Estimating changes in a multi-parameter exponential family

A two-stage procedure is studied for estimating changes in the parameters of the multi-parameter exponential family, given a sample X ₁,…,X _n. The first step is a likelihood ratio test of the hypothesis H_oof no change. Upon rejection of this hypothesis, the change point index and pre- and post-change parameters are estimated by maximum likelihood. The asymptotic (n → ∞) distribution of the log-likelihood ratio statistic is obtained under both H_oand local alternatives. The m.l.e.fs o of the pre- and post-change parameters are shown to be asymptotically jointly normal. The distribution of the change point estimate is obtained under local alternatives. Performance of the procedure for moderate samples is studied by Monte Carlo methods.     

Goodness-of-fit testing of a weak memoryless property of life distributions

A life distribution is said to have a weak memoryless property if its conditional probability of survival beyond a fixed time point is equal to its (unconditional) survival probability at that point. Goodness‐of‐fit testing of this notion is proposed in the current investigation, both when the fixed time point is known and when it is unknown but estimable from the data. The limiting behaviour of the proposed test statistic is obtained and the null variance is explicitly given. The empirical power of the test is evaluated for a commonly known alternative using Monte Carlo methods, showing that the test performs well. The case when the fixed time point t₀ equals a quantile of the distribution F gives a distribution‐free test procedure. The procedure works even if t₀ is unknown but is estimable.     

The discrepancy of P-values and posterior probability: Two-sided hypothesis of variance of normal distribution with unknown mean

This article is concerned with the comparison of P-value and Bayesian measure in point null hypothesis for the variance of Normal distribution with unknown mean. First, using fixed prior for test parameter, the posterior probability is obtained and compared with the P-value when an appropriate prior is used for the mean parameter. In the second, lower bounds of the posterior probability of H₀ under a reasonable class of prior are compared with the P-value. It has been shown that even in the presence of nuisance parameters, these two approaches can lead to different results in the statistical inference.     

Estimating the Hurst parameter in financial time series via heuristic approaches

This research investigates long memory financial equity markets using three heuristic methodologies namely a proposed modified variance time-aggregated plot, modified rescaled-range plot and periodogram approaches. The intensity of the long memory process is quantified in terms of Hurst parameter (H). Five Malaysian equity market indices are selected in the empirical studies with the inclusion of pre- and post-drastic economic events. Our empirical results evidenced dissimilar long memory behaviours in the different regimes of significant economic events. It is also found that after the short-memory adjustment, all the equity markets exhibited substantial reductions in long memory estimations.     

Comparison of ANOVA-F and ANOM tests with regard to type I error rate and test power

A Monte Carlo simulation was conducted to compare the type I error rate and test power of the analysis of means (ANOM) test to the one-way analysis of variance F-test (ANOVA-F). Simulation results showed that as long as the homogeneity of the variance assumption was satisfied, regardless of the shape of the distribution, number of group and the combination of observations, both ANOVA-F and ANOM test have displayed similar type I error rates. However, both tests have been negatively affected from the heterogeneity of the variances. This case became more obvious when the variance ratios increased. The test power values of both tests changed with respect to the effect size (Δ), variance ratio and sample size combinations. As long as the variances are homogeneous, ANOVA-F and ANOM test have similar powers except unbalanced cases. Under unbalanced conditions, the ANOVA-F was observed to be powerful than the ANOM-test. On the other hand, an increase in total number of observations caused the power values of ANOVA-F and ANOM test approach to each other. The relations between effect size (Δ) and the variance ratios affected the test power, especially when the sample sizes are not equal. As ANOVA-F has become to be superior in some of the experimental conditions being considered, ANOM is superior in the others. However, generally, when the populations with large mean have larger variances as well, ANOM test has been seen to be superior. On the other hand, when the populations with large mean have small variances, generally, ANOVA-F has observed to be superior. The situation became clearer when the number of the groups is 4 or 5.     

One-sided tolerance limits for unbalanced one-way anova andomc effects model

In this article, we investigate techniques for constructing tolerance limits such that the probability is γ that at least _p proportion of the population would exceed that limit. We consider the unbalanced case and study the behavior of the limit as a function of n_i 's (where n_i is the number of observations in the ith batch), as well as that of the variance ratio. To construct the tolerance limits we use the approximation given in Thomas and Hultquist (1978). We also discuss the procedure for constructing the tolerance limits when the variance ratio is unknown. An example is given to illustrate the results.     

