Brand Awareness via Online Media: An Evidence Using Instagram Medium with Statistical Analysis

Online marketing refers to the practices of promoting a company's brand to its potential customers. It helps the companies to find new venues and trade worldwide. Numerous online media such as Facebook, YouTube, Twitter, and Instagram are available for marketing to promote and sell a company's product. However, in this study, we use Instagram as a marketing medium to see its impact on sales. To carry out the computational process, the approach of linear regression modeling is adopted. Certain statistical tests are implemented to check the significance of Instagram as a marketing tool. Furthermore, a new statistical model, namely a new generalized inverse Weibull distribution, is introduced. This model is obtained using the inverse Weibull model with the new generalized family approach. Certain mathematical properties of the new generalized inverse Weibull model such as moments, order statistics, and incomplete moments are derived. A complete mathematical treatment of the heavy-tailed characteristics of the new generalized inverse Weibull distribution is also provided. Different estimation methods are discussed to obtain the estimators of the new model. Finally, the applicability of the new generalized inverse Weibull model is established via analyzing Instagram advertising data. The comparison of the new distribution is made with two other models. Based on seven analytical tools, it is observed that the new distribution is a better model to deal with data in the business, finance, and management sectors.

A decade ago, Instagram was founded by Michel Kriger (a software engineer) and Kevin Systrom (a former Google employee). In April 2012, Facebook bought Instagram for $1 billion. It is one of the most influential and biggest social media platforms (SMPs). In June 2018, this platform had hit one thousand million monthly active users [5]. Due to a large number of active users, it is a very useful platform for online marketing; see Salleh et al. [6] and Yu et al. [7].
In this work, we test the significance of online media on the sales of certain products. For this activity, we choose the Instagram medium among the well-known online platforms. We use a simple linear regression (SLR) model to check the significance of the Instagram medium. Two well-known statistical tests such as the (i) t-test and (ii) F-test, along with the correlation test (CT) are considered to perform the regression analysis (RA).
In addition to the RA, a new flexible statistical distribution (SD) is introduced to model the Instagram sales data. e proposed SD may be called a new generalized inverse Weibull (NIG-Weibull) model. e NGI-Weibull is very flexible and offers a close fit to Instagram sales data.

Regression Analysis
Within this section, we provide the RA to see the impact and usefulness of advertising on sales using the Instagram medium. Furthermore, we apply the t-test statistic and F-test statistic to test a hypothesis about the significant role of Instagram advertising in the business sector.

Simple Linear Regression Model.
e SLR model to describe the relationship between Instagram advertising and sales has the following form: (1) By implementing the RA technique, we observe that λ 0 � 5.1030, interpreted as the expected dollar sales (in 1000 s). So, for allocating no advertising budget on Instagram, the expected sale is 5.1030 * 1000 � 5103. e slope of the regression model (regression coefficient) presented in equation (1)  (2) A visual display (graphical illustration) of the positive linear relationship between Instagram medium and sales is presented in Figure 1. From the visual display in Figure 1, we observe that spending money on Instagram advertising is very fruitful and helps to increase the sale.

t-Test.
To carry out the numerical computation using the t-test, we need to find whether the estimate of the regression coefficient (RC) λ 1 is far from 0 or not. If the standard error (SE) of the estimate of the RC λ 1 is very small, then we have sufficient evidence to reject NH H 0 . After implementing the t-test, a summary of the numerical analysis is reported in Table 1.
From the results in Table 1, we can see that for (i) λ 0 , the value of t-statistic is 11.389 and p value is less than 2e-16, and for (ii) λ 1 , the value of t-statistic is 6.259 and p value is less than 8.01e − 09. As we see that the value of the t-statistic (for λ 0 and λ 1 ) is far from zero and the p value is less than 0.05, therefore, we have enough evidence against H 0 , and so, we reject it.

F-Test.
In this part, we implement the F-test to see the impact/significance of Instagram advertising (X) on sales (Y). A larger value of F-test statistic indicates a significant impact of X on Y. After carrying out the computation process, the numerical results are summarized in Table 2. From the presented results in Table 2, we have F-test � 39.17 with p value � 8.011e − 09. From the results in Table 2, we conclude that spending an advertising budget on the Instagram medium will increase the sale. e R square (R 2 ) is an important statistical tool for measuring the fit of the underlined regression model. It deals with the linear relationship between the response variable (sale in this study) and the explanatory variable (Instagram  Computational Intelligence and Neuroscience medium in this study). e value of (R 2 ) ranges between 0 and 1. Its value near to 1 indicates the best fit, and a value near to 0 indicates the poor fit. Corresponding to Instagram advertising data, we observe R 2 � 0.2594. So, using Instagram as an advertising tool, the sale can be increased up to 25.94%.

