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The article presents the R programming language series, which explores probability distributions and introduces statistical tests. It mentions tests, including the Chi-Square test to determine the correlation between two categorical variables, the Beta function mathematically defined by the integral for two inputs, alpha and beta; and the Gamma distribution, which has a shape (z) and scale parameter (k).

Abstract Estimation and hypothesis tests for partial linear multiplicative models are considered in this paper. A profile least product relative error estimation method is proposed to estimate unknown parameters. We employ the smoothly clipped absolute deviation penalty to do variable selection. A Wald-type test statistic is proposed to test a hypothesis on parametric components. The asymptotic properties of the estimators and test statistics are established. We also suggest a score-type test statistic for checking the validity of partial linear multiplicative models. The quadratic form of the scaled test statistic has an asymptotic chi-squared distribution under the null hypothesis and follows a non-central chi-squared distribution under local alternatives, converging to the null hypothesis at a parametric convergence rate. We conduct simulation studies to demonstrate the performance of the proposed procedure and a real data is analyzed to illustrate its practical usage. [ABSTRACT FROM AUTHOR]

This paper proposes a chi-square method (CSM) to obtain a priority vector for group decision making (GDM) problems where decision-makers’ (DMs’) assessment on alternatives is furnished as incomplete reciprocal preference relations with missing values. Relevant theorems and an iterative algorithm about CSM are proposed. Saaty’s consistency ratio concept is adapted to judge whether an incomplete reciprocal preference relation provided by a DM is of acceptable consistency. If its consistency is unacceptable, an algorithm is proposed to repair it until its consistency ratio reaches a satisfactory threshold. The repairing algorithm aims to rectify an inconsistent incomplete reciprocal preference relation to one with acceptable consistency in addition to preserving the initial preference information as much as possible. Finally, four examples are examined to illustrate the applicability and validity of the proposed method, and comparative analyses are provided to show its advantages over existing approaches. [ABSTRACT FROM AUTHOR]

Abstract: Generalized additive models (GAMs) have distinct advantages over generalized linear models as they allow investigators to make inferences about associations between outcomes and predictors without placing parametric restrictions on the associations. The variable of interest is often smoothed using a locally weighted scatterplot smoothing (LOESS) and the optimal span (degree of smoothing) can be determined by minimizing the Akaike Information Criterion (AIC). A natural hypothesis when using GAMs is to test whether the smoothing term is necessary or if a simpler model would suffice. The statistic of interest is the difference in deviances between models including and excluding the smoothed term. As approximate chi-square tests of this hypothesis are known to be biased, permutation tests are a reasonable alternative. We compare the type I error rates of the chi-square test and of three permutation test methods using synthetic data generated under the null hypothesis. In each permutation method a distribution of differences in deviances is obtained from 999 permuted datasets and the null hypothesis is rejected if the observed statistic falls in the upper 5% of the distribution. One test is a conditional permutation test using the optimal span size for the observed data; this span size is held constant for all permutations. This test is shown to have an inflated type I error rate. Alternatively, the span size can be fixed a priori such that the span selection technique is not reliant on the observed data. This test is shown to be unbiased; however, the choice of span size is not clear. A third method is an unconditional permutation test where the optimal span size is selected for observed and permuted datasets. This test is unbiased though computationally intensive. [Copyright &y& Elsevier]

Abstract: Computing the appropriate sample size is one of the most important aspects of designing an experiment. For the Chi-square test of the equality of multinomial populations a very simple method is proposed for calculating the sample size to satisfy specified significance level and power. This method is especially understandable and does not require software nor the non-central Chi-square tables. Simulations are used to verify the accuracy of the new method. [Copyright &y& Elsevier]

Abstract: A numerical maximum likelihood (ML) estimation procedure is developed for the constrained parameters of multinomial distributions. The main difficulty involved in computing the likelihood function is the precise and fast determination of the multinomial coefficients. For this the coefficients are rewritten into a telescopic product. The presented method is applied to the ML estimation of the Zipf–Mandelbrot (ZM) distribution, which provides a true model in many real-life cases. The examples discussed arise from ecological and medical observations. Based on the estimates, the hypothesis that the data is ZM distributed is tested using a chi-square test. The computer code of the presented procedure is available on request by the author. [Copyright &y& Elsevier]

