Higher Order Approximation of the Probability Distribution of the Ratio Estimator for a Regression Model

Abstract

In the present paper, we consider the problem of estimating a ratio ρ = E(Y )/E(X) in a regression model Y = α + βX + U. We obtain the higher or- der approximation of the probability distribution of the usual ratio estimator based on the sample means. In the gamma, lognormal and exponential cases, the approxi- mation is numerically compared with the normal one and the empirical distribution. We also consider the higher order approximation of the percentage point and the construction of the confidence interval by using the approximation. When we want to estimate the ratio ρ of the means of two random variables X and Y , i.e. ρ = E(Y ) /E (X), it often happens that X and Y are correlated. In this case, we can assume a linear regression model Y = α + βX + U, and we can utilize this information in the estimation of ρ. Many authors have studied the estimation of the ratio ρ using the linear regression model. For example, Durbin (1959), Rao (1965), Rao and Webster (1966), Gray and Schucany (1972), Rao (1988), Akahira and Kawai (1990), Kawai and Akahira (1994) have discussed the estimation of ρ from the viewpoint of the jackknife method proposed by Quenouille (1956), which is based on dividing the sample at random into groups. Their papers addressed the problem of the optimum choice of the number of blocks and presented the comparison of the jackknife estimator with other estimators for ρ. In the present work, we consider the probability distribution and the percent- age point of the ratio estimator. Since it is difficult to obtain them exactly, some approximations are needed. One approach is to get the empirical distribution function by running the Monte Carlo simulation in computers. The simulation becomes more accurate when the number of Monte Carlo trials increases, but the cost gets higher. Another approach is to get the approximate formula by the asymptotic expansions of the type of Edgeworth and Cornish-Fisher. The derivation of the approximate formula is complicated but once it is obtained, we only calculate it by substituting the appropriate values. It takes less time and fewer computational complexity than the Monte Carlo simulation.