Some Test Problems In Statistics

Testing hypothesis, as one of the three statistical inferences, always receives many statistician's concern. One method for testing hypothesis is likelihood ratio test (LRT). Generally, the power performance of LRT is satisfying. Although this, sometimes people tends to use other testing methods for special ends. For example, the multiple comparison method will be discussed in chapter 2 and 3. Also other testing methods are tent to be used when the null distribution of LRT is hard to get. Chapter 4, 5 and 6 do not propose LRT for this reason. In this dissertation, we do the work as following.First, we propose a new test procedure for comparing several treatments with a control. This new procedure has good power performance when the treatments and the control are under simple-tree order restriction. It can provide lower simultaneous confidence bounds for the differences between the treatments and the control.Second, for comparing several treatments, a test procedure which try to keep the advantage of Lee & Spurrier's (1995) procedure and promote the power performance of their test procedure is proposed.Third, testing homogeneity of multivariate normal mean vectors under an order restriction when the covariance matrices are common but unknown is discussed. Sasabuchi (2003) proposed a test (Sasabuchi test) when the order restriction is the simple order restriction. Sasabuchi test is complex for computation, and not uniformly superior to classical MANOVA, which is the LRT when the normal mean vectors is without order restriction. This dissertation propose a new test and derive out its asymptotic null distribution. The new test is easy to calculate, and uniformly superior to Sasabuchi test and classical MANOVA.Fourth, testing homogeneity of multivariate normal mean vectors under a general order restriction when the covariance matrices are common but unknown is discussed.Last, testing hypotheses about the regression coefficients of panel data model is discussed. When the model of data is panel data model with one-way or two-way error component, testing hypotheses about the regression coefficients is a difficult thing. Here the generalized p-value (Tusi & Weerahandi, 1989) is used. We find that the generalized p-value method always does better than other two methods usually used in practice, which will be introduced in chapter 6 in more details. We also demonstrate that my conclusions are same as Weerahandi & Berger's (1999) for a simple growth curve model. The dissertation contains six chapters. It is organized in the following way.Chapter 1 introduces some conceptions and recent developments of concerned problems, which are needed by the following chapters. They are the conception of order restriction, likelihood ratio test statistic of testing the equality of order restricted normal means, developments of testing the equality of order restricted multivariate normal mean vectors, conceptions and developments of multiple comparisons. As the generalized p-value will be used, its conception also is introduced.Chapter 2 considers testing the equality of simple-tree order restricted normal mean; that iswith at least one strict inequality. μ_i are the means of random variate X_i with normal distribution, i.e. X_i ～N(μ_i, σ²), i = 0,..., k. In practice, X_i, i = 1,..., k, often denote the effects of several treatments, and X₀ denotes that of a control. Bartholomew (1961) derived the likelihood ratio test (LRT) for this problem. Tang & Lin (1997) proposed an approximate likelihood ratio test (ALR). Mukerjee et al. (1987) proposed an orthogonal contrast test (OC). Dunnett (1955) proposed multiple comparison method (Dunnett test). If the null hypothesis H₀ is rejected, the second question will be which μ_i > μ₀. If this question is answered, the third question will be how much the treatments differ with the control. It will become more clear by considering the following three questions sequentially: (1)Are μ_i = μ₀ for all i? (2)If not, which μ_i > μ₀?(3)If we decide for μ_i > μ₀, how large is the difference at least?LRT, ALRT and OC are designed for the first question. Although using them in closed testing can answer the second question, they can't answer the third. Dunnett test is designed for the second and third question. It does not perform as good as LRT and ALRT for the first question; that is its power performance is not as good as LRT's and ALRT's. This chapter proposes a new test, shows how the new test answers the three questions, and compares it with the tests mentioned above. It's found that the new test has good power performance as LRT and ALRT, and can provide lower simultaneous confidence bounds for μ_i- μ₀. If, regardless of the data, the experimenter is certain of the ordering of the treatments and the control μ_i ≥ μ₀, neither Dunnett confidence intervals nor the proposed confidence intervals can uniformly dominate another one. Chapter 3 considers testing the equality of simple order resticted normal means, i.e. H₀ : μ₁= ... = μ_k; H₁ : μ₁ ≤ ... ≤ μ_kwith at least one strict inequality. Bartholomew (1959) derived out the likelihood ratio test for this problem. To obtain the lower confidence bounds for μ_j - μ_i, j > i, Hayter (1990) proposed one-sided studentized rang test (OSPT). If we are concerned with the following three questions sequentially:(1) Are μ_i = μ_j for all i < j?