In this paper, the effect of fuzzy time series on estimates of the spectral, bispectral and normalized bispectral density functions are studied. This study is conducted for one of the integer autoregressive of order one (INAR(1)) models. The model of interest here is the dependent counting geometric INAR(1) which is symbolized by (DCGINAR(1)). A realization is generated for this model of size n = 500 for estimation. Based on fuzzy time series, the forecasted observations of this model are obtained. The estimators of spectral, bispectral and normalized bispectral density functions are smoothed by different one- and two-dimensional lag windows. Finally, after the smoothing, all estimators are studied in the case of generated and forecasted observations of the DCGINAR(1) model. We investigate the contribution of the fuzzy time series to the smoothing of these estimates through the results.

In this paper, we study the estimation of the spectral density function and properties of the resulting estimator which called the periodogram. The statistical properties for this periodogram in case of actual observations and forecasted observations of the fuzzy time series are studied. Depending on MSEs which based on these statistical properties, the periodogram results can be compared in both cases. For this purpose a program that transfer the observed time series to fuzzy time series with large sizes is constructed.

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Several models for time series with integer values have been published as a result of the substantial demand for the description of process stability having discrete marginal distributions. One of these models is the mixed count geometric integer autoregressive of order one (MCGINAR(1)), which is based on two thinning operators. This study examines how the estimates of the spectral density functions of the MCGINAR(1) model are affected by fuzzy time series Markov chain (FTSMC). Regarding this study’s context, the higher-order moments, central moments and spectral density functions of MCGINAR(1) are computed. The anticipated realizations of the generated realizations for this model are obtained based on FTSMC. In the case of generated and anticipated realizations, several lag windows are used to smooth the spectral density estimators. The generated realization estimates are compared with the anticipated realization estimates using the MSE to ascertain the FTSMC’s role in improving the estimation process.

Assume X1,X2,…,XN are realizations of N observations from a real-valued discrete parameter third-order stationary process Xt,t=0±1,±2,…, with bispectrum fXXX(λ1,λ2) where “−π≤λ1,λ2≤π”. Based on the previous assumption, L different multitapered biperiodograms IXXX(mt)j(λ1,λ2);j=1,2,…,L on overlapped segments (Xt(j);1≤t<N) can be constructed. Further, the mean and variance of the average of these different multitapered biperiodograms can be expressed as asymptotic expressions. According to different bispectral windows/kernels (Wβ(j)(α1,α2), where “−π⩽α1,α2⩽π” andβ is the bandwidth) and IXXX(mt)j(λ1,λ2), the bispectrum fXXX(λ1,λ2) can be estimated. The asymptotic expressions of the first- and second-ordered moments as well as the integrated relative mean squared error (IMSE) of this estimate are derived. Finally, some estimation results based on numerically generated data from the selected process “DCGINAR(1)” are presented and discussed in detail.