Higher order C(α) tests with applications to mixture models

A class of second-order C(α) tests is proposed for testing composite hypotheses when no optimal test among the (first-order) C(α) tests, as defined by Neyman (Probability and Statistics, The Harald Cramer Volume, Almqvist and Wiksell, Uppsala, Sweden, 1959, p. 213), exists. The form of an optimal test of the new subclass is also presented. The procedure is seen to be easily extendable to C(α) tests of third or higher orders. Two examples of univariate normal mixtures are considered where the procedure is successfully applied. For the corresponding multivariate models, union–intersection tests of Roy (Ann. Math. Statist. 24 (1953) 220) are derived by combining the above optimal tests of the univariate problems.     

Analytical proofs of classical inequalities between Spearman's and Kendall's

Short analytical proofs are given for classical inequalities due to Daniels [1950. Rank correlation and population models. J. Roy. Statist. Soc. Ser. B 12, 171–181; 1951. Note on Durbin and Stuart's formula for E(r_s). J. Roy. Statist. Soc. Ser. B 13, 310] and Durbin and Stuart [1951. Inversions and rank correlation coefficients. J. Roy. Statist. Soc. Ser. B 13, 303–309] relating Spearman's ρ and Kendall's τ.     

Independence in multi-way contingency tables: S.N. Roy's breakthroughs and later developments

In the mid-1950s S.N. Roy and his students contributed two landmark articles to the contingency table literature [Roy, S.N., Kastenbaum, M.A., 1956. On the hypothesis of no “interaction” in a multiway contingency table. Ann. Math. Statist. 27, 749–757; Roy, S.N., Mitra, S.K., 1956. An introduction to some nonparametric generalizations of analysis of variance and multivariate analysis. Biometrika 43, 361–376]. The first article generalized concepts of interaction from 2×2×2$2\times 2\times 2$ contingency tables to three-way tables of arbitrary size and to larger tables. In the second article, which is the source of our primary focus, various notions of independence were clarified for three-way contingency tables, Roy's union–intersection test was applied to construct chi-squared tests of hypotheses about the structure of such tables, and the chi-squared statistics were shown not to depend on the distinction between response and explanatory variables. This work pre-dates by many years later developments that expressed such results in the context of loglinear models. It pre-dates by a quarter century the development of graphical models. We summarize the main results in these key articles and discuss the connection between them and the later developments of loglinear modeling and of graphical modeling. We also mention ways in which these later developments have themselves been further generalized.     

Another look at plotting positions

Probability paper was used as early as 1896, and was mentioned in the literature more than 30 times before 1950, mainly by hydrologists, most of whom used the plotting position (i-0.5)/n proposed by Hazen (1914). Gumbel (1942a) considered the modal position (i-1)/(n-1) and the mean position i/(n+1) [the latter proposed by Weibull (1939a,b)], and chose the latter. Lebedev (1952) and others proposed the use of (i-0.3)/(n+0.4), which is approximately the median position advocated by Johnson (1951). Blom (1958) sug-gested (i-α)/(n-2α+1), where a is a constant (usually 0 ≤ α ≤ 1), which includes all of the above plotting positions as special cases. Moreover, by proper choice of α, one can approximate F[E(x_i)], the position proposed by Kimball (1946), for any distri-bution of interest. Gumbel (1954) stated five postulates which plotting positions should satisfy. Chernoff & Lieberman (1954) discussed the optimum choice of plotting positions in various situ-ations. It is clear that the optimum plotting position depends on the use that is to be made of the results and may also depend on the underlying distribution. The author endeavors to formulate recommendations as to the best choice in various situations.     

On multivariate linear regression shrinkage and reduced-rank procedures

An alternate derivation of the canonical analysis shrinkage prediction procedure of Breiman and Friedman (1997. J. Roy. Statist. Soc. B 59, 3–54) is presented for the multivariate linear model. It is based on consideration of prediction mean square error matrix, and bias of the squared sample canonical correlations. A modified procedure involving partial canonical correlation analysis is also introduced and discussed.     

