Methods of estimating parametric and nonparametric components, as well as properties of the corresponding estimators, have been examined in partially linear models by Wahba [1987], Green et al. [1985], Engle et al. [1986], Speckman [1988], Hu et al. [2004], Charnigo et al. [2015] among others. These models are appealing due to their flexibility and wide range of practical applications including the electricity usage study by Engle et al. [1986], gum disease study by Speckman [1988], etc., wherea parametric component explains linear trends and a nonparametric part captures nonlinear relationships. The compound estimator (Charnigo et al. [2015]) has been used to estimate the nonparametric component of such a model with multiple covariates, in conjunction with linear mixed modeling for the parametric component. These authors showed, under a strict orthogonality condition, that parametric and nonparametric component estimators could achieve what appear to be (nearly) optimal rates, even in the presence of subject-specific random effects. We continue with research on partially linear models with subject-specific random intercepts. Inspired by Speckman [1988], we propose estimators of both parametric and nonparametric components of a partially linear model, where consistency is achievable under an orthogonality condition. We also examine a scenario without orthogonality to find that bias could still exist asymptotically. The random intercepts accommodate analysis of individuals on whom repeated measures are taken. We illustrate our estimators in a biomedical case study and assess their finite-sample performance in simulation studies. Jump points have often been found within the domain of nonparametric models (Muller [1992], Loader [1996] and Gijbels et al. [1999]), which may lead to a poor fit when falsely assuming the underlying mean response is continuous. We study a specific type of change-point where the underlying mean response is continuous on both left and right sides of the change-point. We identify the convergence rate of the estimator proposed in Liu [2017] and illustrate the result in simulation studies.

The present study aimed at examining both vocal and laryngeal changes caused by the vocal fry in five female subjects without vocal complaints, signs nor symptoms, with no report of previous voice therapy, and free of inflammatory or allergic processes at the moment of assessment. Initially, a digital recording of sustained emission of /a/ vowel and a recording of a videostroboscopic examination of larynx were performed. Soon after this procedure, the subjects did the vocal fry, in three series of 15 repetitions, with a 30-second interval of passive rest between the series, when the subjects should remain in absolute silence. Immediately after that, both the laryngeal examination and the recording of the vocal emission with the same phonatory task were carried out again. Both laryngeal and vocal data found prior to and after the technique were submitted to acoustic, perceptive-auditive, and videostroboscopic analyses. The perceptive-auditive analysis as well as the videostroboscopic one were performed by six judges (three speech and language pathologists especialized in voice, and three otorhinolaryngologist, respectively). The acoustic analysis was generated by the Multi Speech program of Kay Elemetrics. The analysis of results showed the following effects of the vocal fry: improved vibration of the mucosa of the vocal folds, particularly as to width and symmetry of vibration; improved or sustained voice type and pitch; improved or sustained parameters related to jitter and shimmer as well as the index that suggests glottic noise, suggesting decreased noise; improved index of mild phonation, suggesting greater balance of the degree of coaptation strength and completion of vocal folds closure; sustained or worsened vocal quality and resonance focus, with laryngopharyngeal prevalence; decreased fundamental frequency; and increased frequency variation and width, suggesting phonatory unstableness. Thus, in this study, it was concluded that the vocal fry had a positive effect on the vibration of the mucosa of the vocal folds andon the voice noise as well, and a negative effect on the voice resonance and stability. / O presente estudo teve como objetivo verificar as mudanças vocais e laríngeas ocasionadas pelo som basal em cinco indivíduos adultos do sexo feminino sem queixas, sinais e sintomas vocais e laríngeos, sem relato de terapia vocal anterior e livres de quadro inflamatório ou alérgico no momento da avaliação. Inicialmente, realizou-se gravação digital da emissão sustentada da vogal /a/ e gravação do exame videolaringoestroboscópico da laringe. Imediatamente após, os sujeitos realizaram a técnica do som basal durante três séries de 15 repetições. Em seguida, realizaram-se novamente o exame laríngeo e a gravação da emissão vocal com a mesma tarefa fonatória. Os dados laríngeos e vocais pré e pós-realização da técnica foram submetidos às análises acústica, perceptivo-auditiva e videolaringoestroboscópica, sendo as duas últimas realizadas por seis juízes (três fonoaudiólogas especialistas em voz e três otorrinolaringologistas, respectivamente). A análise acústica foi gerada pelo programa Multi Speech, da Kay Elemetrics. A análise dos resultados evidenciou que o som basal propiciou: melhora da vibração da mucosa das pregas vocais, mais especificamente quanto à amplitude e simetria de vibração; melhora ou manutenção do tipo de voz e do pitch; melhora ou manutenção dos parâmetros relacionados ao jitter e shimmer (PPQ e APQ) e do índice que sugere ruído glótico (NHR), sugerindo diminuição do ruído; melhora do índice de fonação suave (SPI), sugerindo maior equilíbrio do grau de força de coaptação e da completude de fechamento das pregas vocais; manutenção ou piora da qualidade vocal e do foco ressonantal, com predomínio laringofaríngeo;diminuição da freqüência fundamental; e aumento da variação da freqüência (vf0) e amplitude (vAm), sugerindo instabilidade fonatória. Dessa forma, concluiu-se que, neste estudo, o som basal promoveu um efeito positivo sobre a vibração da mucosa das pregas vocais e sobre o ruído na voz, e um efeito negativo sobre a ressonância e a estabilidade da voz.