Multiscale detection and location of multiple variance changes in the presence of long memory

Procedures for detecting change points in sequences of correlated observations (e.g., time series) can help elucidate their complicated structure. Current literature on the detection of multiple change points emphasizes the analysis of sequences of independent random variables. We address the problem of an unknown number of variance changes in the presence of long-range dependence (e.g., long memory processes). Our results are also applicable to time series whose spectrum slowly varies across octave bands. An iterated cumulative sum of squares procedure is introduced in order to look at the multiscale stationarity of a time series; that is, the variance structure of the wavelet coefficients on a scale by scale basis. The discrete wavelet transform enables us to analyze a given time series on a series of physical scales. The result is a partitioning of the wavelet coefficients into locally stationary regions. Simulations are performed to validate the ability of this procedure to detect and locate multiple variance changes. A ‘time’ series of vertical ocean shear measurements is also analyzed, where a variety of nonstationary features are identified.     

Behaviour of skewness, kurtosis and normality tests in long memory data

We establish the limiting distributions for empirical estimators of the coefficient of skewness, kurtosis, and the Jarque–Bera normality test statistic for long memory linear processes. We show that these estimators, contrary to the case of short memory, are neither ${\sqrt{n}}We establish the limiting distributions for empirical estimators of the coefficient of skewness, kurtosis, and the Jarque–Bera normality test statistic for long memory linear processes. We show that these estimators, contrary to the case of short memory, are neither ?n{\sqrt{n}}-consistent nor asymptotically normal. The normalizations needed to obtain the limiting distributions depend on the long memory parameter d. A direct consequence is that if data are long memory then testing normality with the Jarque–Bera test by using the chi-squared critical values is not valid. Therefore, statistical inference based on skewness, kurtosis, and the Jarque–Bera normality test, needs a rescaling of the corresponding statistics and computing new critical values of their nonstandard limiting distributions.     

On Testing Changes in Autoregressive Parameters of a VAR Model

The article deals with the problem of testing a change in autoregressive matrices of the p-th order vector autoregressive process, VAR(p). The proposed test statistics are based on the likelihood ratio concept and are studied under the null hypothesis of no change in parameters. Their asymptotic behavior is derived under minimal moment assumptions in both cases where the time point of possible change is known a priori and is undefined. The Gumbel-type approximation of the test statistic is also developed, which previous papers on VAR(p) models do not cover.     

Monitoring Change in Persistence Against the Null of Difference-Stationarity in Infinite Variance Observations

In this article, we propose a moving kernel-weighted variance ratio statistic to monitor persistence change in infinite variance observations. We focus on I(1) to I(0) persistence change for sequences in the domain of attraction of a stable law and local-to-finite variance sequences. The null distribution of the monitoring statistic and its consistency are proved. In particular, a bootstrap procedure is proposed to determine the critical values for the derived asymptotic distribution depends on unknown tail index. The small sample performances of proposed monitoring procedure are illustrated by both simulation and application to a high frequency financial data.     

Some extensions of zero-inflated models and Bayesian tests for them

Zero-inflated power series distribution is commonly used for modelling count data with extra zeros. Inflation at point zero has been investigated and several tests for zero inflation have been examined. However sometimes, inflation occurs at a point apart from zero. In this case, we say inflation occurs at an arbitrary point j. The j-inflation has been discussed less than zero inflation. In this paper, inflation at an arbitrary point j is studied with more details and a Bayesian test for detecting inflation at point j is presented. The Bayesian method is extended to inflation at arbitrary points i and j. The relationship between the distribution for inflation at point j, inflation at points i and j and missing value imputation is studied. It is shown how to obtain a proper estimate of the population variance if a mean-imputed missing at random data set is used. Some simulation studies are conducted and the proposed Bayesian test is applied on two real data sets.     

Some asymptotic properties of point estimation with n-dimensional fuzzy data

Some asymptotic properties of point estimation with n-dimensional fuzzy data with respect to a special L ₂-metric ρ are investigated in this article. It is shown that the collection of all n-dimensional fuzzy data endowed with the ρ-metric is a complete and separable space. Some criterions for point estimation in such fuzzy environments are proposed, and the sample mean, variance and covariance with n-dimensional fuzzy data under these criteria are further studied.     

Testing for equivalence of variances using Hartley's ratio

Hartley's test for homogeneity of k normal‐distribution variances is based on the ratio between the maximum sample variance and the minimum sample variance. In this paper, the author uses the same statistic to test for equivalence of k variances. Equivalence is defined in terms of the ratio between the maximum and minimum population variances, and one concludes equivalence when Hartley's ratio is small. Exact critical values for this test are obtained by using an integral expression for the power function and some theoretical results about the power function. These exact critical values are available both when sample sizes are equal and when sample sizes are unequal. One related result in the paper is that Hartley's test for homogeneity of variances is no longer unbiased when the sample sizes are unequal. The Canadian Journal of Statistics 38: 647–664; 2010 © 2010 Statistical Society of Canada     

A fully nonparametric diagnostic test for homogeneity of variances

The authors propose a new nonparametric diagnostic test for checking the constancy of the conditional variance function σ²(x) in the regression model Y_i = m(x_i) + σ(x_i)?_i, i = 1,…, m. Their test, which does not assume a known parametric form for the conditional mean function m(x), is inspired by a recent asymptotic theory in the analysis of variance when the number of factor levels is large. The authors demonstrate through simulations the good finite‐sample properties of the test and illustrate its use in a study on the effect of drug utilization on health care costs.     