Correlation Test.
e CT is an important statistical approach to evaluate the association between two variables. In this study, we have considered two variables (Instagram advertising and sales). To check the correlation between X and Y, we consider the Pearson correlation test to check the linear dependency between X and Y. e Pearson correlation coefficient (PCC) expressed by r is calculated as follows: where M Instagram and M Sales represent the mean values of Instagram advertising and sales data, respectively. If the p value is <0.05, then it shows a significant correlation between Instagram medium and sales. After applying the CT, we have r � 0.5471673, showing a positive relationship (PR) between Instagram advertising and sales; see Figure 2. Corresponding to this test, we observe that the value of Spearman's rank correlation is 100 445 and the p value is 6.208e − 10. As the p value is <0.05, therefore, we reject H 0 , which states that Instagram advertisement has no significant impact on sales.

Statistical Modeling
After performing the RA, we move forward and introduce a novel SD for dealing with the Instagram advertising data. is section is organized into six different subsections: (a) the very first part of the section deals with the introduction of the methodology used to obtain the new model, namely NGI-Weibull (new generalized inverse Weibull) model, (b) the new model is fully described in the second subsection, (c) the heavy-tailed (HT) characteristics of the NGI-Weibull are provided in the third subsection, (d) some mathematical properties are obtained in the fourth subsection, (e) the estimators of the NGI-Weibull are obtained in the fifth subsection, and finally (d) the sixth part of this section is devoted to analyzing the Instagram advertising data. e statistical distributions play a useful role to model data in numerous areas such as (a) health sector, (b) finance sector, and (c) reliability engineering. Due to the applicability of the statistical distributions in applied sectors, numerous extensions and modifications of the existing distributions have been proposed; see Tahir and Cordeiro [8].
In the health sector, Wahed et al. [9] proposed a new generalized (NG) form of the Weibull distribution (WD) for modeling breast cancer data. Zhu et al. [10] used the WD for modeling the survival times of patients with gastric cancer. Jan et al. [11] applied the transmuted exponentiated IW (TEIW) distribution of the survival times of patients having bladder cancer. Yoosefi et al. [12] used the exponentiated Weibull (EW) distribution for modeling the survival times of colorectal cancer patients. Mohammed et al. [13] studied a new modified (NM) form of the WD and analyzed the bladder cancer data.
In the finance sector, Nadarajah and Kotz [14] applied the modified Weibull (MW) for asset returns. Bakar et al.   Computational Intelligence and Neuroscience [15] used the composite models for modeling loss data. Bhati and Ravi [16] applied the generalized log-Moyal (GLM) model to the Norwegian fire insurance loss data. Punzo and Bagnato [17] applied the Laplace scale mixtures (LSMs) to data related to cryptocurrencies.
In reliability engineering, Sarhan and Zaindin [18] used the modified Weibull (MW) distribution to model the lifetime of electronic devices. Almalki and Yuan [19] introduced a new modified Weibull (NMW) model for dealing with reliability data. Singh [20] proposed the additive Perks-Weibull (APW) distribution form modeling the Arset data. Okasha et al. [21] analyzed the failure time data by introducing the extended inverse Weibull (EIW) distribution.
According to the findings of Beirlant et al. [29], a statistical model is said to possess the HT characteristics, if its where m > 0. An important property of the HT models is the regular variational property (ReVaPr); see Resnick [30]. A model is termed as a regular varying (ReVa) model, if it obeys where a ∈ 0, ∞ { }. In this work, we move a step further and contribute a new HT model to the literature of DT. Consider that the distribution function (DF) T(z; Φ) of the inverse Weibull (IW) model is given by with probability density function (PDF) t(z; Φ) given by Recently, Wang et al. (2021) proposed a new generalized family of distributions via the DF K(z; λ, ϕ), which is given by where T(z; Φ) � 1 − T(z; Φ). e PDF k(z; λ, ϕ) associated with equation (8) is as follows: We combine the DF of the IW model provided in equation (6) with the DF expressed in equation (8) to generate and study a new model, namely a NGI-Weibull model. e HT characteristics and behaviors of the NGI-Weibull model are obtained. e parameters of the NG-Weibull are estimated via different estimation approaches. Finally, after carrying out the mathematical work, real-life data are analyzed.

e HT Characteristics.
Here, we use a mathematical approach to show that the NGI-Weibull model possesses the HT characteristics.

