Statistica

Rediscovering a Little Known Fact about the t-test and the F-test: Algebraic, Geometric, Distributional and Graphical Considerations

Jennifer A. Sinnott — 2023-04-05

We discuss the role that the null hypothesis should play in the construction of a test statistic used to make a decision about that hypothesis. To construct the test statistic for a point null hypothesis about a binomial proportion, a common recommendation is to act as if the null hypothesis is true. We argue that, on the surface, the one-sample t -test of a point null hypothesis about a Gaussian population mean does not appear to follow the recommendation. We show how simple algebraic manipulations of the usual t-statistic lead to an equivalent test procedure consistent with the recommendation. We provide geometric intuition regarding this equivalence and we consider extensions to testing nested hypotheses in Gaussian linear models. We discuss an application to graphical residual diagnostics where the form of the test statistic makes a practical difference. By examining the formulation of the test statistic from multiple perspectives in this familiar example, we provide simple, concrete illustrations of some important issues that can guide the formulation of effective solutions to more complex statistical problems.

The Marshall-Olkin Extended Unit-Gompertz Distribution: its Properties, Simulations and Applications

Festus Chukubogu Opone — 2023-04-05

In this paper, a newbounded generalization of the unit-Gompertz distribution called the Marshall-Olkin extended unit-Gompertz distribution (MOEUGD) is introduced. The mathematical properties and an associated quantile regression model of the proposed distribution are derived. The maximum likelihood estimation method is employed for estimating the parameters of the proposed distribution, and a Monte Carlo simulation study is carried out to investigate the asymptotic behaviour of the parameter estimates of the proposed distribution. Finally, the applicability of the proposed distribution is illustrated by means of two real data sets defined on a unit-interval andan application of the regression model to a real data set.

McDonald-G Poisson Family of Distributions

Keshav Pokhrel — 2023-04-05

In this article, we utilize the method proposed by Tahir and Cordeiro (2016) to study a new family of distributions called the McDonald Generalized Poisson (McGP) family. This family is defined by using the genesis of the McDonald distribution and the zero truncated Poisson (ZTP) distribution. We provide some mathematical properties and parameter estimation procedures of the McGP family. Three real-life data are analyzed to illustrate the potential applications of the McGP family. Our examples illustrate that the development of new probability distributions is of great interest to capture the nature of the data under study. However, one can’t guarantee a better fit just because a probability distribution possesses a larger number of parameters than its sub-model.

Application of Ranked Set Sampling in Parameter Estimation of Cambanis-Type Bivariate Exponential Distribution

Kirtee Kiran Kamalja — 2023-04-05

Ranked set sampling (RSS) is an efficient technique for estimating parameters and is applicable whenever ranking on a set of sampling units can be done easily by a judgment method or based on an auxiliary variable. In this paper, we assume (X,Y) to have a Cambanis-type bivariate exponential (CTBE) distribution, where a study variable Y is difficult and/or expensive to measure and is correlated with an auxiliary variable X that is readily measurable. The auxiliary variable is used to rank the sampling units. This paper addresses the problem of estimation of the scale parameter associated with the Y-variable based on the RSS scheme and some of the other modified RSS schemes. Comparison between estimators is done through relative efficiency to find the best RSS scheme. The efficiency performance of the estimators under various RSS schemes is presented numerically and graphically through 2-D and 3-D plots. To study the performance of the proposed estimators through a simulation study we develop a Matlab function to simulate data from the CTBE distribution. The results are applied to a real data set on mercury concentration in large mouth bass from Florida.

Efficient Classes of Estimators of Population Variance in Two-Phase Successive Sampling Under Random Non-Response

Zeeshan Basit — 2023-04-05

This paper presents some efficient classes of estimators of population variance in two-phase successive sampling under random non-response. The suggested classes of estimators are for simple random sampling and for different situations of non-response. Up to first-order approximation MSE’s of suggested classes of estimators are derived. The efficiency of the presented estimators is contrasted with the estimators for the complete response scenarios. Usefulness of the presented classes of estimators is checked. To test the efficiency real data sets are used. The proposed classes of estimators are more efficient. Results are interpreted.