(2) If not, which μ_j > μ_i?(3)If we decide for μ_j > μ_i, how large is the difference at least?then OSRT is a good test. It can provide lower simultaneous confidence bounds for μ_j-μ_i,and has slight lower power than LRT.Lee & Spurrier (1995) proposed the successive comparison test (SCT). SCT has obviousadvantages over other tests for the two questions: which μ_j > μ_i? If we decide for μ_j > μ_i,how large is the difference at least?We may be concerned with the following three questions, not only the two ones above:(l')Are μ_i = μ_i+1 for 1 ≤ i ≤ k - 1? (2/)If not, which μ_i+1 > μ_i? (3')If we decide forμ_i+1 > μ_i, how large is the difference at least?As described above, SCT has obvious advantages over OSRT for (2') and (3')- However,SCT does much worse than OSRT for (1'). A test is proposed in chapter 3. The proposedtest has lower power than OSRT, but higher than SCT. The power performance of theproposed test becomes closer to that of OSRT with the increasing of k. The proposedlower confidence bounds for μ_i+1 -μ_i are smaller or same as SCT lower confidence bounds.The differences, if any, between the proposed lower bounds with the SCT lower boundsgo less sharply with the increasing of k. This is different with OSRT. The OSRT lowerbounds are smaller than the SCT lower bounds. The former becomes more smaller thanthe latter with the increasing of k. Although LRT has good power performance, it can'tproduce lower confidence intervals for μ_i+1 - μ_i. We don't discuss it any more.Chapter 4 compares several multivariate normal mean vectors under the simple order restriction.Consider p-variate normal distributions X_ij ～ N_p(μ_i,∑), i = 1,..., k, 1 ≤ j ≤ n_i. Sasabuchi et al. (2003) considered the testing problem with unknown covariance matrices where "μ_i ≤ μ_j" means that each element of "μ_j - μ_i" is not less than zero. The LRT for this test problem is infeasible for its null distribution being hard to get. Sasabuchi et al. (2003) gave a test (Sasabuchi test). Sasabuchi test is complex for computation, and not uniformly superior to classical MANOVA, which is the LRT when the normal mean vectors is without order restriction. This dissertation proposes a new test and derives out its asymptotic null distribution. The test is easy to calculate, and uniformly superior to Sasabuchi test and classical MANOVA.Chapter 5 considers the following test problem: where dimension is γ, μ_i(2)'s dimension is p - γ. Obviously if γ = 0, this test problem is the one in Sasabuchi et al. (2003). So it is an extended test problem of that in Sasabuchi et al. (2003). Also the LRT is infeasible. This chapter proposes a test and discusses the relationship between the proposed test and the likelihood function.Chapter 6 considers testing the regression coefficients of panel data model. Suppose we observed N individuals, each was observed T times repeatedly. If the data model is asthen we call the data is a panel data, where μ_i denotes the unobservable individual specific effect and v_it denotes the remainder disturbance. Generally suppose μ_i ～ N(0, σ_μ²), v_it ～ N(0, σ_v²). α is a scalar, β is K × 1 and X_it is the itth observation on K explanatory variables. Rewrite it in matrix notationwhere Y is NT× 1, X is NT×K, Z = [l_NT,X],δ' = (α,β'),u = Z_μμ+v,Z_μ = I_N(?)l_T, and l_NT,l_T is a vector of ones of dimension NT, T respectively. Suppose μ_i ～ N(0, σ_μ²), v_it ～ where . Exact inference is difficult to do in models involving unknown variance components. However, some recent simulation studies have indicated the importance of making exact inference. For example, according to the discussion by Chi and Weerahandi (1998), in typical applications of mixed models, the Type I error of MLE-based asymptotic tests on variance components can be as large as 40% when the intended level is 5%. Hence, there is a great need for encouraging the development of exact and size-guaranteed tests and intervals for models involving variance components. In this chapter exact inferences using generalized p-values (Tusi & Weerahandi, 1989) are obtained.