state. Ery?lmaz (2011 Ery?lmaz, S. (2011). A new perspective to stress–strength models. Ann. Inst. Stat. Math. 63(1):101–115.[Crossref], [Web of Science ®] , [Google Scholar]) introduced the stress–strength reliability in a different framework assigning more than two states to the system depending on the difference between strength and stress values. Unlike Ery?lmaz (2011 Ery?lmaz, S. (2011). A new perspective to stress–strength models. Ann. Inst. Stat. Math. 63(1):101–115.[Crossref], [Web of Science ®] , [Google Scholar]), the present article deals with the ratio of the strength and stress values when the stress and strength follow independent exponential distributions. This article presents in detail the estimation aspect of the multistate stress–strength reliability function.     

Adaptive Procedures for the Wilcoxon–Mann–Whitney Test: Seven Decades of Advances

The Wilcoxon–Mann–Whitney test has dominated non parametric analyses in behavioral sciences for the past seven decades. Its widespread use masks the fact that there exist simple “adaptive” procedures which use data-dependent statistical decision rules to select an optimal non parametric test. This paper discusses key adaptive approaches for testing differences in locations in two-sample environments. Our Monte Carlo analysis shows that adaptive procedures often perform substantially better than t-tests, even with moderately sized samples (80 observations). We illustrate adaptive approaches using data from Gneezy and Smorodinsky (2006 Gneezy, U., Smorodinsky, R. (2006). All-pay auctions: an experimental study. J. Economic Behav. Organizat. 61(2): 255275.[Crossref], [Web of Science ®] , [Google Scholar]), and offer a Stata package to researchers interested in taking advantage of these techniques.     

Recurrence relations for single and product moments of generalized order statistics from Marshall–Olkin extended Pareto distributions

Marshall and Olkin (1997 Marshall, A.W., Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84(3):641–652.[Crossref], [Web of Science ®] , [Google Scholar]) introduced a new method of adding parameter to expand a family of distributions. Using this concept, in this article, the Marshall–Olkin extended Pareto distribution is introduced and some recurrence relations for single and product moments of generalized order statistics are studied. Also the results are deduced for record values and order statistics.     

An efficient use of moment's ratios of scrambling variables in a randomized response technique

This paper aimed at providing an efficient new unbiased estimator for estimating the proportion of a potentially sensitive attribute in survey sampling. The suggested randomization device makes use of the means, variances of scrambling variables, and the two scalars lie between “zero” and “one.” Thus, the same amount of information has been used at the estimation stage. The variance formula of the suggested estimator has been obtained. We have compared the proposed unbiased estimator with that of Kuk (1990 Kuk, A.Y.C. (1990). Asking sensitive questions inderectely. Biometrika 77:436–438.[Crossref], [Web of Science ®] , [Google Scholar]) and Franklin (1989 Franklin, L.A. (1989). A comparision of estimators for randomized response sampling with continuous distribution s from a dichotomous population. Commun. Stat. Theor. Methods 18:489–505.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), and Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) estimators. Relevant conditions are obtained in which the proposed estimator is more efficient than Kuk (1990 Kuk, A.Y.C. (1990). Asking sensitive questions inderectely. Biometrika 77:436–438.[Crossref], [Web of Science ®] , [Google Scholar]) and Franklin (1989 Franklin, L.A. (1989). A comparision of estimators for randomized response sampling with continuous distribution s from a dichotomous population. Commun. Stat. Theor. Methods 18:489–505.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) estimators. The optimum estimator (OE) in the proposed class of estimators has been identified which finally depends on moments ratios of the scrambling variables. The variance of the optimum estimator has been obtained and compared with that of the Kuk (1990 Kuk, A.Y.C. (1990). Asking sensitive questions inderectely. Biometrika 77:436–438.[Crossref], [Web of Science ®] , [Google Scholar]) and Franklin (1989 Franklin, L.A. (1989). A comparision of estimators for randomized response sampling with continuous distribution s from a dichotomous population. Commun. Stat. Theor. Methods 18:489–505.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) estimator and Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) estimator. It is interesting to mention that the “optimum estimator” of the class of estimators due to Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) depends on the parameter π under investigation which limits the use of Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) OE in practice while the proposed OE in this paper is free from such a constraint. The proposed OE depends only on the moments ratios of scrambling variables. This is an advantage over the Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) estimator. Numerical illustrations are given in the support of the present study when the scrambling variables follow normal distribution. Theoretical and empirical results are very sound and quite illuminating in the favor of the present study.