Modified entropy estimators for testing normality

In this paper, we present two new estimators for the entropy of absolutely continuous random variables and consider some of their properties. Consistency of the first estimator is shown by Monte Carlo method, and the consistency of the second estimator is proved theoretically. Using these estimators, two new tests for normality are presented and their powers are compared with the other entropy-based tests. Simulation results show that the proposed estimators and test statistics perform very well. Finally, a real example is presented and analysed.     

Testing normality based on new entropy estimators

In this paper, we first introduce two new estimators for estimating the entropy of absolutely continuous random variables. We then compare the introduced estimators with the existing entropy estimators, including the first of such estimators proposed by Dimitriev and Tarasenko [On the estimation functions of the probability density and its derivatives, Theory Probab. Appl. 18 (1973), pp. 628–633]. We next propose goodness-of-fit tests for normality based on the introduced entropy estimators and compare their powers with the powers of other entropy-based tests for normality. Our simulation results show that the introduced estimators perform well in estimating entropy and testing normality.     

Test for normality based on two new estimators of entropy

In this article, two new consistent estimators are introduced of Shannon's entropy that compares root of mean-square error with other estimators. Then we define new tests for normality based on these new estimators. Finally, by simulation, the powers of the proposed tests are compared under different alternatives with other entropy tests for normality.     

A new estimator of entropy and its application in testing normality

In this paper, we introduce a new estimator of entropy of a continuous random variable. We compare the proposed estimator with the existing estimators, namely, Vasicek [A test for normality based on sample entropy, J. Roy. Statist. Soc. Ser. B 38 (1976), pp. 54–59], van Es [Estimating functionals related to a density by class of statistics based on spacings, Scand. J. Statist. 19 (1992), pp. 61–72], Correa [A new estimator of entropy, Commun. Statist. Theory and Methods 24 (1995), pp. 2439–2449] and Wieczorkowski-Grzegorewski [Entropy estimators improvements and comparisons, Commun. Statist. Simulation and Computation 28 (1999), pp. 541–567]. We next introduce a new test for normality. By simulation, the powers of the proposed test under various alternatives are compared with normality tests proposed by Vasicek (1976) and Esteban et al. [Monte Carlo comparison of four normality tests using different entropy estimates, Commun. Statist.–Simulation and Computation 30(4) (2001), pp. 761–785].     

Efficiency of ranked set sampling in tests for normality

In this article, we consider the ranked set sampling (RSS) and investigate seven tests for normality under RSS. Each test is described and then power of each test is obtained by Monte Carlo simulations under various alternatives. Finally, the powers of the tests based on RSS are compared with the powers of the tests based on the simple random sampling and the results are discussed.     

On stratified bivariate ranked set sampling for regression estimators

We investigate the relative performance of stratified bivariate ranked set sampling (SBVRSS), with respect to stratified simple random sampling (SSRS) for estimating the population mean with regression methods. The mean and variance of the proposed estimators are derived with the mean being shown to be unbiased. We perform a simulation study to compare the relative efficiency of SBVRSS to SSRS under various data-generating scenarios. We also compare the two sampling schemes on a real data set from trauma victims in a hospital setting. The results of our simulation study and the real data illustration indicate that using SBVRSS for regression estimation provides more efficiency than SSRS in most cases.     

On Entropy-Based Test of Exponentiality in Ranked Set Sampling

When the sampling units can be easily ranked than quantified, ranked set sampling (RSS) is a viable alternative to the traditional simple random sampling (SRS). Much effort has been made for modifying basic RSS protocol with the aim of deriving more efficient estimators of the population attributes. Entropy has been seminal in developing measures of distributional disparities as a tool for statistical inference. This article is concerned with testing exponentiality based on sample entropy under some RSS-based designs. A simulation study shows that the proposed tests possess good power properties against several alternatives as compared with the ordinary test based on SRS.     

Modified Maximum Likelihood Estimator of Scale Parameter Using Moving Extremes Ranked Set Sampling

A modified maximum likelihood estimator (MMLE) of scale parameter is considered under moving extremes ranked set sampling (MERSS), and its properties are obtained. For some usual scale distributions, we obtain explicit form of the MMLE and prove the MMLE is an unbiased estimator under MERSS. The simulation results show that the MMLE using MERSS is always more efficient than the MLE using simple random sampling, when the same sample size is used. The simulation results also show that the loss of efficiency in using the MMLE instead of the MLE is very small for small sample.     

Ratio estimator for the population mean using ranked set sampling

We adapt the ratio estimation using ranked set sampling, suggested by Samawi and Muttlak (Biometr J 38:753–764, 1996), to the ratio estimator for the population mean, based on Prasad (Commun Stat Theory Methods 18:379–392, 1989), in simple random sampling. Theoretically, we show that the proposed ratio estimator for the population mean is more efficient than the ratio estimator, in Prasad (1989), in all conditions. In addition, we support this theoretical result with the aid of a numerical example.      

A Horvitz-Thompson Estimator of the Population Mean Using Inclusion Probabilities of Ranked Set Sampling

In this study, we define the Horvitz-Thompson estimator of the population mean using the inclusion probabilities of a ranked set sample in a finite population setting. The second-order inclusion probabilities that are required to calculate the variance of the Horvitz-Thompson estimator were obtained. The Horvitz-Thompson estimator, using the inclusion probabilities of ranked set sample, tends to be more efficient than the classical ranked set sampling estimator especially in a positively skewed population with small sizes. Also, we present a real data example with the volatility of gasoline to illustrate the Horvitz-Thompson estimator based on ranked set sampling.     