A new financial risk ratio

Randomness in financial markets has been recognized for over a century: Bachelier (1900), Cowles (1932), Kendall (1953), and Samuelson (1959). Risk thus enters into efficient portfolio design: Fisher (1906), Williams (1936), Working (1948), Markowitz (1952). Reward versus risk decisions then depend upon utility to the investor: Bernoulli (1738), Kelly (1956), Sharpe (1964), and Modigliani (1997). Returns of a portfolio adjusted to risk are measured by a number of ratios: Treynor, Sharpe, Sortino, M2, among others. I will propose a refinement of such ratios. This possibility was mentioned in my recent book: Antieigenvalue analysis, World-Scientific (2011). The result is a new set of growth-to-return risk-based financial ratios of ratios.     

A new functional statistic for multivariate normality

The test statistics of assessing multivariate normality based on Roy’s union-intersection principle (Roy, Some Aspects of Multivariate Analysis, Wiley, New York, 1953) are generalizations of univariate normality, and are formed as the optimal value of a nonlinear multivariate function. Due to the difficulty of solving multivariate optimization problems, researchers have proposed various approximations. However, this paper shows that the (nearly) global solution contrarily results in unsatisfactory power performance in Monte Carlo simulations. Thus, instead of searching for a true optimal solution, this study proposes a functional statistic constructed by the q% quantile of the objective function values. A comparative Monte Carlo analysis shows that the proposed method is superior to two highly recommended tests when detecting widely-selected alternatives that characterize the various properties of multivariate normality.     

On a fisherian detour of the step-down procedure for manova

The well known step-down procedure for MANOVA given by J. Roy (1958) can be modified by combining the step-down tests using a B-optimal combination method such as Fisher's. The Fisherian detour of the stepwise MANOVA is shown to be asymptotically equivalent to the likelihood ratio test.     

Serial correlation and distributions on the shere

Estimation and tests for serial correlation in recation and regression models with normal error have been derive from various points of view; for example: Anderson (1948), Durbi for Watson (1950, 1951, 1971), Theil (1965), Durbin (1970), Haq (1970), Kadiyala (1970), Abrahamse & Louter (1971), Levenbac (1972), Berenblut & Webb (1973), Phillips & Harvey (1974), a Sims (1975). In this paper we derive likelihood functions and most powerful tests for serial correclation in Locationa and regression models with arbitrary but specificed error; the methods extend to include the determination of the likelihood for the parameter of the error distribution.

In Section 2, we survey the modthods that have been used in deriving the various tests and estimates in the literature. In Section 2, we introduce the stataistical model that directly describes the error distribution and we obtain the likelihood function for error correlation and determine locally and specifically kost powerful tests for correlation. In Section 3 we consider the case with normal error derive a normal distribution on the sphere by radial projection. The likelihood function and test are then specialized to the case of normal error in Section 4. The computational procedures for the tests and related power functions are examined in Section 5. Power comparisons for the textile data of Theil and Nagar (1961), the consumption data of Kelin (1950), and the plums and the wheat data of Hildreth & Lu (1960) are presented in Section 6, while the likelihood functions for correlation in these data are given in Section 7.     

Mixtures,embedding, and ancillarity

Ancillary statistics, proposed by Fisher (1925), can be constructed by forming a mixture model (Birnbaum 1962) or can be extracted or derived from a transformation-parameter model (Peisakoff 1951, Fraser 1961) or from the corresponding error-based structural model (Fraser 1968, 1979); these latter models involve an implicit mixture structure. Compound models with ancillaries can also be formed by a cross embedding, discussed from a technical viewpoint in this paper. Of the 25 examples in Buehler (1982), 22 are mixtures or implicit mixtures and 3 correspond to cross embedding. The cross embedding examples exemplify the nonuniqueness difficulties with ancillaries. This paper discusses a simple and two generalized versions of cross embedding but makes no general valuations of these for statistical inference; their role within inference is discussed in Evans, Fraser, and Monette (1984, 1985).     

