The Proposed Buys-Ballot Estimates for Multiplicative Model with the Error Variances

Abstract

This article presents the condition(s) under which the multiplicative model with the error variances best describes the pattern in an observed time series, while comparing it with those of the additive and mixed models.  The method of estimation is based on the periodic, seasonal and overall averages and variances of time series data arranged in a Buys-Ballot table. The method assumes that (1) the underlying distribution of the variable,  X i j ,   i =  1, 2,  ...,  m ,  j  = 1 , 2 ,  ..., s  , under study is normal. (2) the trending curve is linear  (3) the decomposition method is  either additive or multiplicative or mixed. For multiplicative model, the error variance is not known and needs to be estimated with time series data. For additive and mixed models, the error variances are known and assumed to be equal to 1. Result shows that, under the stated assumptions, the seasonal variances of the Buys-Ballot table, for multiplicative model, a function of column ( j ) through the seasonal component  S2j with error variance.