What is the correct order for compressing, rebuilding indexes and updating statistics in ArcGIS?

Looks good to me. This recommended ESRI workflow should help you, but it sounds like you have it set. If you do any reconciling I would do that first obviously.

1. Reconcile
2. Compress
3. Rebuild Indexes
4. Analyze Datasets / Update Stats

ESRI has put a great python code example on their site that does just this:

import arcpy, time, smtplib

# set the workspace 
arcpy.env.workspace = 'Database Connections/admin.sde'

# set a variable for the workspace
workspace = arcpy.env.workspace

# get a list of connected users.
userList = arcpy.ListUsers("Database Connections/admin.sde")

# get a list of usernames of users currently connected and make email addresses
emailList = [u.Name + "@yourcompany.com" for user in arcpy.ListUsers("Database Connections/admin.sde")]

# take the email list and use it to send an email to connected users.
SERVER = "mailserver.yourcompany.com"
FROM = "SDE Admin <[email protected]>"
TO = emailList
SUBJECT = "Maintenance is about to be performed"
MSG = "Auto generated Message.nrServer maintenance will be performed in 15 minutes. Please log off."

# Prepare actual message
MESSAGE = """
From: %s
To: %s
Subject: %s

%s
""" % (FROM, ", ".join(TO), SUBJECT, MSG)

# Send the mail
server = smtplib.SMTP(SERVER)
server.sendmail(FROM, TO, MESSAGE)
server.quit()

#block new connections to the database.
arcpy.AcceptConnections('Database Connections/admin.sde', False)

# wait 15 minutes
time.sleep(900)

#disconnect all users from the database.
arcpy.DisconnectUser('Database Connections/admin.sde', "ALL")

# Get a list of versions to pass into the ReconcileVersions tool.
versionList = arcpy.ListVersions('Database Connections/admin.sde')

# Execute the ReconcileVersions tool.
arcpy.ReconcileVersions_management('Database Connections/admin.sde', "ALL_VERSIONS", "sde.DEFAULT", versionList, "LOCK_ACQUIRED", "NO_ABORT", "BY_OBJECT", "FAVOR_TARGET_VERSION", "POST", "DELETE_VERSION", "c:/temp/reconcilelog.txt")

# Run the compress tool. 
arcpy.Compress_management('Database Connections/admin.sde')

#Allow the database to begin accepting connections again
arcpy.AcceptConnections('Database Connections/admin.sde', True)

#Get a list of datasets owned by the admin user

# Get the user name for the workspace
# this assumes you are using database authentication.
# OS authentication connection files do not have a 'user' property.
userName = arcpy.Describe(arcpy.env.workspace).connectionProperties.user

# Get a list of all the datasets the user has access to.
# First, get all the stand alone tables, feature classes and rasters.
dataList = arcpy.ListTables('*.' + userName + '.*') + arcpy.ListFeatureClasses('*.' + userName + '.*') + arcpy.ListRasters('*.' + userName + '.*')

# Next, for feature datasets get all of the featureclasses
# from the list and add them to the master list.
for dataset in arcpy.ListDatasets('*.' + userName + '.*'):
    dataList += arcpy.ListFeatureClasses(feature_dataset=dataset)

# pass in the list of datasets owned by the admin to the rebuild indexes and analyze datasets tools
# Note: to use the "SYSTEM" option the user must be an administrator.
arcpy.RebuildIndexes_management(workspace, "SYSTEM", dataList, "ALL")

arcpy.AnalyzeDatasets_management(workspace, "SYSTEM", dataList, "ANALYZE_BASE", "ANALYZE_DELTA", "ANALYZE_ARCHIVE")