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where L(z) represents the slowly varying function (SVF). From (8), we have where L(z) � (1/e T(z;Φ) ). So, if we are able to show that L(z) is a SVF, then the variational result obtained in (16) is true.
To prove that L(z) is SVF, we have shown that So, Applying the limit, we get

Mathematical Properties.
Here, we derive some mathematical properties of the NGI-Weibull distribution.

Moments and Incomplete Moments.
Let Z follows the NGI-Weibull with parameters (λ, λ 1 , λ 2 ), and then, the n th moment of Z is given by After using generalized binomial expansion and the Taylor expansion, we can obtain

Computational Intelligence and Neuroscience
For incomplete moments, we have After using generalized binomial expansion and the Taylor expansion, we obtain Using t − λ 1 � u transformation, we obtain where Γ(s, z) � ∞ z u s− 1 e − u du is the upper incomplete gamma function.

Order Statistics.
Let Z 1 , Z 2 , . . . , Z n be random variables with size n from (10); then, the DF of i th order statistics (OS) is given by By differentiating equation (27), we get the PDF of the i th OS as given by Computational Intelligence and Neuroscience
e CMEs are obtained by minimizing the following functions: where S CME λ, λ 1 ,

Application to Sales Data.
is subsection deals with the implementation of the NGI-Weibull distribution to Instagram advertising data given by 11 e NGI-Weibull is applied to Instagram advertising data to establish its flexibility and best fitting capability. For this practical demonstration, the results of the NGI-Weibull model are compared with the parent model (IW distribution) and exponentiated Lomax (Exp-Lomax) distributions.
Singh et al. [32] showed that the IW model fits the financial data sets closely than the other well-known competitors.
e Exp-Lomax is a flexible modification of the Lomax model, which was basically introduced for dealing with data in the finance sector. e survival functions (SFs) of the competitive distributions are given by e graphs of boxplot (BP) and total time test (TTT) for the Instagram sales data are provided in Figure 4.
To show the usefulness of the NGI-Weibull model, certain statistical tools (STs) are considered.
ese STs consist of four information criteria (IC) and three goodnessof-fit measures (GFMs) along with the p value. e values of the IC are calculated as follows: (ii) e Bayesian IC (BIC) (iii) e corrected AIC (CAIC) (iv) e Hannan-Quinn IC (HQIC) where the terms p, n, and Δ(Φ) represent the number of parameters, sample size, and LL function, respectively. e values of the GFMs are calculated as follows: (i) e Anderson-Darling (AD) test statistic Computational Intelligence and Neuroscience 9 (ii) e Cramér-von Mises (CM) test Corresponding to the Instagram sales data, the numerical estimates (NEs) of the model parameters are obtained via implementing the R-script with the method SANN; see Appendix.
e NEs of the fitted models are provided in Table 3, whereas the values of IC measures and GFMs of the fitted models are, respectively, given in Tables 4  and 5.
For the underline data, a model with a larger p value and smaller values of IC and GFMs is considered a better model. From the presented results in Tables 4 and 5, it is obvious that the NGI-Weibull model is the best, as it has the smallest values of the IC and GFMs and a larger p value. is fact reveals the applicability and importance of the NGI-Weibull distribution to deal with Instagram sales data and other data sets in the business management and finance sectors.
Besides the numerical illustration, a visual display of the performances of the competing models is presented in         Computational Intelligence and Neuroscience

Concluding Remarks
is work explored the impact of online marketing on sales. Among the available online marketing media, a wellknown online medium called Instagram is considered. e data sets related to Instagram advertising and sales were studied and analyzed scientifically. To carry out the analysis, we implemented the linear regression approach along with the F-test and t-test. Based on these tests, it showed that there is a positive impact of Instagram advertising on sales. According to the finding of this study, it showed that spending money on Instagram advertising can increase the sale. In addition to the F-test and t-test, we also performed the CT. Based on the results of the CT, it was observed that there is a positive correlation between Instagram advertising and sales.
Finally, a new statistical model named a NGI-Weibull was introduced and studied in detail. Certain mathematical properties along the HT characteristics of the NGI-Weibull distribution were obtained. e NGI-Weibull was applied to model the Instagram advertising sales data. e comparison of the NGI-Weibull model was made with the other models. Certain statistical tools (AIC, CM, BIC, AD, CAIC, KS, and HQIC) were considered for comparative purposes to see which model provides the best description of the Instagram advertising sales data. Using these statistical tools, it showed that the NGI-Weibull model is the best model for taking care of financial data sets.