Empirical likelihood ratio under infinite second moment

In this article, we show that the log empirical likelihood ratio statistic for the population mean converges in distribution to χ²₍₁₎ as n → ∞ when the population is in the domain of attraction of normal law but has infinite variance. The simulation results show that the empirical likelihood ratio method is applicable under the infinite second moment condition.     

Estimation of Long Memory in Integrated Variance

A stylized fact is that realized variance has long memory. We show that, when the instantaneous volatility is a long memory process of order d, the integrated variance is characterized by the same long-range dependence. We prove that the spectral density of realized variance is given by the sum of the spectral density of the integrated variance plus that of a measurement error, due to the sparse sampling and market microstructure noise. Hence, the realized volatility has the same degree of long memory as the integrated variance. The additional term in the spectral density induces a finite-sample bias in the semiparametric estimates of the long memory. A Monte Carlo simulation provides evidence that the corrected local Whittle estimator of Hurvich et al. (2005 Hurvich , C. M. , Moulines , E. , Soulier , P. ( 2005 ). Estimating long memory in volatility . Econometrica 73 ( 4 ): 1283 – 1328 .[Crossref], [Web of Science ®] , [Google Scholar]) is much less biased than the standard local Whittle estimator and the empirical application shows that it is robust to the choice of the sampling frequency used to compute the realized variance. Finally, the empirical results suggest that the volatility series are more likely to be generated by a nonstationary fractional process.     

Multiparameter estimation of discrete exponential distributions

Let X₁, …, X_p be independent random variables, all having the same distribution up to a possibly varying unspecified parameter, where each of the p distributions belongs to the family of one parameter discrete exponential distributions. The problem is to estimate the unknown parameters simultaneously. Hudson (1978) shows that the minimum variance unbiased estimator (MVUE) of the parameters is inadmissible under squared error loss, and estimators better than the MVUE are proposed. Essentially, these estimators shrink the MVUE towards the origin. In this paper, we indicate that estimators shifting the MVUE towards a point different from the origin or a point determined by the observations can be obtained.     

Bayes Estimation of Change Point in Non Standard Mixture of Left-Truncated Exponential With Unknown Proportions

A sequence of independent lifetimes X ₁, X ₂,…, X _m , X _m+1,…, X _n were observed from the mixture of a degenerate and left-truncated exponential (LTE) distribution, with reliability R _1τ at time τ and minimum life length η with unknown proportion p ₁ and θ₁ but later it was found that there was a change in the system at some point of time m and it is reflected in the sequence after X _m by change in reliability R _2τ at time τ and unknown proportion p ₂ and θ₂. This distribution occurs in many practical situations, for instance; life of a unit may have a LTE distribution but some of the units fail instantaneously. Apart from mixture distributions, the phenomenon of change point is also observed in several situations in life testing and reliability estimation problems. It may happen that at some point of time instability in the sequence of failure times is observed. The problem of study is: When and where this change has started occurring. This is called change point inference problem. The estimators of m, R ₁(t ₀), R ₂(t ₀), p ₁, and p ₂ are derived under asymmetric loss functions namely Linex loss & general entropy loss functions. Both the non informative and informative prior are considered. The effects of prior consideration on Bayes estimates of change point are also studied.     

ON THE CHOICE OF ESTIMATOR IN SURVERY SAMPLING*

We re-examine the criteria of “hyper-admissibility” and “necessary bestness”, for the choice of estimator, from the point of view of their relevance to the design of actual surveys. Both these criteria give rise to a unique choice of estimator (viz. the Horvitz-Thompson estimator ?_HT) whatever be the character under investigation or sample design. However, we show here that the “principal hyper-surfaces” (or “domains”) of dimension one (which are practically uninteresting)play the key role in arriving at the unique choice. A variance estimator v₁(?_HT) (due to Horvitz-Thompson), which takes negative values “often”, is shown to be uniquely “hyperadmissible” in a wide class of unbiased estimators of the variance of ?_HT. Extensive empirical evidence on the superiority of the Sen-Yates-Grundy variance estimator v₂(?_HT) over v₁(?_HT) is presented.