Statistical inference on asymptotic properties of two estimators for the partially linear single-index models

The outer product of gradients (OPG) estimation procedure based on least squares (LS) approach has been presented by Xia et al. [An adaptive estimation of dimension reduction space. J Roy Statist Soc Ser B. 2002;64:363–410] to estimate the single-index parameter in partially linear single-index models (PLSIM). However, its asymptotic property has not been established yet and the efficiency of LS-based method can be significantly affected by outliers and heavy-tailed distributions. In this paper, we firstly derive the asymptotic property of OPG estimator developed by Xia et al. [An adaptive estimation of dimension reduction space. J Roy Statist Soc Ser B. 2002;64:363–410] in theory, and a novel robust estimation procedure combining the ideas of OPG and local rank (LR) inference is further developed for PLSIM along with its theoretical property. Then, we theoretically derive the asymptotic relative efficiency (ARE) of the proposed LR-based procedure with respect to LS-based method, which is shown to possess an expression that is closely related to that of the signed-rank Wilcoxon test in comparison with the t-test. Moreover, we demonstrate that the new proposed estimator has a great efficiency gain across a wide spectrum of non-normal error distributions and almost not lose any efficiency for the normal error. Even in the worst case scenarios, the ARE owns a lower bound equalling to 0.864 for estimating the single-index parameter and a lower bound being 0.8896 for estimating the nonparametric function respectively, versus the LS-based estimators. Finally, some Monte Carlo simulations and a real data analysis are conducted to illustrate the finite sample performance of the estimators.     

Modified kolmogorov-smirnov tests of goodness of fit

The Kolmogorov-Smirnov (K–S) one-sided and two-sided tests of goodness of fit based on the test statistics D⁺ _n D^? _n and D_n are equivalent to tests based on taking the cumulative probability of the i–th order statistic of a sample of size n to be (i–.5)/n. Modified test statistics C⁺ _n, C^? _n and C_n are obtained by taking the cumulative probability to be i/(n+l). More generally, the cumula-tive probability may be taken to be (i?δ)/(n+l?2δ), as suggested by Blom (1958), where 0 less than or equal δ less than or equal .5. Critical values of the test statis-tics can be found by interpolating inversely in tables of the proba-bility integrals obtained by setting a=l/(n+l?2δ) in an expression given by Pyke (1959). Critical values for the D's (corresponding to δ=.5) have been tabulated to 5DP by Miller (1956) for n=1(1)100. The authors have made analogous tabulations for the C's (corresponding to δ=0) [previously tabulated by Durbin (1969) for n=1(1)60(2)100] and for the test statistics E⁺ _n, E^? _n and E_n corresponding to δ f.3. They have also made a Monte Carlo comparison of the power of the modified tests with that of the K–S test for several hypothetical distributions. In a number of cases, the power of the modified tests is greater than that of the K–S test, especially when the standard deviation is greater under the alternative than under the null hypo-thesis.     

On the analysis of circular balanced crossover designs

The paper considers Azaïs' (J. Roy. Statist. Soc. B, 49 (1987) 334–345) randomization procedure for circular balanced crossover designs. It is shown that this randomization does not justify the assumption of independent identically distributed errors when the estimates are corrected for carryover effects. This might lead to underestimation of the variance of treatment estimates. Similar to the results of Kunert (Biometrics, 43 (1987) 833–845) and Kunert and Utzig (J. Roy. Statist. Soc. B, 55 (1993) 919–927), we give constants, such that multiplication with this constant makes the usual estimate of variance conservative.     

Improved likelihood inference for the shape parameter in Weibull regression

We obtain adjustments to the profile likelihood function in Weibull regression models with and without censoring. Specifically, we consider two different modified profile likelihoods: (i) the one proposed by Cox and Reid [Cox, D.R. and Reid, N., 1987, Parameter orthogonality and approximate conditional inference. Journal of the Royal Statistical Society B, 49, 1–39.], and (ii) an approximation to the one proposed by Barndorff–Nielsen [Barndorff–Nielsen, O.E., 1983, On a formula for the distribution of the maximum likelihood estimator. Biometrika, 70, 343–365.], the approximation having been obtained using the results by Fraser and Reid [Fraser, D.A.S. and Reid, N., 1995, Ancillaries and third-order significance. Utilitas Mathematica, 47, 33–53.] and by Fraser et al. [Fraser, D.A.S., Reid, N. and Wu, J., 1999, A simple formula for tail probabilities for frequentist and Bayesian inference. Biometrika, 86, 655–661.]. We focus on point estimation and likelihood ratio tests on the shape parameter in the class of Weibull regression models. We derive some distributional properties of the different maximum likelihood estimators and likelihood ratio tests. The numerical evidence presented in the paper favors the approximation to Barndorff–Nielsen's adjustment.     