Abstract: The asymptotic robustness of the normal theory asymptotic biases of the least-squares estimators of the parameters in covariance structures against the violation of normality is shown, which is obtained under the conditions required for the asymptotic robustness for the normal theory standard errors and the usual chi-square statistic. The asymptotic robustness holds not only for the estimators of the parameters whose normal theory asymptotic standard errors are asymptotically robust, but also for the non-robust ones. The Wishart maximum likelihood estimators are also shown to have the asymptotic robustness. A numerical illustration for the factor analysis model shows that the empirical biases of robust estimators under non-normality are close to their corresponding normal theory asymptotic biases. [Copyright &y& Elsevier]

Abstract: The testing for goodness-of-fit in multinomial sampling contexts is usually based on the asymptotic distribution of Pearson-type chi-squared statistics. However, approximations are not justified for those cases where sample size and number of cells permit the use of adequate algorithms to calculate the exact distribution of test statistics in a reasonable time. In particular, Rukhin statistics, containing and Neyman''s modified statistics, are considered for testing uniformity. Their exact distributions are calculated for different sample sizes and number of cells. Several exact power comparisons are carried out to analyse the behaviour of selected statistics. As a result of the numerical study some recommendations are given. Conclusions may be extended to testing the goodness of fit to a given absolutely continuous cumulative distribution function. [Copyright &y& Elsevier]

An accurate normal approximation for the cumulative distribution function of the chi-square distribution with n degrees of freedom is proposed. This considers a linear combination of appropriate fractional powers of chi-square. Numerical results show that the maximum absolute error associated with the new transformation is substantially lower than that found for other power transformations of a chi-square random variable for all the degrees of freedom considered (1⩽n⩽1000). [Copyright &y& Elsevier]

Six methods are studied for combining p-values from independent tests into a new test of the combined hypothesis. The methods—minimum (The Method of Statistics, Williams and Norgate, London, 1931), chi-square (2)(Statistical Methods for Research Workers, 4th Edition, Oliver and Boyd, London, 1932), normal (Magyar Tudományos Akadémia Matematikai Kutató Intezetenek Kozlemenyei 3 (1958) 1971), maximum (Wilkinson, Psycholog. Bull. 48 (1951) 156), uniform (J. Pyschol. 80 (1972) 351), and logistic (in: Rustagi (Ed.), Symposium on Optimizing Methods in Statistics, Academic Press, New York, 1979, pp. 345–366)—are compared heuristically and through simulation. Plots of the rejection regions for combining two tests reveal much about the tests’ relative strengths. The simulations compare methods using different numbers of tests, different patterns of evidence against the combined null hypothesis, and different total strengths of the evidence, allowing broader recommendations than have been made from past simulations. The results indicate that the most difficult kind of problem for a combined test is one in which the total evidence against the combined null is concentrated in one or very few of the tests being combined. For this case alone is the minimum combining function useful. The normal combining function does well in problems where evidence against the combined null is spread among more than a small fraction of the individual tests, or when the total evidence is weak. The chi-square (2) does best when the evidence is at least moderately strong and is concentrated in a relatively small fraction of the individual tests. The logistic combination provides a compromise between these two. The maximum and uniform combinations have generally very poor power and cannot be recommended for use. [Copyright &y& Elsevier]

A common approach to split selection in classification trees is to search through all possible splits generated by predictor variables. A splitting criterion is then used to evaluate those splits and the one with the largest criterion value is usually chosen to actually channel samples into corresponding subnodes. However, this greedy method is biased in variable selection when the numbers of the available split points for each variable are different. Such result may thus hamper the intuitively appealing nature of classification trees. The problem of the split selection bias for two-class tasks with numerical predictors is examined. The statistical explanation of its existence is given and a solution based on the P-values is provided, when the Pearson chi-square statistic is used as the splitting criterion. [Copyright &y& Elsevier]