Tests of fit for the Laplace distribution based on correcting moments of entropy estimators

ABSTRACT

In this paper, we first consider the entropy estimators introduced by Vasicek [A test for normality based on sample entropy. J R Statist Soc, Ser B. 1976;38:54–59], Ebrahimi et al. [Two measures of sample entropy. Stat Probab Lett. 1994;20:225–234], Yousefzadeh and Arghami [Testing exponentiality based on type II censored data and a new cdf estimator. Commun Stat – Simul Comput. 2008;37:1479–1499], Alizadeh Noughabi and Arghami [A new estimator of entropy. J Iran Statist Soc. 2010;9:53–64], and Zamanzade and Arghami [Goodness-of-fit test based on correcting moments of modified entropy estimator. J Statist Comput Simul. 2011;81:2077–2093], and the nonparametric distribution functions corresponding to them. We next introduce goodness-of-fit test statistics for the Laplace distribution based on the moments of nonparametric distribution functions of the aforementioned estimators. We obtain power estimates of the proposed test statistics with Monte Carlo simulation and compare them with the competing test statistics against various alternatives. Performance of the proposed new test statistics is illustrated in real cases.     

Methods improving the estimate of diagnostic odds ratio

Diagnostic odds ratio is defined as the ratio of the odds of the positivity of a diagnostic test results in the diseased population relative to that in the non-diseased population. It is a function of sensitivity and specificity, which can be seen as an indicator of the diagnostic accuracy for the evaluation of a biomarker/test. The naïve estimator of diagnostic odds ratio fails when either sensitivity or specificity is close to one, which leads the denominator of diagnostic odds ratio equal to zero. We propose several methods to adjust for such situation. Agresti and Coull’s adjustment is a common and straightforward way for extreme binomial proportions. Alternatively, estimation methods based on a more advanced sampling design can be applied, which systematically selects samples from underlying population based on judgment ranks. Under such design, the odds can be estimated by the sum of indicator functions and thus avoid the situation of dividing by zero and provide a valid estimation. The asymptotic mean and variance of the proposed estimators are derived. All methods are readily applied for the confidence interval estimation and hypothesis testing for diagnostic odds ratio. A simulation study is conducted to compare the efficiency of the proposed methods. Finally, the proposed methods are illustrated using a real dataset.     

On the asymptotic normality of the kernel estimators of the density function and its derivatives under censoring

In this paper, we study asymptotic normality of the kernel estimators of the density function and its derivatives as well as the mode in the randomly right censorship model. The mode estimator is defined as the random variable that maximizes the kernel density estimator. Our results are stated under some suitable conditions upon the kernel function, the smoothing parameter and both distributions functions that appear in this model. Here, the Kaplan–Meier estimator of the distribution function is used to build the estimates. We carry out a simulation study which shows how good the normality works.     

Asymptotic normality for plug-in estimators of diversity indices on countable alphabets

The plug-in estimator is one of the most popular approaches to the estimation of diversity indices. In this paper, we study its asymptotic distribution for a large class of diversity indices on countable alphabets. In particular, we give conditions for the plug-in estimator to be asymptotically normal, and in the case of uniform distributions, where asymptotic normality fails, we give conditions for the asymptotic distribution to be chi-squared. Our results cover some of the most commonly used indices, including Simpson's index, Reńyi's entropy and Shannon's entropy.     

Estimation of a Parameter of Bivariate Pareto Distribution by Ranked Set Sampling

Ranked set sampling is applicable whenever ranking of a set of sampling units can be done easily by a judgement method or based on the measurement of an auxiliary variable on the units selected. In this work, we derive different estimators of a parameter associated with the distribution of the study variate Y, based on a ranked-set sample obtained by using an auxiliary variable X correlated with Y for ranking the sample units, when (X, Y) follows a bivariate Pareto distribution. Efficiency comparisons among these estimators are also made. Real-life data have been used to illustrate the application of the results obtained.     

Asymptotic normality of generalized maximum spacing estimators for multivariate observations

In this paper, the maximum spacing method is considered for multivariate observations. Nearest neighbor balls are used as a multidimensional analogue to univariate spacings. A class of information-type measures is used to generalize the concept of maximum spacing estimators of model parameters. Asymptotic normality of these generalized maximum spacing estimators is proved when the assigned model class is correct, that is, the true density is a member of the model class.     

Asymptotic normality of estimators for parameters of a multivariate skew-normal distribution

In this paper, asymptotic normality is established for the parameters of the multivariate skew-normal distribution under two parametrizations. Also, an analytic expression and an asymptotic normal law are derived for the skewness vector of the skew-normal distribution. The estimates are derived using the method of moments. Convergence to the asymptotic distributions is examined both computationally and in a simulation experiment.     

The global minimum variance unbiased estimator of the parameter for a truncated parameter family under the optimal ranked set sampling

The minimum variance unbiased estimators (MVUEs) of the parameters for various distributions are extensively studied under ranked set sampling (RSS). However, the results in existing literatures are only locally MVUEs, i.e. the MVUE in a class of some unbiased estimators is obtained. In this paper, the global MVUE of the parameter in a truncated parameter family is obtained, that is to say, it is the MVUE in the class of all unbiased estimators. Firstly we find the optimal RSS according to the character of a truncated parameter family, i.e. arrange RSS based on complete and sufficient statistics of independent and identically distributed samples. Then under this RSS, the global MVUE of the parameter in a truncated parameter family is found. Numerical simulations for some usual distributions in this family fully support the result from the above two-step optimizations. A real data set is used for illustration.