Bounds for the t-tail area

The bounds of Birnbaum (1942) and Sampford (1953) for the upper tail area of the normal distribution are extended to the upper tail of the t-distribution. Numerical and theoretical comparisons are made with the bounds of Peizer and Pratt (1968), Wallace (1959) and Soms (1977).     

Influence diagnostics in a vector autoregressive model

In this paper, we use a likelihood approach and the local influence method introduced by Cook [Assessment of local influence (with discussion). J Roy Statist Soc Ser B. 1986;48:133–149] to study a vector autoregressive (VAR) model. We present the maximum likelihood estimators and the information matrix. We establish the normal curvature and slope diagnostics for the VAR model under several perturbation schemes and use the Monte Carlo method to obtain benchmark values for determining the influence of directional diagnostics and possible influential observations. An empirical study using the VAR model to fit real data of monthly returns of IBM and S&P500 index illustrates the effectiveness of our proposed diagnostics.     

Subset selection for normal means in multi-factor experiments

The procedure of Gupta [1956], [1965] for selecting a random sized subset of k ≧ 2 normal populations which contains the population with the largest population mean when the populations have a common variance is generalized to multi-factor experiments. Two-factor experiments with equal replication on each factor-level combination are discussed in detail. The cases of zero and non-zero interactions between factor levels are considered. For the two-factor, zero interaction case with a common number of observations at each factor-level combination, a table of constants necessary to implement the procedure is provided for experiments having selected levels per factor; the constants are equi-coordinate upper percentage points of a multivariate Student t distribution.     

On Test Statistics in Profile Analysis with High-dimensional Data

We consider profile analysis with unequal covariance matrices under multivariate normality. In particular, we discuss this problem for high-dimensional data where the dimension is larger than the sample size. We propose three test statistics based on Bennett’s (1951) transformation and the Dempster trace criterion proposed by Dempster (1958 Dempster, A.P. (1958). A high dimensional two samples significance test. Annals of Mathematical Statistics 29:995–1010.[Crossref] , [Google Scholar]). We derive the null distributions as well as the nonnull distributions of the test statistics. Finally, in order to investigate the accuracy of the proposed statistics, we perform Monte Carlo simulations for some selected values of parameters.     

A note on the effect of simple equicorrelation in detecting a spurious multivariate observation

This paper examines the robustness of the multivariate version of Grubs' (1950) procedure for detecting an outlier in a sample of n independent observations against equicorrelation of the observations. It is shown that the robustness of the univariate test to equicorrelation extends to the multivariate test in that the distribution of the maximum squared radii-test for a multivariate oulier in identical for both the independent and siaply equicorrelated data models.     

Multivariate attribute control chart using Mahalanobis D2 statistic

Process control involves repeated hypothesis testing based on several samples. However, process control is not exactly hypothesis testing as such since it deals with detection of non-random patterns of variation as well in a fleeting kind of population. Compare this with hypothesis testing which is principally meant for a stagnant population. Dr Walter A. Shewhart introduced a graphic method for doing this testing in a fleeting population in 1924. This graphic method came to be known as control chart and is widely used throughout the world today for process management purposes. Subsequently there was much advancement in process control techniques. In particular, when more than one variable was involved, process control techniques were developed mainly by Hicks (1955), Jackson (1956 and 1959) and Montgomery and Wadsworth (1972) based on the pioneering work of Hotelling in 1931. Most of them have worked in the area of multivariate variable control chart with the underlying distribution as multivariate normal. When more than one attribute variables are involved some works relating to test of hypothesis was done by Mahalanobis (1946). These works were also based on the Hotelling T² test. This paper expands the concept of 'Mahalanobis Distance' in case of a multinomial distribution and thereby proposes a multivariate attribute control chart.