is replaced by μ0.
The likelihood ratio test statistic is the ratio of the maximum likelihood over the subset of the parameter space specified by H0 and the maximum likelihood over the entire parameter space. Specifically, the likelihood ratio test statistic of H0 : μ = μ0 is
The test based on the T2-statistic in (3.21) and the likelihood ratio test is equivalent because it can be shown that
Example 3.2: Hot rolling is among the key steel-making processes that convert cast or semi-finished steel into finished products. A typical hot rolling process usually includes a melting division and a rolling division. The melting division is a continuous casting process that melts scrapped metals and solidifies the molten steel into semi-finished steel billet; the rolling division will further squeeze the steel billet by a sequence of stands in the hot rolling process. Each stand is composed of several rolls. The side_temp_defect data set contains the side temperature measurements on 139 defective steel billets at Stand 5 of a hot rolling process where the side temperatures are measured at 79 equally spaced locations spread along the stand. In this example, we focus on the three measurements at locations 2, 40, and 78, which correspond to locations close to the middle and the two ends of the stands. The nominal mean temperature values at the three locations are 1926, 1851, and 1872, which are obtained based on a large sample of billets without defects. We want to check if the defective billets have significantly different mean side temperature from the nominal values. We can, therefore, test the hypothesis
The following R codes calculate the sample mean, sample covariance matrix, and the T2-statistic for the three side temperature measurements.
side.temp.defect <- read.csv("side_temp_defect.csv",
header = F) X <- side.temp.defect[, c(2, 40, 78)] mu0 <- c(1926, 1851, 1872) x.bar <- apply(X, 2, mean) # sample mean S <- cov(X) # sample var-cov matrix n <- nrow(X) p <- ncol(X) alpha = 0.05 T2 <- n*t(x.bar-mu0)%*%solve(S)%*%(x.bar -mu0) F0 <- (n-1)*p/(n-p)*qf(1-alpha, p, n-p) p.value <- 1 - pf((n-p)/((n-1)*p)*T2, p, n-p)
Using the above R codes, the sample mean and sample covariance matrix are obtained as
The T2-statistic is obtained by (3.21) as T2 = 19.71. The right-hand side of (3.22) at α = 0.05 is obtained as F0 = 8.13. Since the observed value of T2 exceeds the critical value F0, we reject the null hypothesis H0 and conclude that the mean vector of the three side temperatures of the defective billets is significantly different from the nominal mean vector. In addition, the p-value is 0.0004 < α =0.05, which further confirms that H0 should be rejected.
3.5 Bayesian Inference for Normal Distribution
Let D = {x1, x2,…, xn} denote the observed data set. In the maximum likelihood estimation, the distribution parameters are considered as fixed. The estimation errors are obtained by considering the random distribution of possible data sets D. By contrast, in Bayesian inference, we treat the observed data set D as the only data set. The uncertainty in the parameters is characterized through a probability distribution over the parameters.
In this subsection, we focus on Bayesian inference of normal distribution when the mean μ is unknown and the covariance matrix Σ is assumed as known. The Bayesian inference is based on the Bayes’ theorem. In general, the Bayes’ theorem is about the conditional probability of an event A given that an event B occurs:
Applying Bayes’ theorem for Bayesian inference of μ, we have
where g(μ) is the prior distribution of μ, which is the distribution before observing the data, and f(μ|D) is called as the posterior distribution, which is the distribution after we have observed D. The function f(D|μ) on the right-hand side of (3.25) is the density function for the observed data set D. If it is viewed as a function of the unknown parameter μ, f(D|μ) is exactly the likelihood function of μ. Therefore the Bayes’ theorem can be stated in words as
where ∝ stands for “is proportional to”. Note the denominator p(D) in the right-hand side of (3.25) is a constant which does not depend on the parameter μ. It plays the normalization role to ensure the left-hand side is a valid probability density function and integrates to one. Taking the integral of the right-hand side of (
